{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261007-advance-20261009-r06-definite-integral-0075",
  "pilot_number": "R0006-03875",
  "number": 3875,
  "family_id": "qf-density-integral-left-error",
  "title": {
    "zh-Hant": "第 3875 題：精確矩積分減左端有限和",
    "en": "Question 3875: Exact Moment Integral Minus a Finite Left Sum"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261007-advance-20261009-r06-definite-integral-0075",
    "batch_id": "batch-20261007-advance-20261009-r06-definite-integral-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-definite-integral"
    ],
    "title_zh": "精確矩積分減左端有限和",
    "stem_zh": "全實數論域；升冪係數的受限多項式模型。f(x)=25x+(12)，其升冪係數=[12, 25]；a=-2、b=-1，N=50等份左端和S_N。求 I(a,b)−S_N。",
    "answer": {
      "kind": "expression",
      "value": "1/4",
      "explanation_zh": "多項式矩減左端有限和=1/4。產生端用等差級數導出的精確誤差m(b−a)²/(2N)；S_N仍是有限和，負斜率的差可負，不當作近似證據。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 3,
      "skills": [
        "精確矩積分減左端有限和"
      ]
    },
    "variant": {
      "family_id": "qf-density-integral-left-error",
      "seed": 202610096075,
      "parameters": {
        "m": 25,
        "c": 12,
        "a": -2,
        "b": -1,
        "N": 50,
        "operation": "density_integral_left_error",
        "domain": "reals",
        "semantic_template": "qf-density-integral-left-error/[25,\"1\",50]"
      }
    },
    "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-0075",
    "batch_id": "batch-20261007-advance-20261009-r06-definite-integral-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "f9d5dd056a03791cfe650b3b96d8afcde27f070e51be8c93db8ef551efbc068e",
      "family_id": "qf-density-integral-left-error",
      "seed": 202610096075,
      "semantic_template": "qf-density-integral-left-error/[25,\"1\",50]"
    },
    "fields": {
      "title": "Exact Moment Integral Minus a Finite Left Sum",
      "stem": "On all R, f(x)=(25)x+(12). On a=-2≤b=-1, use N=50 uniform left cells. Find the signed error I−S_N, exact moment integral minus the finite left sum. Use reduced positive-denominator fractions, retaining /1 for integers.",
      "answer": {
        "kind": "expression",
        "value": "1/4",
        "explanation": "Compute the polynomial moment and finite left sum separately, giving I−S_N=1/4. The error may be negative; equality in a constant or zero-width case does not make every finite sum an integral."
      },
      "skills": [
        "Exact Moment Integral Minus a Finite Left Sum"
      ],
      "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."
    }
  },
  "givens": [
    {
      "id": "G001",
      "text": {
        "zh-Hant": "全實數論域；升冪係數的受限多項式模型。f(x)=25x+(12)，其升冪係數=[12, 25]；a=-2、b=-1，N=50等份左端和S_N。求 I(a,b)−S_N。",
        "en": "On all R, f(x)=(25)x+(12). On a=-2≤b=-1, use N=50 uniform left cells. Find the signed error I−S_N, exact moment integral minus the finite left sum. 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": "interval width := sub(-1, -2)",
      "after": "interval width = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先求非負區間寬度，允許零寬。",
        "en": "Compute the nonnegative interval width,allowing zero width."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 interval width，不以小數近似替代。",
          "en": "Recompute interval width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-1",
          "-2"
        ],
        "output": "1",
        "label": "interval width"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "partition step := div(1, 50)",
      "after": "partition step = 1/50",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "除以正整數N得到精確步長。",
        "en": "Divide by positive integer N to get the exact step."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 partition step，不以小數近似替代。",
          "en": "Recompute partition step with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "1",
          "50"
        ],
        "output": "1/50",
        "label": "partition step"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "left sample0 offset := mul(0, 1/50)",
      "after": "left sample0 offset = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。",
        "en": "Left samples use i=0,...,N−1,not the final right endpoint."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample0 offset，不以小數近似替代。",
          "en": "Recompute left sample0 offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "1/50"
        ],
        "output": "0",
        "label": "left sample0 offset"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "left sample0 x := sum(-2, 0)",
      "after": "left sample0 x = -2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "起點加上第i個偏移得到採樣位置。",
        "en": "Add the indexed offset to the start."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample0 x，不以小數近似替代。",
          "en": "Recompute left sample0 x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "0"
        ],
        "output": "-2",
        "label": "left sample0 x"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "left sample0 mx := mul(25, -2)",
      "after": "left sample0 mx = -50",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "代入仿射函數乘法部分。",
        "en": "Evaluate the product term of the affine integrand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample0 mx，不以小數近似替代。",
          "en": "Recompute left sample0 mx with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-2"
        ],
        "output": "-50",
        "label": "left sample0 mx"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "left sample0 value := sum(-50, 12)",
      "after": "left sample0 value = -38",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加常數c得到本採樣值。",
        "en": "Add c to obtain this sample value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample0 value，不以小數近似替代。",
          "en": "Recompute left sample0 value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-50",
          "12"
        ],
        "output": "-38",
        "label": "left sample0 value"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "left sample0 rectangle := mul(-38, 1/50)",
      "after": "left sample0 rectangle = -19/25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。",
        "en": "Multiply by the step to get the signed rectangle contribution,without absolute values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample0 rectangle，不以小數近似替代。",
          "en": "Recompute left sample0 rectangle with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-38",
          "1/50"
        ],
        "output": "-19/25",
        "label": "left sample0 rectangle"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "[-2, -38, -19/25]",
      "after": "finite left-sum term index=0",
      "relation": "implies",
      "rule_id": "left_partition_term",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。",
        "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "left sample1 offset := mul(1, 1/50)",
      "after": "left sample1 offset = 1/50",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。",
        "en": "Left samples use i=0,...,N−1,not the final right endpoint."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample1 offset，不以小數近似替代。",
          "en": "Recompute left sample1 offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1/50"
        ],
        "output": "1/50",
        "label": "left sample1 offset"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "left sample1 x := sum(-2, 1/50)",
      "after": "left sample1 x = -99/50",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "起點加上第i個偏移得到採樣位置。",
        "en": "Add the indexed offset to the start."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample1 x，不以小數近似替代。",
          "en": "Recompute left sample1 x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "1/50"
        ],
        "output": "-99/50",
        "label": "left sample1 x"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "left sample1 mx := mul(25, -99/50)",
      "after": "left sample1 mx = -99/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "代入仿射函數乘法部分。",
        "en": "Evaluate the product term of the affine integrand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample1 mx，不以小數近似替代。",
          "en": "Recompute left sample1 mx with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-99/50"
        ],
        "output": "-99/2",
        "label": "left sample1 mx"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "left sample1 value := sum(-99/2, 12)",
      "after": "left sample1 value = -75/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加常數c得到本採樣值。",
        "en": "Add c to obtain this sample value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample1 value，不以小數近似替代。",
          "en": "Recompute left sample1 value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-99/2",
          "12"
        ],
        "output": "-75/2",
        "label": "left sample1 value"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "left sample1 rectangle := mul(-75/2, 1/50)",
      "after": "left sample1 rectangle = -3/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。",
        "en": "Multiply by the step to get the signed rectangle contribution,without absolute values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample1 rectangle，不以小數近似替代。",
          "en": "Recompute left sample1 rectangle with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-75/2",
          "1/50"
        ],
        "output": "-3/4",
        "label": "left sample1 rectangle"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "[-99/50, -75/2, -3/4]",
      "after": "finite left-sum term index=1",
      "relation": "implies",
      "rule_id": "left_partition_term",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。",
        "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof."
      },
      "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": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "left sample2 offset := mul(2, 1/50)",
      "after": "left sample2 offset = 1/25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。",
        "en": "Left samples use i=0,...,N−1,not the final right endpoint."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample2 offset，不以小數近似替代。",
          "en": "Recompute left sample2 offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "1/50"
        ],
        "output": "1/25",
        "label": "left sample2 offset"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "left sample2 x := sum(-2, 1/25)",
      "after": "left sample2 x = -49/25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "起點加上第i個偏移得到採樣位置。",
        "en": "Add the indexed offset to the start."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample2 x，不以小數近似替代。",
          "en": "Recompute left sample2 x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "1/25"
        ],
        "output": "-49/25",
        "label": "left sample2 x"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "left sample2 mx := mul(25, -49/25)",
      "after": "left sample2 mx = -49",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "代入仿射函數乘法部分。",
        "en": "Evaluate the product term of the affine integrand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample2 mx，不以小數近似替代。",
          "en": "Recompute left sample2 mx with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-49/25"
        ],
        "output": "-49",
        "label": "left sample2 mx"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "left sample2 value := sum(-49, 12)",
      "after": "left sample2 value = -37",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加常數c得到本採樣值。",
        "en": "Add c to obtain this sample value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample2 value，不以小數近似替代。",
          "en": "Recompute left sample2 value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-49",
          "12"
        ],
        "output": "-37",
        "label": "left sample2 value"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "left sample2 rectangle := mul(-37, 1/50)",
      "after": "left sample2 rectangle = -37/50",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。",
        "en": "Multiply by the step to get the signed rectangle contribution,without absolute values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample2 rectangle，不以小數近似替代。",
          "en": "Recompute left sample2 rectangle with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-37",
          "1/50"
        ],
        "output": "-37/50",
        "label": "left sample2 rectangle"
      }
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "[-49/25, -37, -37/50]",
      "after": "finite left-sum term index=2",
      "relation": "implies",
      "rule_id": "left_partition_term",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。",
        "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof."
      },
      "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": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "left sample3 offset := mul(3, 1/50)",
      "after": "left sample3 offset = 3/50",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。",
        "en": "Left samples use i=0,...,N−1,not the final right endpoint."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample3 offset，不以小數近似替代。",
          "en": "Recompute left sample3 offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3",
          "1/50"
        ],
        "output": "3/50",
        "label": "left sample3 offset"
      }
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "left sample3 x := sum(-2, 3/50)",
      "after": "left sample3 x = -97/50",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "起點加上第i個偏移得到採樣位置。",
        "en": "Add the indexed offset to the start."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample3 x，不以小數近似替代。",
          "en": "Recompute left sample3 x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "3/50"
        ],
        "output": "-97/50",
        "label": "left sample3 x"
      }
    },
    {
      "id": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "left sample3 mx := mul(25, -97/50)",
      "after": "left sample3 mx = -97/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "代入仿射函數乘法部分。",
        "en": "Evaluate the product term of the affine integrand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample3 mx，不以小數近似替代。",
          "en": "Recompute left sample3 mx with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-97/50"
        ],
        "output": "-97/2",
        "label": "left sample3 mx"
      }
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "left sample3 value := sum(-97/2, 12)",
      "after": "left sample3 value = -73/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加常數c得到本採樣值。",
        "en": "Add c to obtain this sample value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample3 value，不以小數近似替代。",
          "en": "Recompute left sample3 value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-97/2",
          "12"
        ],
        "output": "-73/2",
        "label": "left sample3 value"
      }
    },
    {
      "id": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "left sample3 rectangle := mul(-73/2, 1/50)",
      "after": "left sample3 rectangle = -73/100",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。",
        "en": "Multiply by the step to get the signed rectangle contribution,without absolute values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample3 rectangle，不以小數近似替代。",
          "en": "Recompute left sample3 rectangle with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-73/2",
          "1/50"
        ],
        "output": "-73/100",
        "label": "left sample3 rectangle"
      }
    },
    {
      "id": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "[-97/50, -73/2, -73/100]",
      "after": "finite left-sum term index=3",
      "relation": "implies",
      "rule_id": "left_partition_term",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。",
        "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof."
      },
      "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": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "left sample4 offset := mul(4, 1/50)",
      "after": "left sample4 offset = 2/25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。",
        "en": "Left samples use i=0,...,N−1,not the final right endpoint."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample4 offset，不以小數近似替代。",
          "en": "Recompute left sample4 offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "4",
          "1/50"
        ],
        "output": "2/25",
        "label": "left sample4 offset"
      }
    },
    {
      "id": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "left sample4 x := sum(-2, 2/25)",
      "after": "left sample4 x = -48/25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "起點加上第i個偏移得到採樣位置。",
        "en": "Add the indexed offset to the start."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample4 x，不以小數近似替代。",
          "en": "Recompute left sample4 x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "2/25"
        ],
        "output": "-48/25",
        "label": "left sample4 x"
      }
    },
    {
      "id": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "left sample4 mx := mul(25, -48/25)",
      "after": "left sample4 mx = -48",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "代入仿射函數乘法部分。",
        "en": "Evaluate the product term of the affine integrand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample4 mx，不以小數近似替代。",
          "en": "Recompute left sample4 mx with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-48/25"
        ],
        "output": "-48",
        "label": "left sample4 mx"
      }
    },
    {
      "id": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "left sample4 value := sum(-48, 12)",
      "after": "left sample4 value = -36",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加常數c得到本採樣值。",
        "en": "Add c to obtain this sample value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample4 value，不以小數近似替代。",
          "en": "Recompute left sample4 value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-48",
          "12"
        ],
        "output": "-36",
        "label": "left sample4 value"
      }
    },
    {
      "id": "S032",
      "uses": [
        "G001",
        "S031"
      ],
      "before": "left sample4 rectangle := mul(-36, 1/50)",
      "after": "left sample4 rectangle = -18/25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。",
        "en": "Multiply by the step to get the signed rectangle contribution,without absolute values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample4 rectangle，不以小數近似替代。",
          "en": "Recompute left sample4 rectangle with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-36",
          "1/50"
        ],
        "output": "-18/25",
        "label": "left sample4 rectangle"
      }
    },
    {
      "id": "S033",
      "uses": [
        "G001",
        "S032"
      ],
      "before": "[-48/25, -36, -18/25]",
      "after": "finite left-sum term index=4",
      "relation": "implies",
      "rule_id": "left_partition_term",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。",
        "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof."
      },
      "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": "S034",
      "uses": [
        "G001",
        "S033"
      ],
      "before": "left sample5 offset := mul(5, 1/50)",
      "after": "left sample5 offset = 1/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。",
        "en": "Left samples use i=0,...,N−1,not the final right endpoint."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample5 offset，不以小數近似替代。",
          "en": "Recompute left sample5 offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "5",
          "1/50"
        ],
        "output": "1/10",
        "label": "left sample5 offset"
      }
    },
    {
      "id": "S035",
      "uses": [
        "G001",
        "S034"
      ],
      "before": "left sample5 x := sum(-2, 1/10)",
      "after": "left sample5 x = -19/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "起點加上第i個偏移得到採樣位置。",
        "en": "Add the indexed offset to the start."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample5 x，不以小數近似替代。",
          "en": "Recompute left sample5 x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "1/10"
        ],
        "output": "-19/10",
        "label": "left sample5 x"
      }
    },
    {
      "id": "S036",
      "uses": [
        "G001",
        "S035"
      ],
      "before": "left sample5 mx := mul(25, -19/10)",
      "after": "left sample5 mx = -95/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "代入仿射函數乘法部分。",
        "en": "Evaluate the product term of the affine integrand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample5 mx，不以小數近似替代。",
          "en": "Recompute left sample5 mx with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-19/10"
        ],
        "output": "-95/2",
        "label": "left sample5 mx"
      }
    },
    {
      "id": "S037",
      "uses": [
        "G001",
        "S036"
      ],
      "before": "left sample5 value := sum(-95/2, 12)",
      "after": "left sample5 value = -71/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加常數c得到本採樣值。",
        "en": "Add c to obtain this sample value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample5 value，不以小數近似替代。",
          "en": "Recompute left sample5 value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-95/2",
          "12"
        ],
        "output": "-71/2",
        "label": "left sample5 value"
      }
    },
    {
      "id": "S038",
      "uses": [
        "G001",
        "S037"
      ],
      "before": "left sample5 rectangle := mul(-71/2, 1/50)",
      "after": "left sample5 rectangle = -71/100",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。",
        "en": "Multiply by the step to get the signed rectangle contribution,without absolute values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample5 rectangle，不以小數近似替代。",
          "en": "Recompute left sample5 rectangle with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-71/2",
          "1/50"
        ],
        "output": "-71/100",
        "label": "left sample5 rectangle"
      }
    },
    {
      "id": "S039",
      "uses": [
        "G001",
        "S038"
      ],
      "before": "[-19/10, -71/2, -71/100]",
      "after": "finite left-sum term index=5",
      "relation": "implies",
      "rule_id": "left_partition_term",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。",
        "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof."
      },
      "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": "S040",
      "uses": [
        "G001",
        "S039"
      ],
      "before": "left sample6 offset := mul(6, 1/50)",
      "after": "left sample6 offset = 3/25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。",
        "en": "Left samples use i=0,...,N−1,not the final right endpoint."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample6 offset，不以小數近似替代。",
          "en": "Recompute left sample6 offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "6",
          "1/50"
        ],
        "output": "3/25",
        "label": "left sample6 offset"
      }
    },
    {
      "id": "S041",
      "uses": [
        "G001",
        "S040"
      ],
      "before": "left sample6 x := sum(-2, 3/25)",
      "after": "left sample6 x = -47/25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "起點加上第i個偏移得到採樣位置。",
        "en": "Add the indexed offset to the start."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample6 x，不以小數近似替代。",
          "en": "Recompute left sample6 x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "3/25"
        ],
        "output": "-47/25",
        "label": "left sample6 x"
      }
    },
    {
      "id": "S042",
      "uses": [
        "G001",
        "S041"
      ],
      "before": "left sample6 mx := mul(25, -47/25)",
      "after": "left sample6 mx = -47",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "代入仿射函數乘法部分。",
        "en": "Evaluate the product term of the affine integrand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample6 mx，不以小數近似替代。",
          "en": "Recompute left sample6 mx with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-47/25"
        ],
        "output": "-47",
        "label": "left sample6 mx"
      }
    },
    {
      "id": "S043",
      "uses": [
        "G001",
        "S042"
      ],
      "before": "left sample6 value := sum(-47, 12)",
      "after": "left sample6 value = -35",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加常數c得到本採樣值。",
        "en": "Add c to obtain this sample value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample6 value，不以小數近似替代。",
          "en": "Recompute left sample6 value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-47",
          "12"
        ],
        "output": "-35",
        "label": "left sample6 value"
      }
    },
    {
      "id": "S044",
      "uses": [
        "G001",
        "S043"
      ],
      "before": "left sample6 rectangle := mul(-35, 1/50)",
      "after": "left sample6 rectangle = -7/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。",
        "en": "Multiply by the step to get the signed rectangle contribution,without absolute values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample6 rectangle，不以小數近似替代。",
          "en": "Recompute left sample6 rectangle with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-35",
          "1/50"
        ],
        "output": "-7/10",
        "label": "left sample6 rectangle"
      }
    },
    {
      "id": "S045",
      "uses": [
        "G001",
        "S044"
      ],
      "before": "[-47/25, -35, -7/10]",
      "after": "finite left-sum term index=6",
      "relation": "implies",
      "rule_id": "left_partition_term",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。",
        "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof."
      },
      "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": "S046",
      "uses": [
        "G001",
        "S045"
      ],
      "before": "left sample7 offset := mul(7, 1/50)",
      "after": "left sample7 offset = 7/50",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。",
        "en": "Left samples use i=0,...,N−1,not the final right endpoint."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample7 offset，不以小數近似替代。",
          "en": "Recompute left sample7 offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "7",
          "1/50"
        ],
        "output": "7/50",
        "label": "left sample7 offset"
      }
    },
    {
      "id": "S047",
      "uses": [
        "G001",
        "S046"
      ],
      "before": "left sample7 x := sum(-2, 7/50)",
      "after": "left sample7 x = -93/50",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "起點加上第i個偏移得到採樣位置。",
        "en": "Add the indexed offset to the start."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample7 x，不以小數近似替代。",
          "en": "Recompute left sample7 x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "7/50"
        ],
        "output": "-93/50",
        "label": "left sample7 x"
      }
    },
    {
      "id": "S048",
      "uses": [
        "G001",
        "S047"
      ],
      "before": "left sample7 mx := mul(25, -93/50)",
      "after": "left sample7 mx = -93/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "代入仿射函數乘法部分。",
        "en": "Evaluate the product term of the affine integrand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample7 mx，不以小數近似替代。",
          "en": "Recompute left sample7 mx with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-93/50"
        ],
        "output": "-93/2",
        "label": "left sample7 mx"
      }
    },
    {
      "id": "S049",
      "uses": [
        "G001",
        "S048"
      ],
      "before": "left sample7 value := sum(-93/2, 12)",
      "after": "left sample7 value = -69/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加常數c得到本採樣值。",
        "en": "Add c to obtain this sample value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample7 value，不以小數近似替代。",
          "en": "Recompute left sample7 value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-93/2",
          "12"
        ],
        "output": "-69/2",
        "label": "left sample7 value"
      }
    },
    {
      "id": "S050",
      "uses": [
        "G001",
        "S049"
      ],
      "before": "left sample7 rectangle := mul(-69/2, 1/50)",
      "after": "left sample7 rectangle = -69/100",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。",
        "en": "Multiply by the step to get the signed rectangle contribution,without absolute values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample7 rectangle，不以小數近似替代。",
          "en": "Recompute left sample7 rectangle with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-69/2",
          "1/50"
        ],
        "output": "-69/100",
        "label": "left sample7 rectangle"
      }
    },
    {
      "id": "S051",
      "uses": [
        "G001",
        "S050"
      ],
      "before": "[-93/50, -69/2, -69/100]",
      "after": "finite left-sum term index=7",
      "relation": "implies",
      "rule_id": "left_partition_term",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。",
        "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof."
      },
      "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": "S052",
      "uses": [
        "G001",
        "S051"
      ],
      "before": "left sample8 offset := mul(8, 1/50)",
      "after": "left sample8 offset = 4/25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。",
        "en": "Left samples use i=0,...,N−1,not the final right endpoint."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample8 offset，不以小數近似替代。",
          "en": "Recompute left sample8 offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "8",
          "1/50"
        ],
        "output": "4/25",
        "label": "left sample8 offset"
      }
    },
    {
      "id": "S053",
      "uses": [
        "G001",
        "S052"
      ],
      "before": "left sample8 x := sum(-2, 4/25)",
      "after": "left sample8 x = -46/25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "起點加上第i個偏移得到採樣位置。",
        "en": "Add the indexed offset to the start."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample8 x，不以小數近似替代。",
          "en": "Recompute left sample8 x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "4/25"
        ],
        "output": "-46/25",
        "label": "left sample8 x"
      }
    },
    {
      "id": "S054",
      "uses": [
        "G001",
        "S053"
      ],
      "before": "left sample8 mx := mul(25, -46/25)",
      "after": "left sample8 mx = -46",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "代入仿射函數乘法部分。",
        "en": "Evaluate the product term of the affine integrand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample8 mx，不以小數近似替代。",
          "en": "Recompute left sample8 mx with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-46/25"
        ],
        "output": "-46",
        "label": "left sample8 mx"
      }
    },
    {
      "id": "S055",
      "uses": [
        "G001",
        "S054"
      ],
      "before": "left sample8 value := sum(-46, 12)",
      "after": "left sample8 value = -34",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加常數c得到本採樣值。",
        "en": "Add c to obtain this sample value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample8 value，不以小數近似替代。",
          "en": "Recompute left sample8 value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-46",
          "12"
        ],
        "output": "-34",
        "label": "left sample8 value"
      }
    },
    {
      "id": "S056",
      "uses": [
        "G001",
        "S055"
      ],
      "before": "left sample8 rectangle := mul(-34, 1/50)",
      "after": "left sample8 rectangle = -17/25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。",
        "en": "Multiply by the step to get the signed rectangle contribution,without absolute values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample8 rectangle，不以小數近似替代。",
          "en": "Recompute left sample8 rectangle with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-34",
          "1/50"
        ],
        "output": "-17/25",
        "label": "left sample8 rectangle"
      }
    },
    {
      "id": "S057",
      "uses": [
        "G001",
        "S056"
      ],
      "before": "[-46/25, -34, -17/25]",
      "after": "finite left-sum term index=8",
      "relation": "implies",
      "rule_id": "left_partition_term",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。",
        "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof."
      },
      "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": "S058",
      "uses": [
        "G001",
        "S057"
      ],
      "before": "left sample9 offset := mul(9, 1/50)",
      "after": "left sample9 offset = 9/50",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。",
        "en": "Left samples use i=0,...,N−1,not the final right endpoint."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample9 offset，不以小數近似替代。",
          "en": "Recompute left sample9 offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "9",
          "1/50"
        ],
        "output": "9/50",
        "label": "left sample9 offset"
      }
    },
    {
      "id": "S059",
      "uses": [
        "G001",
        "S058"
      ],
      "before": "left sample9 x := sum(-2, 9/50)",
      "after": "left sample9 x = -91/50",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "起點加上第i個偏移得到採樣位置。",
        "en": "Add the indexed offset to the start."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample9 x，不以小數近似替代。",
          "en": "Recompute left sample9 x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "9/50"
        ],
        "output": "-91/50",
        "label": "left sample9 x"
      }
    },
    {
      "id": "S060",
      "uses": [
        "G001",
        "S059"
      ],
      "before": "left sample9 mx := mul(25, -91/50)",
      "after": "left sample9 mx = -91/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "代入仿射函數乘法部分。",
        "en": "Evaluate the product term of the affine integrand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample9 mx，不以小數近似替代。",
          "en": "Recompute left sample9 mx with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-91/50"
        ],
        "output": "-91/2",
        "label": "left sample9 mx"
      }
    },
    {
      "id": "S061",
      "uses": [
        "G001",
        "S060"
      ],
      "before": "left sample9 value := sum(-91/2, 12)",
      "after": "left sample9 value = -67/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加常數c得到本採樣值。",
        "en": "Add c to obtain this sample value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample9 value，不以小數近似替代。",
          "en": "Recompute left sample9 value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-91/2",
          "12"
        ],
        "output": "-67/2",
        "label": "left sample9 value"
      }
    },
    {
      "id": "S062",
      "uses": [
        "G001",
        "S061"
      ],
      "before": "left sample9 rectangle := mul(-67/2, 1/50)",
      "after": "left sample9 rectangle = -67/100",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。",
        "en": "Multiply by the step to get the signed rectangle contribution,without absolute values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample9 rectangle，不以小數近似替代。",
          "en": "Recompute left sample9 rectangle with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-67/2",
          "1/50"
        ],
        "output": "-67/100",
        "label": "left sample9 rectangle"
      }
    },
    {
      "id": "S063",
      "uses": [
        "G001",
        "S062"
      ],
      "before": "[-91/50, -67/2, -67/100]",
      "after": "finite left-sum term index=9",
      "relation": "implies",
      "rule_id": "left_partition_term",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。",
        "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof."
      },
      "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": "S064",
      "uses": [
        "G001",
        "S063"
      ],
      "before": "left sample10 offset := mul(10, 1/50)",
      "after": "left sample10 offset = 1/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。",
        "en": "Left samples use i=0,...,N−1,not the final right endpoint."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample10 offset，不以小數近似替代。",
          "en": "Recompute left sample10 offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "10",
          "1/50"
        ],
        "output": "1/5",
        "label": "left sample10 offset"
      }
    },
    {
      "id": "S065",
      "uses": [
        "G001",
        "S064"
      ],
      "before": "left sample10 x := sum(-2, 1/5)",
      "after": "left sample10 x = -9/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "起點加上第i個偏移得到採樣位置。",
        "en": "Add the indexed offset to the start."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample10 x，不以小數近似替代。",
          "en": "Recompute left sample10 x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "1/5"
        ],
        "output": "-9/5",
        "label": "left sample10 x"
      }
    },
    {
      "id": "S066",
      "uses": [
        "G001",
        "S065"
      ],
      "before": "left sample10 mx := mul(25, -9/5)",
      "after": "left sample10 mx = -45",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "代入仿射函數乘法部分。",
        "en": "Evaluate the product term of the affine integrand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample10 mx，不以小數近似替代。",
          "en": "Recompute left sample10 mx with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-9/5"
        ],
        "output": "-45",
        "label": "left sample10 mx"
      }
    },
    {
      "id": "S067",
      "uses": [
        "G001",
        "S066"
      ],
      "before": "left sample10 value := sum(-45, 12)",
      "after": "left sample10 value = -33",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加常數c得到本採樣值。",
        "en": "Add c to obtain this sample value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample10 value，不以小數近似替代。",
          "en": "Recompute left sample10 value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-45",
          "12"
        ],
        "output": "-33",
        "label": "left sample10 value"
      }
    },
    {
      "id": "S068",
      "uses": [
        "G001",
        "S067"
      ],
      "before": "left sample10 rectangle := mul(-33, 1/50)",
      "after": "left sample10 rectangle = -33/50",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。",
        "en": "Multiply by the step to get the signed rectangle contribution,without absolute values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample10 rectangle，不以小數近似替代。",
          "en": "Recompute left sample10 rectangle with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-33",
          "1/50"
        ],
        "output": "-33/50",
        "label": "left sample10 rectangle"
      }
    },
    {
      "id": "S069",
      "uses": [
        "G001",
        "S068"
      ],
      "before": "[-9/5, -33, -33/50]",
      "after": "finite left-sum term index=10",
      "relation": "implies",
      "rule_id": "left_partition_term",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。",
        "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof."
      },
      "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": "S070",
      "uses": [
        "G001",
        "S069"
      ],
      "before": "left sample11 offset := mul(11, 1/50)",
      "after": "left sample11 offset = 11/50",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。",
        "en": "Left samples use i=0,...,N−1,not the final right endpoint."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample11 offset，不以小數近似替代。",
          "en": "Recompute left sample11 offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "11",
          "1/50"
        ],
        "output": "11/50",
        "label": "left sample11 offset"
      }
    },
    {
      "id": "S071",
      "uses": [
        "G001",
        "S070"
      ],
      "before": "left sample11 x := sum(-2, 11/50)",
      "after": "left sample11 x = -89/50",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "起點加上第i個偏移得到採樣位置。",
        "en": "Add the indexed offset to the start."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample11 x，不以小數近似替代。",
          "en": "Recompute left sample11 x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "11/50"
        ],
        "output": "-89/50",
        "label": "left sample11 x"
      }
    },
    {
      "id": "S072",
      "uses": [
        "G001",
        "S071"
      ],
      "before": "left sample11 mx := mul(25, -89/50)",
      "after": "left sample11 mx = -89/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "代入仿射函數乘法部分。",
        "en": "Evaluate the product term of the affine integrand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample11 mx，不以小數近似替代。",
          "en": "Recompute left sample11 mx with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-89/50"
        ],
        "output": "-89/2",
        "label": "left sample11 mx"
      }
    },
    {
      "id": "S073",
      "uses": [
        "G001",
        "S072"
      ],
      "before": "left sample11 value := sum(-89/2, 12)",
      "after": "left sample11 value = -65/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加常數c得到本採樣值。",
        "en": "Add c to obtain this sample value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample11 value，不以小數近似替代。",
          "en": "Recompute left sample11 value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-89/2",
          "12"
        ],
        "output": "-65/2",
        "label": "left sample11 value"
      }
    },
    {
      "id": "S074",
      "uses": [
        "G001",
        "S073"
      ],
      "before": "left sample11 rectangle := mul(-65/2, 1/50)",
      "after": "left sample11 rectangle = -13/20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。",
        "en": "Multiply by the step to get the signed rectangle contribution,without absolute values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample11 rectangle，不以小數近似替代。",
          "en": "Recompute left sample11 rectangle with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-65/2",
          "1/50"
        ],
        "output": "-13/20",
        "label": "left sample11 rectangle"
      }
    },
    {
      "id": "S075",
      "uses": [
        "G001",
        "S074"
      ],
      "before": "[-89/50, -65/2, -13/20]",
      "after": "finite left-sum term index=11",
      "relation": "implies",
      "rule_id": "left_partition_term",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。",
        "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof."
      },
      "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": "S076",
      "uses": [
        "G001",
        "S075"
      ],
      "before": "left sample12 offset := mul(12, 1/50)",
      "after": "left sample12 offset = 6/25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。",
        "en": "Left samples use i=0,...,N−1,not the final right endpoint."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample12 offset，不以小數近似替代。",
          "en": "Recompute left sample12 offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "12",
          "1/50"
        ],
        "output": "6/25",
        "label": "left sample12 offset"
      }
    },
    {
      "id": "S077",
      "uses": [
        "G001",
        "S076"
      ],
      "before": "left sample12 x := sum(-2, 6/25)",
      "after": "left sample12 x = -44/25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "起點加上第i個偏移得到採樣位置。",
        "en": "Add the indexed offset to the start."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample12 x，不以小數近似替代。",
          "en": "Recompute left sample12 x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "6/25"
        ],
        "output": "-44/25",
        "label": "left sample12 x"
      }
    },
    {
      "id": "S078",
      "uses": [
        "G001",
        "S077"
      ],
      "before": "left sample12 mx := mul(25, -44/25)",
      "after": "left sample12 mx = -44",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "代入仿射函數乘法部分。",
        "en": "Evaluate the product term of the affine integrand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample12 mx，不以小數近似替代。",
          "en": "Recompute left sample12 mx with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-44/25"
        ],
        "output": "-44",
        "label": "left sample12 mx"
      }
    },
    {
      "id": "S079",
      "uses": [
        "G001",
        "S078"
      ],
      "before": "left sample12 value := sum(-44, 12)",
      "after": "left sample12 value = -32",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加常數c得到本採樣值。",
        "en": "Add c to obtain this sample value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample12 value，不以小數近似替代。",
          "en": "Recompute left sample12 value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-44",
          "12"
        ],
        "output": "-32",
        "label": "left sample12 value"
      }
    },
    {
      "id": "S080",
      "uses": [
        "G001",
        "S079"
      ],
      "before": "left sample12 rectangle := mul(-32, 1/50)",
      "after": "left sample12 rectangle = -16/25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。",
        "en": "Multiply by the step to get the signed rectangle contribution,without absolute values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample12 rectangle，不以小數近似替代。",
          "en": "Recompute left sample12 rectangle with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-32",
          "1/50"
        ],
        "output": "-16/25",
        "label": "left sample12 rectangle"
      }
    },
    {
      "id": "S081",
      "uses": [
        "G001",
        "S080"
      ],
      "before": "[-44/25, -32, -16/25]",
      "after": "finite left-sum term index=12",
      "relation": "implies",
      "rule_id": "left_partition_term",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。",
        "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof."
      },
      "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": "S082",
      "uses": [
        "G001",
        "S081"
      ],
      "before": "left sample13 offset := mul(13, 1/50)",
      "after": "left sample13 offset = 13/50",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。",
        "en": "Left samples use i=0,...,N−1,not the final right endpoint."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample13 offset，不以小數近似替代。",
          "en": "Recompute left sample13 offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "13",
          "1/50"
        ],
        "output": "13/50",
        "label": "left sample13 offset"
      }
    },
    {
      "id": "S083",
      "uses": [
        "G001",
        "S082"
      ],
      "before": "left sample13 x := sum(-2, 13/50)",
      "after": "left sample13 x = -87/50",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "起點加上第i個偏移得到採樣位置。",
        "en": "Add the indexed offset to the start."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample13 x，不以小數近似替代。",
          "en": "Recompute left sample13 x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "13/50"
        ],
        "output": "-87/50",
        "label": "left sample13 x"
      }
    },
    {
      "id": "S084",
      "uses": [
        "G001",
        "S083"
      ],
      "before": "left sample13 mx := mul(25, -87/50)",
      "after": "left sample13 mx = -87/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "代入仿射函數乘法部分。",
        "en": "Evaluate the product term of the affine integrand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample13 mx，不以小數近似替代。",
          "en": "Recompute left sample13 mx with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-87/50"
        ],
        "output": "-87/2",
        "label": "left sample13 mx"
      }
    },
    {
      "id": "S085",
      "uses": [
        "G001",
        "S084"
      ],
      "before": "left sample13 value := sum(-87/2, 12)",
      "after": "left sample13 value = -63/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加常數c得到本採樣值。",
        "en": "Add c to obtain this sample value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample13 value，不以小數近似替代。",
          "en": "Recompute left sample13 value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-87/2",
          "12"
        ],
        "output": "-63/2",
        "label": "left sample13 value"
      }
    },
    {
      "id": "S086",
      "uses": [
        "G001",
        "S085"
      ],
      "before": "left sample13 rectangle := mul(-63/2, 1/50)",
      "after": "left sample13 rectangle = -63/100",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。",
        "en": "Multiply by the step to get the signed rectangle contribution,without absolute values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample13 rectangle，不以小數近似替代。",
          "en": "Recompute left sample13 rectangle with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-63/2",
          "1/50"
        ],
        "output": "-63/100",
        "label": "left sample13 rectangle"
      }
    },
    {
      "id": "S087",
      "uses": [
        "G001",
        "S086"
      ],
      "before": "[-87/50, -63/2, -63/100]",
      "after": "finite left-sum term index=13",
      "relation": "implies",
      "rule_id": "left_partition_term",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。",
        "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof."
      },
      "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": "S088",
      "uses": [
        "G001",
        "S087"
      ],
      "before": "left sample14 offset := mul(14, 1/50)",
      "after": "left sample14 offset = 7/25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。",
        "en": "Left samples use i=0,...,N−1,not the final right endpoint."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample14 offset，不以小數近似替代。",
          "en": "Recompute left sample14 offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "14",
          "1/50"
        ],
        "output": "7/25",
        "label": "left sample14 offset"
      }
    },
    {
      "id": "S089",
      "uses": [
        "G001",
        "S088"
      ],
      "before": "left sample14 x := sum(-2, 7/25)",
      "after": "left sample14 x = -43/25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "起點加上第i個偏移得到採樣位置。",
        "en": "Add the indexed offset to the start."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample14 x，不以小數近似替代。",
          "en": "Recompute left sample14 x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "7/25"
        ],
        "output": "-43/25",
        "label": "left sample14 x"
      }
    },
    {
      "id": "S090",
      "uses": [
        "G001",
        "S089"
      ],
      "before": "left sample14 mx := mul(25, -43/25)",
      "after": "left sample14 mx = -43",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "代入仿射函數乘法部分。",
        "en": "Evaluate the product term of the affine integrand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample14 mx，不以小數近似替代。",
          "en": "Recompute left sample14 mx with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-43/25"
        ],
        "output": "-43",
        "label": "left sample14 mx"
      }
    },
    {
      "id": "S091",
      "uses": [
        "G001",
        "S090"
      ],
      "before": "left sample14 value := sum(-43, 12)",
      "after": "left sample14 value = -31",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加常數c得到本採樣值。",
        "en": "Add c to obtain this sample value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample14 value，不以小數近似替代。",
          "en": "Recompute left sample14 value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-43",
          "12"
        ],
        "output": "-31",
        "label": "left sample14 value"
      }
    },
    {
      "id": "S092",
      "uses": [
        "G001",
        "S091"
      ],
      "before": "left sample14 rectangle := mul(-31, 1/50)",
      "after": "left sample14 rectangle = -31/50",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。",
        "en": "Multiply by the step to get the signed rectangle contribution,without absolute values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample14 rectangle，不以小數近似替代。",
          "en": "Recompute left sample14 rectangle with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-31",
          "1/50"
        ],
        "output": "-31/50",
        "label": "left sample14 rectangle"
      }
    },
    {
      "id": "S093",
      "uses": [
        "G001",
        "S092"
      ],
      "before": "[-43/25, -31, -31/50]",
      "after": "finite left-sum term index=14",
      "relation": "implies",
      "rule_id": "left_partition_term",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。",
        "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof."
      },
      "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": "S094",
      "uses": [
        "G001",
        "S093"
      ],
      "before": "left sample15 offset := mul(15, 1/50)",
      "after": "left sample15 offset = 3/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。",
        "en": "Left samples use i=0,...,N−1,not the final right endpoint."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample15 offset，不以小數近似替代。",
          "en": "Recompute left sample15 offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "15",
          "1/50"
        ],
        "output": "3/10",
        "label": "left sample15 offset"
      }
    },
    {
      "id": "S095",
      "uses": [
        "G001",
        "S094"
      ],
      "before": "left sample15 x := sum(-2, 3/10)",
      "after": "left sample15 x = -17/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "起點加上第i個偏移得到採樣位置。",
        "en": "Add the indexed offset to the start."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample15 x，不以小數近似替代。",
          "en": "Recompute left sample15 x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "3/10"
        ],
        "output": "-17/10",
        "label": "left sample15 x"
      }
    },
    {
      "id": "S096",
      "uses": [
        "G001",
        "S095"
      ],
      "before": "left sample15 mx := mul(25, -17/10)",
      "after": "left sample15 mx = -85/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "代入仿射函數乘法部分。",
        "en": "Evaluate the product term of the affine integrand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample15 mx，不以小數近似替代。",
          "en": "Recompute left sample15 mx with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-17/10"
        ],
        "output": "-85/2",
        "label": "left sample15 mx"
      }
    },
    {
      "id": "S097",
      "uses": [
        "G001",
        "S096"
      ],
      "before": "left sample15 value := sum(-85/2, 12)",
      "after": "left sample15 value = -61/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加常數c得到本採樣值。",
        "en": "Add c to obtain this sample value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample15 value，不以小數近似替代。",
          "en": "Recompute left sample15 value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-85/2",
          "12"
        ],
        "output": "-61/2",
        "label": "left sample15 value"
      }
    },
    {
      "id": "S098",
      "uses": [
        "G001",
        "S097"
      ],
      "before": "left sample15 rectangle := mul(-61/2, 1/50)",
      "after": "left sample15 rectangle = -61/100",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。",
        "en": "Multiply by the step to get the signed rectangle contribution,without absolute values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample15 rectangle，不以小數近似替代。",
          "en": "Recompute left sample15 rectangle with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-61/2",
          "1/50"
        ],
        "output": "-61/100",
        "label": "left sample15 rectangle"
      }
    },
    {
      "id": "S099",
      "uses": [
        "G001",
        "S098"
      ],
      "before": "[-17/10, -61/2, -61/100]",
      "after": "finite left-sum term index=15",
      "relation": "implies",
      "rule_id": "left_partition_term",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。",
        "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof."
      },
      "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": "S100",
      "uses": [
        "G001",
        "S099"
      ],
      "before": "left sample16 offset := mul(16, 1/50)",
      "after": "left sample16 offset = 8/25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。",
        "en": "Left samples use i=0,...,N−1,not the final right endpoint."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample16 offset，不以小數近似替代。",
          "en": "Recompute left sample16 offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "16",
          "1/50"
        ],
        "output": "8/25",
        "label": "left sample16 offset"
      }
    },
    {
      "id": "S101",
      "uses": [
        "G001",
        "S100"
      ],
      "before": "left sample16 x := sum(-2, 8/25)",
      "after": "left sample16 x = -42/25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "起點加上第i個偏移得到採樣位置。",
        "en": "Add the indexed offset to the start."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample16 x，不以小數近似替代。",
          "en": "Recompute left sample16 x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "8/25"
        ],
        "output": "-42/25",
        "label": "left sample16 x"
      }
    },
    {
      "id": "S102",
      "uses": [
        "G001",
        "S101"
      ],
      "before": "left sample16 mx := mul(25, -42/25)",
      "after": "left sample16 mx = -42",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "代入仿射函數乘法部分。",
        "en": "Evaluate the product term of the affine integrand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample16 mx，不以小數近似替代。",
          "en": "Recompute left sample16 mx with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-42/25"
        ],
        "output": "-42",
        "label": "left sample16 mx"
      }
    },
    {
      "id": "S103",
      "uses": [
        "G001",
        "S102"
      ],
      "before": "left sample16 value := sum(-42, 12)",
      "after": "left sample16 value = -30",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加常數c得到本採樣值。",
        "en": "Add c to obtain this sample value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample16 value，不以小數近似替代。",
          "en": "Recompute left sample16 value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-42",
          "12"
        ],
        "output": "-30",
        "label": "left sample16 value"
      }
    },
    {
      "id": "S104",
      "uses": [
        "G001",
        "S103"
      ],
      "before": "left sample16 rectangle := mul(-30, 1/50)",
      "after": "left sample16 rectangle = -3/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。",
        "en": "Multiply by the step to get the signed rectangle contribution,without absolute values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample16 rectangle，不以小數近似替代。",
          "en": "Recompute left sample16 rectangle with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-30",
          "1/50"
        ],
        "output": "-3/5",
        "label": "left sample16 rectangle"
      }
    },
    {
      "id": "S105",
      "uses": [
        "G001",
        "S104"
      ],
      "before": "[-42/25, -30, -3/5]",
      "after": "finite left-sum term index=16",
      "relation": "implies",
      "rule_id": "left_partition_term",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。",
        "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof."
      },
      "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": "S106",
      "uses": [
        "G001",
        "S105"
      ],
      "before": "left sample17 offset := mul(17, 1/50)",
      "after": "left sample17 offset = 17/50",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。",
        "en": "Left samples use i=0,...,N−1,not the final right endpoint."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample17 offset，不以小數近似替代。",
          "en": "Recompute left sample17 offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "17",
          "1/50"
        ],
        "output": "17/50",
        "label": "left sample17 offset"
      }
    },
    {
      "id": "S107",
      "uses": [
        "G001",
        "S106"
      ],
      "before": "left sample17 x := sum(-2, 17/50)",
      "after": "left sample17 x = -83/50",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "起點加上第i個偏移得到採樣位置。",
        "en": "Add the indexed offset to the start."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample17 x，不以小數近似替代。",
          "en": "Recompute left sample17 x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "17/50"
        ],
        "output": "-83/50",
        "label": "left sample17 x"
      }
    },
    {
      "id": "S108",
      "uses": [
        "G001",
        "S107"
      ],
      "before": "left sample17 mx := mul(25, -83/50)",
      "after": "left sample17 mx = -83/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "代入仿射函數乘法部分。",
        "en": "Evaluate the product term of the affine integrand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample17 mx，不以小數近似替代。",
          "en": "Recompute left sample17 mx with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-83/50"
        ],
        "output": "-83/2",
        "label": "left sample17 mx"
      }
    },
    {
      "id": "S109",
      "uses": [
        "G001",
        "S108"
      ],
      "before": "left sample17 value := sum(-83/2, 12)",
      "after": "left sample17 value = -59/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加常數c得到本採樣值。",
        "en": "Add c to obtain this sample value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample17 value，不以小數近似替代。",
          "en": "Recompute left sample17 value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-83/2",
          "12"
        ],
        "output": "-59/2",
        "label": "left sample17 value"
      }
    },
    {
      "id": "S110",
      "uses": [
        "G001",
        "S109"
      ],
      "before": "left sample17 rectangle := mul(-59/2, 1/50)",
      "after": "left sample17 rectangle = -59/100",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。",
        "en": "Multiply by the step to get the signed rectangle contribution,without absolute values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample17 rectangle，不以小數近似替代。",
          "en": "Recompute left sample17 rectangle with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-59/2",
          "1/50"
        ],
        "output": "-59/100",
        "label": "left sample17 rectangle"
      }
    },
    {
      "id": "S111",
      "uses": [
        "G001",
        "S110"
      ],
      "before": "[-83/50, -59/2, -59/100]",
      "after": "finite left-sum term index=17",
      "relation": "implies",
      "rule_id": "left_partition_term",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。",
        "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof."
      },
      "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": "S112",
      "uses": [
        "G001",
        "S111"
      ],
      "before": "left sample18 offset := mul(18, 1/50)",
      "after": "left sample18 offset = 9/25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。",
        "en": "Left samples use i=0,...,N−1,not the final right endpoint."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample18 offset，不以小數近似替代。",
          "en": "Recompute left sample18 offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "18",
          "1/50"
        ],
        "output": "9/25",
        "label": "left sample18 offset"
      }
    },
    {
      "id": "S113",
      "uses": [
        "G001",
        "S112"
      ],
      "before": "left sample18 x := sum(-2, 9/25)",
      "after": "left sample18 x = -41/25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "起點加上第i個偏移得到採樣位置。",
        "en": "Add the indexed offset to the start."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample18 x，不以小數近似替代。",
          "en": "Recompute left sample18 x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "9/25"
        ],
        "output": "-41/25",
        "label": "left sample18 x"
      }
    },
    {
      "id": "S114",
      "uses": [
        "G001",
        "S113"
      ],
      "before": "left sample18 mx := mul(25, -41/25)",
      "after": "left sample18 mx = -41",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "代入仿射函數乘法部分。",
        "en": "Evaluate the product term of the affine integrand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample18 mx，不以小數近似替代。",
          "en": "Recompute left sample18 mx with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-41/25"
        ],
        "output": "-41",
        "label": "left sample18 mx"
      }
    },
    {
      "id": "S115",
      "uses": [
        "G001",
        "S114"
      ],
      "before": "left sample18 value := sum(-41, 12)",
      "after": "left sample18 value = -29",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加常數c得到本採樣值。",
        "en": "Add c to obtain this sample value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample18 value，不以小數近似替代。",
          "en": "Recompute left sample18 value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-41",
          "12"
        ],
        "output": "-29",
        "label": "left sample18 value"
      }
    },
    {
      "id": "S116",
      "uses": [
        "G001",
        "S115"
      ],
      "before": "left sample18 rectangle := mul(-29, 1/50)",
      "after": "left sample18 rectangle = -29/50",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。",
        "en": "Multiply by the step to get the signed rectangle contribution,without absolute values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample18 rectangle，不以小數近似替代。",
          "en": "Recompute left sample18 rectangle with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-29",
          "1/50"
        ],
        "output": "-29/50",
        "label": "left sample18 rectangle"
      }
    },
    {
      "id": "S117",
      "uses": [
        "G001",
        "S116"
      ],
      "before": "[-41/25, -29, -29/50]",
      "after": "finite left-sum term index=18",
      "relation": "implies",
      "rule_id": "left_partition_term",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。",
        "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof."
      },
      "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": "S118",
      "uses": [
        "G001",
        "S117"
      ],
      "before": "left sample19 offset := mul(19, 1/50)",
      "after": "left sample19 offset = 19/50",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。",
        "en": "Left samples use i=0,...,N−1,not the final right endpoint."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample19 offset，不以小數近似替代。",
          "en": "Recompute left sample19 offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "19",
          "1/50"
        ],
        "output": "19/50",
        "label": "left sample19 offset"
      }
    },
    {
      "id": "S119",
      "uses": [
        "G001",
        "S118"
      ],
      "before": "left sample19 x := sum(-2, 19/50)",
      "after": "left sample19 x = -81/50",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "起點加上第i個偏移得到採樣位置。",
        "en": "Add the indexed offset to the start."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample19 x，不以小數近似替代。",
          "en": "Recompute left sample19 x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "19/50"
        ],
        "output": "-81/50",
        "label": "left sample19 x"
      }
    },
    {
      "id": "S120",
      "uses": [
        "G001",
        "S119"
      ],
      "before": "left sample19 mx := mul(25, -81/50)",
      "after": "left sample19 mx = -81/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "代入仿射函數乘法部分。",
        "en": "Evaluate the product term of the affine integrand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample19 mx，不以小數近似替代。",
          "en": "Recompute left sample19 mx with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-81/50"
        ],
        "output": "-81/2",
        "label": "left sample19 mx"
      }
    },
    {
      "id": "S121",
      "uses": [
        "G001",
        "S120"
      ],
      "before": "left sample19 value := sum(-81/2, 12)",
      "after": "left sample19 value = -57/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加常數c得到本採樣值。",
        "en": "Add c to obtain this sample value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample19 value，不以小數近似替代。",
          "en": "Recompute left sample19 value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-81/2",
          "12"
        ],
        "output": "-57/2",
        "label": "left sample19 value"
      }
    },
    {
      "id": "S122",
      "uses": [
        "G001",
        "S121"
      ],
      "before": "left sample19 rectangle := mul(-57/2, 1/50)",
      "after": "left sample19 rectangle = -57/100",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。",
        "en": "Multiply by the step to get the signed rectangle contribution,without absolute values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample19 rectangle，不以小數近似替代。",
          "en": "Recompute left sample19 rectangle with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-57/2",
          "1/50"
        ],
        "output": "-57/100",
        "label": "left sample19 rectangle"
      }
    },
    {
      "id": "S123",
      "uses": [
        "G001",
        "S122"
      ],
      "before": "[-81/50, -57/2, -57/100]",
      "after": "finite left-sum term index=19",
      "relation": "implies",
      "rule_id": "left_partition_term",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。",
        "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof."
      },
      "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": "S124",
      "uses": [
        "G001",
        "S123"
      ],
      "before": "left sample20 offset := mul(20, 1/50)",
      "after": "left sample20 offset = 2/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。",
        "en": "Left samples use i=0,...,N−1,not the final right endpoint."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample20 offset，不以小數近似替代。",
          "en": "Recompute left sample20 offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "1/50"
        ],
        "output": "2/5",
        "label": "left sample20 offset"
      }
    },
    {
      "id": "S125",
      "uses": [
        "G001",
        "S124"
      ],
      "before": "left sample20 x := sum(-2, 2/5)",
      "after": "left sample20 x = -8/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "起點加上第i個偏移得到採樣位置。",
        "en": "Add the indexed offset to the start."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample20 x，不以小數近似替代。",
          "en": "Recompute left sample20 x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "2/5"
        ],
        "output": "-8/5",
        "label": "left sample20 x"
      }
    },
    {
      "id": "S126",
      "uses": [
        "G001",
        "S125"
      ],
      "before": "left sample20 mx := mul(25, -8/5)",
      "after": "left sample20 mx = -40",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "代入仿射函數乘法部分。",
        "en": "Evaluate the product term of the affine integrand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample20 mx，不以小數近似替代。",
          "en": "Recompute left sample20 mx with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-8/5"
        ],
        "output": "-40",
        "label": "left sample20 mx"
      }
    },
    {
      "id": "S127",
      "uses": [
        "G001",
        "S126"
      ],
      "before": "left sample20 value := sum(-40, 12)",
      "after": "left sample20 value = -28",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加常數c得到本採樣值。",
        "en": "Add c to obtain this sample value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample20 value，不以小數近似替代。",
          "en": "Recompute left sample20 value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-40",
          "12"
        ],
        "output": "-28",
        "label": "left sample20 value"
      }
    },
    {
      "id": "S128",
      "uses": [
        "G001",
        "S127"
      ],
      "before": "left sample20 rectangle := mul(-28, 1/50)",
      "after": "left sample20 rectangle = -14/25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。",
        "en": "Multiply by the step to get the signed rectangle contribution,without absolute values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample20 rectangle，不以小數近似替代。",
          "en": "Recompute left sample20 rectangle with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-28",
          "1/50"
        ],
        "output": "-14/25",
        "label": "left sample20 rectangle"
      }
    },
    {
      "id": "S129",
      "uses": [
        "G001",
        "S128"
      ],
      "before": "[-8/5, -28, -14/25]",
      "after": "finite left-sum term index=20",
      "relation": "implies",
      "rule_id": "left_partition_term",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。",
        "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof."
      },
      "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": "S130",
      "uses": [
        "G001",
        "S129"
      ],
      "before": "left sample21 offset := mul(21, 1/50)",
      "after": "left sample21 offset = 21/50",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。",
        "en": "Left samples use i=0,...,N−1,not the final right endpoint."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample21 offset，不以小數近似替代。",
          "en": "Recompute left sample21 offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "21",
          "1/50"
        ],
        "output": "21/50",
        "label": "left sample21 offset"
      }
    },
    {
      "id": "S131",
      "uses": [
        "G001",
        "S130"
      ],
      "before": "left sample21 x := sum(-2, 21/50)",
      "after": "left sample21 x = -79/50",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "起點加上第i個偏移得到採樣位置。",
        "en": "Add the indexed offset to the start."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample21 x，不以小數近似替代。",
          "en": "Recompute left sample21 x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "21/50"
        ],
        "output": "-79/50",
        "label": "left sample21 x"
      }
    },
    {
      "id": "S132",
      "uses": [
        "G001",
        "S131"
      ],
      "before": "left sample21 mx := mul(25, -79/50)",
      "after": "left sample21 mx = -79/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "代入仿射函數乘法部分。",
        "en": "Evaluate the product term of the affine integrand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample21 mx，不以小數近似替代。",
          "en": "Recompute left sample21 mx with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-79/50"
        ],
        "output": "-79/2",
        "label": "left sample21 mx"
      }
    },
    {
      "id": "S133",
      "uses": [
        "G001",
        "S132"
      ],
      "before": "left sample21 value := sum(-79/2, 12)",
      "after": "left sample21 value = -55/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加常數c得到本採樣值。",
        "en": "Add c to obtain this sample value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample21 value，不以小數近似替代。",
          "en": "Recompute left sample21 value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-79/2",
          "12"
        ],
        "output": "-55/2",
        "label": "left sample21 value"
      }
    },
    {
      "id": "S134",
      "uses": [
        "G001",
        "S133"
      ],
      "before": "left sample21 rectangle := mul(-55/2, 1/50)",
      "after": "left sample21 rectangle = -11/20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。",
        "en": "Multiply by the step to get the signed rectangle contribution,without absolute values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample21 rectangle，不以小數近似替代。",
          "en": "Recompute left sample21 rectangle with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-55/2",
          "1/50"
        ],
        "output": "-11/20",
        "label": "left sample21 rectangle"
      }
    },
    {
      "id": "S135",
      "uses": [
        "G001",
        "S134"
      ],
      "before": "[-79/50, -55/2, -11/20]",
      "after": "finite left-sum term index=21",
      "relation": "implies",
      "rule_id": "left_partition_term",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。",
        "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof."
      },
      "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": "S136",
      "uses": [
        "G001",
        "S135"
      ],
      "before": "left sample22 offset := mul(22, 1/50)",
      "after": "left sample22 offset = 11/25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。",
        "en": "Left samples use i=0,...,N−1,not the final right endpoint."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample22 offset，不以小數近似替代。",
          "en": "Recompute left sample22 offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "22",
          "1/50"
        ],
        "output": "11/25",
        "label": "left sample22 offset"
      }
    },
    {
      "id": "S137",
      "uses": [
        "G001",
        "S136"
      ],
      "before": "left sample22 x := sum(-2, 11/25)",
      "after": "left sample22 x = -39/25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "起點加上第i個偏移得到採樣位置。",
        "en": "Add the indexed offset to the start."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample22 x，不以小數近似替代。",
          "en": "Recompute left sample22 x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "11/25"
        ],
        "output": "-39/25",
        "label": "left sample22 x"
      }
    },
    {
      "id": "S138",
      "uses": [
        "G001",
        "S137"
      ],
      "before": "left sample22 mx := mul(25, -39/25)",
      "after": "left sample22 mx = -39",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "代入仿射函數乘法部分。",
        "en": "Evaluate the product term of the affine integrand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample22 mx，不以小數近似替代。",
          "en": "Recompute left sample22 mx with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-39/25"
        ],
        "output": "-39",
        "label": "left sample22 mx"
      }
    },
    {
      "id": "S139",
      "uses": [
        "G001",
        "S138"
      ],
      "before": "left sample22 value := sum(-39, 12)",
      "after": "left sample22 value = -27",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加常數c得到本採樣值。",
        "en": "Add c to obtain this sample value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample22 value，不以小數近似替代。",
          "en": "Recompute left sample22 value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-39",
          "12"
        ],
        "output": "-27",
        "label": "left sample22 value"
      }
    },
    {
      "id": "S140",
      "uses": [
        "G001",
        "S139"
      ],
      "before": "left sample22 rectangle := mul(-27, 1/50)",
      "after": "left sample22 rectangle = -27/50",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。",
        "en": "Multiply by the step to get the signed rectangle contribution,without absolute values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample22 rectangle，不以小數近似替代。",
          "en": "Recompute left sample22 rectangle with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-27",
          "1/50"
        ],
        "output": "-27/50",
        "label": "left sample22 rectangle"
      }
    },
    {
      "id": "S141",
      "uses": [
        "G001",
        "S140"
      ],
      "before": "[-39/25, -27, -27/50]",
      "after": "finite left-sum term index=22",
      "relation": "implies",
      "rule_id": "left_partition_term",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。",
        "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof."
      },
      "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": "S142",
      "uses": [
        "G001",
        "S141"
      ],
      "before": "left sample23 offset := mul(23, 1/50)",
      "after": "left sample23 offset = 23/50",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。",
        "en": "Left samples use i=0,...,N−1,not the final right endpoint."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample23 offset，不以小數近似替代。",
          "en": "Recompute left sample23 offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "23",
          "1/50"
        ],
        "output": "23/50",
        "label": "left sample23 offset"
      }
    },
    {
      "id": "S143",
      "uses": [
        "G001",
        "S142"
      ],
      "before": "left sample23 x := sum(-2, 23/50)",
      "after": "left sample23 x = -77/50",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "起點加上第i個偏移得到採樣位置。",
        "en": "Add the indexed offset to the start."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample23 x，不以小數近似替代。",
          "en": "Recompute left sample23 x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "23/50"
        ],
        "output": "-77/50",
        "label": "left sample23 x"
      }
    },
    {
      "id": "S144",
      "uses": [
        "G001",
        "S143"
      ],
      "before": "left sample23 mx := mul(25, -77/50)",
      "after": "left sample23 mx = -77/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "代入仿射函數乘法部分。",
        "en": "Evaluate the product term of the affine integrand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample23 mx，不以小數近似替代。",
          "en": "Recompute left sample23 mx with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-77/50"
        ],
        "output": "-77/2",
        "label": "left sample23 mx"
      }
    },
    {
      "id": "S145",
      "uses": [
        "G001",
        "S144"
      ],
      "before": "left sample23 value := sum(-77/2, 12)",
      "after": "left sample23 value = -53/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加常數c得到本採樣值。",
        "en": "Add c to obtain this sample value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample23 value，不以小數近似替代。",
          "en": "Recompute left sample23 value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-77/2",
          "12"
        ],
        "output": "-53/2",
        "label": "left sample23 value"
      }
    },
    {
      "id": "S146",
      "uses": [
        "G001",
        "S145"
      ],
      "before": "left sample23 rectangle := mul(-53/2, 1/50)",
      "after": "left sample23 rectangle = -53/100",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。",
        "en": "Multiply by the step to get the signed rectangle contribution,without absolute values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample23 rectangle，不以小數近似替代。",
          "en": "Recompute left sample23 rectangle with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-53/2",
          "1/50"
        ],
        "output": "-53/100",
        "label": "left sample23 rectangle"
      }
    },
    {
      "id": "S147",
      "uses": [
        "G001",
        "S146"
      ],
      "before": "[-77/50, -53/2, -53/100]",
      "after": "finite left-sum term index=23",
      "relation": "implies",
      "rule_id": "left_partition_term",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。",
        "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof."
      },
      "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": "S148",
      "uses": [
        "G001",
        "S147"
      ],
      "before": "left sample24 offset := mul(24, 1/50)",
      "after": "left sample24 offset = 12/25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。",
        "en": "Left samples use i=0,...,N−1,not the final right endpoint."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample24 offset，不以小數近似替代。",
          "en": "Recompute left sample24 offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "24",
          "1/50"
        ],
        "output": "12/25",
        "label": "left sample24 offset"
      }
    },
    {
      "id": "S149",
      "uses": [
        "G001",
        "S148"
      ],
      "before": "left sample24 x := sum(-2, 12/25)",
      "after": "left sample24 x = -38/25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "起點加上第i個偏移得到採樣位置。",
        "en": "Add the indexed offset to the start."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample24 x，不以小數近似替代。",
          "en": "Recompute left sample24 x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "12/25"
        ],
        "output": "-38/25",
        "label": "left sample24 x"
      }
    },
    {
      "id": "S150",
      "uses": [
        "G001",
        "S149"
      ],
      "before": "left sample24 mx := mul(25, -38/25)",
      "after": "left sample24 mx = -38",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "代入仿射函數乘法部分。",
        "en": "Evaluate the product term of the affine integrand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample24 mx，不以小數近似替代。",
          "en": "Recompute left sample24 mx with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-38/25"
        ],
        "output": "-38",
        "label": "left sample24 mx"
      }
    },
    {
      "id": "S151",
      "uses": [
        "G001",
        "S150"
      ],
      "before": "left sample24 value := sum(-38, 12)",
      "after": "left sample24 value = -26",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加常數c得到本採樣值。",
        "en": "Add c to obtain this sample value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample24 value，不以小數近似替代。",
          "en": "Recompute left sample24 value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-38",
          "12"
        ],
        "output": "-26",
        "label": "left sample24 value"
      }
    },
    {
      "id": "S152",
      "uses": [
        "G001",
        "S151"
      ],
      "before": "left sample24 rectangle := mul(-26, 1/50)",
      "after": "left sample24 rectangle = -13/25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。",
        "en": "Multiply by the step to get the signed rectangle contribution,without absolute values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample24 rectangle，不以小數近似替代。",
          "en": "Recompute left sample24 rectangle with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-26",
          "1/50"
        ],
        "output": "-13/25",
        "label": "left sample24 rectangle"
      }
    },
    {
      "id": "S153",
      "uses": [
        "G001",
        "S152"
      ],
      "before": "[-38/25, -26, -13/25]",
      "after": "finite left-sum term index=24",
      "relation": "implies",
      "rule_id": "left_partition_term",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。",
        "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof."
      },
      "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": "S154",
      "uses": [
        "G001",
        "S153"
      ],
      "before": "left sample25 offset := mul(25, 1/50)",
      "after": "left sample25 offset = 1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。",
        "en": "Left samples use i=0,...,N−1,not the final right endpoint."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample25 offset，不以小數近似替代。",
          "en": "Recompute left sample25 offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "1/50"
        ],
        "output": "1/2",
        "label": "left sample25 offset"
      }
    },
    {
      "id": "S155",
      "uses": [
        "G001",
        "S154"
      ],
      "before": "left sample25 x := sum(-2, 1/2)",
      "after": "left sample25 x = -3/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "起點加上第i個偏移得到採樣位置。",
        "en": "Add the indexed offset to the start."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample25 x，不以小數近似替代。",
          "en": "Recompute left sample25 x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "1/2"
        ],
        "output": "-3/2",
        "label": "left sample25 x"
      }
    },
    {
      "id": "S156",
      "uses": [
        "G001",
        "S155"
      ],
      "before": "left sample25 mx := mul(25, -3/2)",
      "after": "left sample25 mx = -75/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "代入仿射函數乘法部分。",
        "en": "Evaluate the product term of the affine integrand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample25 mx，不以小數近似替代。",
          "en": "Recompute left sample25 mx with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-3/2"
        ],
        "output": "-75/2",
        "label": "left sample25 mx"
      }
    },
    {
      "id": "S157",
      "uses": [
        "G001",
        "S156"
      ],
      "before": "left sample25 value := sum(-75/2, 12)",
      "after": "left sample25 value = -51/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加常數c得到本採樣值。",
        "en": "Add c to obtain this sample value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample25 value，不以小數近似替代。",
          "en": "Recompute left sample25 value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-75/2",
          "12"
        ],
        "output": "-51/2",
        "label": "left sample25 value"
      }
    },
    {
      "id": "S158",
      "uses": [
        "G001",
        "S157"
      ],
      "before": "left sample25 rectangle := mul(-51/2, 1/50)",
      "after": "left sample25 rectangle = -51/100",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。",
        "en": "Multiply by the step to get the signed rectangle contribution,without absolute values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample25 rectangle，不以小數近似替代。",
          "en": "Recompute left sample25 rectangle with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-51/2",
          "1/50"
        ],
        "output": "-51/100",
        "label": "left sample25 rectangle"
      }
    },
    {
      "id": "S159",
      "uses": [
        "G001",
        "S158"
      ],
      "before": "[-3/2, -51/2, -51/100]",
      "after": "finite left-sum term index=25",
      "relation": "implies",
      "rule_id": "left_partition_term",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。",
        "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof."
      },
      "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": "S160",
      "uses": [
        "G001",
        "S159"
      ],
      "before": "left sample26 offset := mul(26, 1/50)",
      "after": "left sample26 offset = 13/25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。",
        "en": "Left samples use i=0,...,N−1,not the final right endpoint."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample26 offset，不以小數近似替代。",
          "en": "Recompute left sample26 offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "26",
          "1/50"
        ],
        "output": "13/25",
        "label": "left sample26 offset"
      }
    },
    {
      "id": "S161",
      "uses": [
        "G001",
        "S160"
      ],
      "before": "left sample26 x := sum(-2, 13/25)",
      "after": "left sample26 x = -37/25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "起點加上第i個偏移得到採樣位置。",
        "en": "Add the indexed offset to the start."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample26 x，不以小數近似替代。",
          "en": "Recompute left sample26 x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "13/25"
        ],
        "output": "-37/25",
        "label": "left sample26 x"
      }
    },
    {
      "id": "S162",
      "uses": [
        "G001",
        "S161"
      ],
      "before": "left sample26 mx := mul(25, -37/25)",
      "after": "left sample26 mx = -37",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "代入仿射函數乘法部分。",
        "en": "Evaluate the product term of the affine integrand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample26 mx，不以小數近似替代。",
          "en": "Recompute left sample26 mx with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-37/25"
        ],
        "output": "-37",
        "label": "left sample26 mx"
      }
    },
    {
      "id": "S163",
      "uses": [
        "G001",
        "S162"
      ],
      "before": "left sample26 value := sum(-37, 12)",
      "after": "left sample26 value = -25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加常數c得到本採樣值。",
        "en": "Add c to obtain this sample value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample26 value，不以小數近似替代。",
          "en": "Recompute left sample26 value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-37",
          "12"
        ],
        "output": "-25",
        "label": "left sample26 value"
      }
    },
    {
      "id": "S164",
      "uses": [
        "G001",
        "S163"
      ],
      "before": "left sample26 rectangle := mul(-25, 1/50)",
      "after": "left sample26 rectangle = -1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。",
        "en": "Multiply by the step to get the signed rectangle contribution,without absolute values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample26 rectangle，不以小數近似替代。",
          "en": "Recompute left sample26 rectangle with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-25",
          "1/50"
        ],
        "output": "-1/2",
        "label": "left sample26 rectangle"
      }
    },
    {
      "id": "S165",
      "uses": [
        "G001",
        "S164"
      ],
      "before": "[-37/25, -25, -1/2]",
      "after": "finite left-sum term index=26",
      "relation": "implies",
      "rule_id": "left_partition_term",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。",
        "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof."
      },
      "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": "S166",
      "uses": [
        "G001",
        "S165"
      ],
      "before": "left sample27 offset := mul(27, 1/50)",
      "after": "left sample27 offset = 27/50",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。",
        "en": "Left samples use i=0,...,N−1,not the final right endpoint."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample27 offset，不以小數近似替代。",
          "en": "Recompute left sample27 offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "27",
          "1/50"
        ],
        "output": "27/50",
        "label": "left sample27 offset"
      }
    },
    {
      "id": "S167",
      "uses": [
        "G001",
        "S166"
      ],
      "before": "left sample27 x := sum(-2, 27/50)",
      "after": "left sample27 x = -73/50",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "起點加上第i個偏移得到採樣位置。",
        "en": "Add the indexed offset to the start."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample27 x，不以小數近似替代。",
          "en": "Recompute left sample27 x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "27/50"
        ],
        "output": "-73/50",
        "label": "left sample27 x"
      }
    },
    {
      "id": "S168",
      "uses": [
        "G001",
        "S167"
      ],
      "before": "left sample27 mx := mul(25, -73/50)",
      "after": "left sample27 mx = -73/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "代入仿射函數乘法部分。",
        "en": "Evaluate the product term of the affine integrand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample27 mx，不以小數近似替代。",
          "en": "Recompute left sample27 mx with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-73/50"
        ],
        "output": "-73/2",
        "label": "left sample27 mx"
      }
    },
    {
      "id": "S169",
      "uses": [
        "G001",
        "S168"
      ],
      "before": "left sample27 value := sum(-73/2, 12)",
      "after": "left sample27 value = -49/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加常數c得到本採樣值。",
        "en": "Add c to obtain this sample value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample27 value，不以小數近似替代。",
          "en": "Recompute left sample27 value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-73/2",
          "12"
        ],
        "output": "-49/2",
        "label": "left sample27 value"
      }
    },
    {
      "id": "S170",
      "uses": [
        "G001",
        "S169"
      ],
      "before": "left sample27 rectangle := mul(-49/2, 1/50)",
      "after": "left sample27 rectangle = -49/100",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。",
        "en": "Multiply by the step to get the signed rectangle contribution,without absolute values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample27 rectangle，不以小數近似替代。",
          "en": "Recompute left sample27 rectangle with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-49/2",
          "1/50"
        ],
        "output": "-49/100",
        "label": "left sample27 rectangle"
      }
    },
    {
      "id": "S171",
      "uses": [
        "G001",
        "S170"
      ],
      "before": "[-73/50, -49/2, -49/100]",
      "after": "finite left-sum term index=27",
      "relation": "implies",
      "rule_id": "left_partition_term",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。",
        "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof."
      },
      "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": "S172",
      "uses": [
        "G001",
        "S171"
      ],
      "before": "left sample28 offset := mul(28, 1/50)",
      "after": "left sample28 offset = 14/25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。",
        "en": "Left samples use i=0,...,N−1,not the final right endpoint."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample28 offset，不以小數近似替代。",
          "en": "Recompute left sample28 offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "28",
          "1/50"
        ],
        "output": "14/25",
        "label": "left sample28 offset"
      }
    },
    {
      "id": "S173",
      "uses": [
        "G001",
        "S172"
      ],
      "before": "left sample28 x := sum(-2, 14/25)",
      "after": "left sample28 x = -36/25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "起點加上第i個偏移得到採樣位置。",
        "en": "Add the indexed offset to the start."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample28 x，不以小數近似替代。",
          "en": "Recompute left sample28 x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "14/25"
        ],
        "output": "-36/25",
        "label": "left sample28 x"
      }
    },
    {
      "id": "S174",
      "uses": [
        "G001",
        "S173"
      ],
      "before": "left sample28 mx := mul(25, -36/25)",
      "after": "left sample28 mx = -36",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "代入仿射函數乘法部分。",
        "en": "Evaluate the product term of the affine integrand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample28 mx，不以小數近似替代。",
          "en": "Recompute left sample28 mx with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-36/25"
        ],
        "output": "-36",
        "label": "left sample28 mx"
      }
    },
    {
      "id": "S175",
      "uses": [
        "G001",
        "S174"
      ],
      "before": "left sample28 value := sum(-36, 12)",
      "after": "left sample28 value = -24",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加常數c得到本採樣值。",
        "en": "Add c to obtain this sample value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample28 value，不以小數近似替代。",
          "en": "Recompute left sample28 value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-36",
          "12"
        ],
        "output": "-24",
        "label": "left sample28 value"
      }
    },
    {
      "id": "S176",
      "uses": [
        "G001",
        "S175"
      ],
      "before": "left sample28 rectangle := mul(-24, 1/50)",
      "after": "left sample28 rectangle = -12/25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。",
        "en": "Multiply by the step to get the signed rectangle contribution,without absolute values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample28 rectangle，不以小數近似替代。",
          "en": "Recompute left sample28 rectangle with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-24",
          "1/50"
        ],
        "output": "-12/25",
        "label": "left sample28 rectangle"
      }
    },
    {
      "id": "S177",
      "uses": [
        "G001",
        "S176"
      ],
      "before": "[-36/25, -24, -12/25]",
      "after": "finite left-sum term index=28",
      "relation": "implies",
      "rule_id": "left_partition_term",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。",
        "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof."
      },
      "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": "S178",
      "uses": [
        "G001",
        "S177"
      ],
      "before": "left sample29 offset := mul(29, 1/50)",
      "after": "left sample29 offset = 29/50",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。",
        "en": "Left samples use i=0,...,N−1,not the final right endpoint."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample29 offset，不以小數近似替代。",
          "en": "Recompute left sample29 offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "29",
          "1/50"
        ],
        "output": "29/50",
        "label": "left sample29 offset"
      }
    },
    {
      "id": "S179",
      "uses": [
        "G001",
        "S178"
      ],
      "before": "left sample29 x := sum(-2, 29/50)",
      "after": "left sample29 x = -71/50",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "起點加上第i個偏移得到採樣位置。",
        "en": "Add the indexed offset to the start."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample29 x，不以小數近似替代。",
          "en": "Recompute left sample29 x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "29/50"
        ],
        "output": "-71/50",
        "label": "left sample29 x"
      }
    },
    {
      "id": "S180",
      "uses": [
        "G001",
        "S179"
      ],
      "before": "left sample29 mx := mul(25, -71/50)",
      "after": "left sample29 mx = -71/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "代入仿射函數乘法部分。",
        "en": "Evaluate the product term of the affine integrand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample29 mx，不以小數近似替代。",
          "en": "Recompute left sample29 mx with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-71/50"
        ],
        "output": "-71/2",
        "label": "left sample29 mx"
      }
    },
    {
      "id": "S181",
      "uses": [
        "G001",
        "S180"
      ],
      "before": "left sample29 value := sum(-71/2, 12)",
      "after": "left sample29 value = -47/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加常數c得到本採樣值。",
        "en": "Add c to obtain this sample value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample29 value，不以小數近似替代。",
          "en": "Recompute left sample29 value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-71/2",
          "12"
        ],
        "output": "-47/2",
        "label": "left sample29 value"
      }
    },
    {
      "id": "S182",
      "uses": [
        "G001",
        "S181"
      ],
      "before": "left sample29 rectangle := mul(-47/2, 1/50)",
      "after": "left sample29 rectangle = -47/100",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。",
        "en": "Multiply by the step to get the signed rectangle contribution,without absolute values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample29 rectangle，不以小數近似替代。",
          "en": "Recompute left sample29 rectangle with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-47/2",
          "1/50"
        ],
        "output": "-47/100",
        "label": "left sample29 rectangle"
      }
    },
    {
      "id": "S183",
      "uses": [
        "G001",
        "S182"
      ],
      "before": "[-71/50, -47/2, -47/100]",
      "after": "finite left-sum term index=29",
      "relation": "implies",
      "rule_id": "left_partition_term",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。",
        "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof."
      },
      "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": "S184",
      "uses": [
        "G001",
        "S183"
      ],
      "before": "left sample30 offset := mul(30, 1/50)",
      "after": "left sample30 offset = 3/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。",
        "en": "Left samples use i=0,...,N−1,not the final right endpoint."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample30 offset，不以小數近似替代。",
          "en": "Recompute left sample30 offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "30",
          "1/50"
        ],
        "output": "3/5",
        "label": "left sample30 offset"
      }
    },
    {
      "id": "S185",
      "uses": [
        "G001",
        "S184"
      ],
      "before": "left sample30 x := sum(-2, 3/5)",
      "after": "left sample30 x = -7/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "起點加上第i個偏移得到採樣位置。",
        "en": "Add the indexed offset to the start."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample30 x，不以小數近似替代。",
          "en": "Recompute left sample30 x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "3/5"
        ],
        "output": "-7/5",
        "label": "left sample30 x"
      }
    },
    {
      "id": "S186",
      "uses": [
        "G001",
        "S185"
      ],
      "before": "left sample30 mx := mul(25, -7/5)",
      "after": "left sample30 mx = -35",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "代入仿射函數乘法部分。",
        "en": "Evaluate the product term of the affine integrand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample30 mx，不以小數近似替代。",
          "en": "Recompute left sample30 mx with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-7/5"
        ],
        "output": "-35",
        "label": "left sample30 mx"
      }
    },
    {
      "id": "S187",
      "uses": [
        "G001",
        "S186"
      ],
      "before": "left sample30 value := sum(-35, 12)",
      "after": "left sample30 value = -23",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加常數c得到本採樣值。",
        "en": "Add c to obtain this sample value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample30 value，不以小數近似替代。",
          "en": "Recompute left sample30 value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-35",
          "12"
        ],
        "output": "-23",
        "label": "left sample30 value"
      }
    },
    {
      "id": "S188",
      "uses": [
        "G001",
        "S187"
      ],
      "before": "left sample30 rectangle := mul(-23, 1/50)",
      "after": "left sample30 rectangle = -23/50",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。",
        "en": "Multiply by the step to get the signed rectangle contribution,without absolute values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample30 rectangle，不以小數近似替代。",
          "en": "Recompute left sample30 rectangle with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-23",
          "1/50"
        ],
        "output": "-23/50",
        "label": "left sample30 rectangle"
      }
    },
    {
      "id": "S189",
      "uses": [
        "G001",
        "S188"
      ],
      "before": "[-7/5, -23, -23/50]",
      "after": "finite left-sum term index=30",
      "relation": "implies",
      "rule_id": "left_partition_term",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。",
        "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof."
      },
      "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": "S190",
      "uses": [
        "G001",
        "S189"
      ],
      "before": "left sample31 offset := mul(31, 1/50)",
      "after": "left sample31 offset = 31/50",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。",
        "en": "Left samples use i=0,...,N−1,not the final right endpoint."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample31 offset，不以小數近似替代。",
          "en": "Recompute left sample31 offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "31",
          "1/50"
        ],
        "output": "31/50",
        "label": "left sample31 offset"
      }
    },
    {
      "id": "S191",
      "uses": [
        "G001",
        "S190"
      ],
      "before": "left sample31 x := sum(-2, 31/50)",
      "after": "left sample31 x = -69/50",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "起點加上第i個偏移得到採樣位置。",
        "en": "Add the indexed offset to the start."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample31 x，不以小數近似替代。",
          "en": "Recompute left sample31 x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "31/50"
        ],
        "output": "-69/50",
        "label": "left sample31 x"
      }
    },
    {
      "id": "S192",
      "uses": [
        "G001",
        "S191"
      ],
      "before": "left sample31 mx := mul(25, -69/50)",
      "after": "left sample31 mx = -69/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "代入仿射函數乘法部分。",
        "en": "Evaluate the product term of the affine integrand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample31 mx，不以小數近似替代。",
          "en": "Recompute left sample31 mx with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-69/50"
        ],
        "output": "-69/2",
        "label": "left sample31 mx"
      }
    },
    {
      "id": "S193",
      "uses": [
        "G001",
        "S192"
      ],
      "before": "left sample31 value := sum(-69/2, 12)",
      "after": "left sample31 value = -45/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加常數c得到本採樣值。",
        "en": "Add c to obtain this sample value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample31 value，不以小數近似替代。",
          "en": "Recompute left sample31 value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-69/2",
          "12"
        ],
        "output": "-45/2",
        "label": "left sample31 value"
      }
    },
    {
      "id": "S194",
      "uses": [
        "G001",
        "S193"
      ],
      "before": "left sample31 rectangle := mul(-45/2, 1/50)",
      "after": "left sample31 rectangle = -9/20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。",
        "en": "Multiply by the step to get the signed rectangle contribution,without absolute values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample31 rectangle，不以小數近似替代。",
          "en": "Recompute left sample31 rectangle with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-45/2",
          "1/50"
        ],
        "output": "-9/20",
        "label": "left sample31 rectangle"
      }
    },
    {
      "id": "S195",
      "uses": [
        "G001",
        "S194"
      ],
      "before": "[-69/50, -45/2, -9/20]",
      "after": "finite left-sum term index=31",
      "relation": "implies",
      "rule_id": "left_partition_term",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。",
        "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof."
      },
      "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": "S196",
      "uses": [
        "G001",
        "S195"
      ],
      "before": "left sample32 offset := mul(32, 1/50)",
      "after": "left sample32 offset = 16/25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。",
        "en": "Left samples use i=0,...,N−1,not the final right endpoint."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample32 offset，不以小數近似替代。",
          "en": "Recompute left sample32 offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "32",
          "1/50"
        ],
        "output": "16/25",
        "label": "left sample32 offset"
      }
    },
    {
      "id": "S197",
      "uses": [
        "G001",
        "S196"
      ],
      "before": "left sample32 x := sum(-2, 16/25)",
      "after": "left sample32 x = -34/25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "起點加上第i個偏移得到採樣位置。",
        "en": "Add the indexed offset to the start."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample32 x，不以小數近似替代。",
          "en": "Recompute left sample32 x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "16/25"
        ],
        "output": "-34/25",
        "label": "left sample32 x"
      }
    },
    {
      "id": "S198",
      "uses": [
        "G001",
        "S197"
      ],
      "before": "left sample32 mx := mul(25, -34/25)",
      "after": "left sample32 mx = -34",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "代入仿射函數乘法部分。",
        "en": "Evaluate the product term of the affine integrand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample32 mx，不以小數近似替代。",
          "en": "Recompute left sample32 mx with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-34/25"
        ],
        "output": "-34",
        "label": "left sample32 mx"
      }
    },
    {
      "id": "S199",
      "uses": [
        "G001",
        "S198"
      ],
      "before": "left sample32 value := sum(-34, 12)",
      "after": "left sample32 value = -22",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加常數c得到本採樣值。",
        "en": "Add c to obtain this sample value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample32 value，不以小數近似替代。",
          "en": "Recompute left sample32 value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-34",
          "12"
        ],
        "output": "-22",
        "label": "left sample32 value"
      }
    },
    {
      "id": "S200",
      "uses": [
        "G001",
        "S199"
      ],
      "before": "left sample32 rectangle := mul(-22, 1/50)",
      "after": "left sample32 rectangle = -11/25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。",
        "en": "Multiply by the step to get the signed rectangle contribution,without absolute values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample32 rectangle，不以小數近似替代。",
          "en": "Recompute left sample32 rectangle with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-22",
          "1/50"
        ],
        "output": "-11/25",
        "label": "left sample32 rectangle"
      }
    },
    {
      "id": "S201",
      "uses": [
        "G001",
        "S200"
      ],
      "before": "[-34/25, -22, -11/25]",
      "after": "finite left-sum term index=32",
      "relation": "implies",
      "rule_id": "left_partition_term",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。",
        "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof."
      },
      "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": "S202",
      "uses": [
        "G001",
        "S201"
      ],
      "before": "left sample33 offset := mul(33, 1/50)",
      "after": "left sample33 offset = 33/50",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。",
        "en": "Left samples use i=0,...,N−1,not the final right endpoint."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample33 offset，不以小數近似替代。",
          "en": "Recompute left sample33 offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "33",
          "1/50"
        ],
        "output": "33/50",
        "label": "left sample33 offset"
      }
    },
    {
      "id": "S203",
      "uses": [
        "G001",
        "S202"
      ],
      "before": "left sample33 x := sum(-2, 33/50)",
      "after": "left sample33 x = -67/50",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "起點加上第i個偏移得到採樣位置。",
        "en": "Add the indexed offset to the start."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample33 x，不以小數近似替代。",
          "en": "Recompute left sample33 x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "33/50"
        ],
        "output": "-67/50",
        "label": "left sample33 x"
      }
    },
    {
      "id": "S204",
      "uses": [
        "G001",
        "S203"
      ],
      "before": "left sample33 mx := mul(25, -67/50)",
      "after": "left sample33 mx = -67/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "代入仿射函數乘法部分。",
        "en": "Evaluate the product term of the affine integrand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample33 mx，不以小數近似替代。",
          "en": "Recompute left sample33 mx with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-67/50"
        ],
        "output": "-67/2",
        "label": "left sample33 mx"
      }
    },
    {
      "id": "S205",
      "uses": [
        "G001",
        "S204"
      ],
      "before": "left sample33 value := sum(-67/2, 12)",
      "after": "left sample33 value = -43/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加常數c得到本採樣值。",
        "en": "Add c to obtain this sample value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample33 value，不以小數近似替代。",
          "en": "Recompute left sample33 value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-67/2",
          "12"
        ],
        "output": "-43/2",
        "label": "left sample33 value"
      }
    },
    {
      "id": "S206",
      "uses": [
        "G001",
        "S205"
      ],
      "before": "left sample33 rectangle := mul(-43/2, 1/50)",
      "after": "left sample33 rectangle = -43/100",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。",
        "en": "Multiply by the step to get the signed rectangle contribution,without absolute values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample33 rectangle，不以小數近似替代。",
          "en": "Recompute left sample33 rectangle with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-43/2",
          "1/50"
        ],
        "output": "-43/100",
        "label": "left sample33 rectangle"
      }
    },
    {
      "id": "S207",
      "uses": [
        "G001",
        "S206"
      ],
      "before": "[-67/50, -43/2, -43/100]",
      "after": "finite left-sum term index=33",
      "relation": "implies",
      "rule_id": "left_partition_term",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。",
        "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof."
      },
      "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": "S208",
      "uses": [
        "G001",
        "S207"
      ],
      "before": "left sample34 offset := mul(34, 1/50)",
      "after": "left sample34 offset = 17/25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。",
        "en": "Left samples use i=0,...,N−1,not the final right endpoint."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample34 offset，不以小數近似替代。",
          "en": "Recompute left sample34 offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "34",
          "1/50"
        ],
        "output": "17/25",
        "label": "left sample34 offset"
      }
    },
    {
      "id": "S209",
      "uses": [
        "G001",
        "S208"
      ],
      "before": "left sample34 x := sum(-2, 17/25)",
      "after": "left sample34 x = -33/25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "起點加上第i個偏移得到採樣位置。",
        "en": "Add the indexed offset to the start."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample34 x，不以小數近似替代。",
          "en": "Recompute left sample34 x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "17/25"
        ],
        "output": "-33/25",
        "label": "left sample34 x"
      }
    },
    {
      "id": "S210",
      "uses": [
        "G001",
        "S209"
      ],
      "before": "left sample34 mx := mul(25, -33/25)",
      "after": "left sample34 mx = -33",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "代入仿射函數乘法部分。",
        "en": "Evaluate the product term of the affine integrand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample34 mx，不以小數近似替代。",
          "en": "Recompute left sample34 mx with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-33/25"
        ],
        "output": "-33",
        "label": "left sample34 mx"
      }
    },
    {
      "id": "S211",
      "uses": [
        "G001",
        "S210"
      ],
      "before": "left sample34 value := sum(-33, 12)",
      "after": "left sample34 value = -21",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加常數c得到本採樣值。",
        "en": "Add c to obtain this sample value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample34 value，不以小數近似替代。",
          "en": "Recompute left sample34 value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-33",
          "12"
        ],
        "output": "-21",
        "label": "left sample34 value"
      }
    },
    {
      "id": "S212",
      "uses": [
        "G001",
        "S211"
      ],
      "before": "left sample34 rectangle := mul(-21, 1/50)",
      "after": "left sample34 rectangle = -21/50",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。",
        "en": "Multiply by the step to get the signed rectangle contribution,without absolute values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample34 rectangle，不以小數近似替代。",
          "en": "Recompute left sample34 rectangle with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-21",
          "1/50"
        ],
        "output": "-21/50",
        "label": "left sample34 rectangle"
      }
    },
    {
      "id": "S213",
      "uses": [
        "G001",
        "S212"
      ],
      "before": "[-33/25, -21, -21/50]",
      "after": "finite left-sum term index=34",
      "relation": "implies",
      "rule_id": "left_partition_term",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。",
        "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof."
      },
      "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": "S214",
      "uses": [
        "G001",
        "S213"
      ],
      "before": "left sample35 offset := mul(35, 1/50)",
      "after": "left sample35 offset = 7/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。",
        "en": "Left samples use i=0,...,N−1,not the final right endpoint."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample35 offset，不以小數近似替代。",
          "en": "Recompute left sample35 offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "35",
          "1/50"
        ],
        "output": "7/10",
        "label": "left sample35 offset"
      }
    },
    {
      "id": "S215",
      "uses": [
        "G001",
        "S214"
      ],
      "before": "left sample35 x := sum(-2, 7/10)",
      "after": "left sample35 x = -13/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "起點加上第i個偏移得到採樣位置。",
        "en": "Add the indexed offset to the start."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample35 x，不以小數近似替代。",
          "en": "Recompute left sample35 x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "7/10"
        ],
        "output": "-13/10",
        "label": "left sample35 x"
      }
    },
    {
      "id": "S216",
      "uses": [
        "G001",
        "S215"
      ],
      "before": "left sample35 mx := mul(25, -13/10)",
      "after": "left sample35 mx = -65/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "代入仿射函數乘法部分。",
        "en": "Evaluate the product term of the affine integrand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample35 mx，不以小數近似替代。",
          "en": "Recompute left sample35 mx with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-13/10"
        ],
        "output": "-65/2",
        "label": "left sample35 mx"
      }
    },
    {
      "id": "S217",
      "uses": [
        "G001",
        "S216"
      ],
      "before": "left sample35 value := sum(-65/2, 12)",
      "after": "left sample35 value = -41/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加常數c得到本採樣值。",
        "en": "Add c to obtain this sample value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample35 value，不以小數近似替代。",
          "en": "Recompute left sample35 value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-65/2",
          "12"
        ],
        "output": "-41/2",
        "label": "left sample35 value"
      }
    },
    {
      "id": "S218",
      "uses": [
        "G001",
        "S217"
      ],
      "before": "left sample35 rectangle := mul(-41/2, 1/50)",
      "after": "left sample35 rectangle = -41/100",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。",
        "en": "Multiply by the step to get the signed rectangle contribution,without absolute values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample35 rectangle，不以小數近似替代。",
          "en": "Recompute left sample35 rectangle with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-41/2",
          "1/50"
        ],
        "output": "-41/100",
        "label": "left sample35 rectangle"
      }
    },
    {
      "id": "S219",
      "uses": [
        "G001",
        "S218"
      ],
      "before": "[-13/10, -41/2, -41/100]",
      "after": "finite left-sum term index=35",
      "relation": "implies",
      "rule_id": "left_partition_term",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。",
        "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof."
      },
      "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": "S220",
      "uses": [
        "G001",
        "S219"
      ],
      "before": "left sample36 offset := mul(36, 1/50)",
      "after": "left sample36 offset = 18/25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。",
        "en": "Left samples use i=0,...,N−1,not the final right endpoint."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample36 offset，不以小數近似替代。",
          "en": "Recompute left sample36 offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "36",
          "1/50"
        ],
        "output": "18/25",
        "label": "left sample36 offset"
      }
    },
    {
      "id": "S221",
      "uses": [
        "G001",
        "S220"
      ],
      "before": "left sample36 x := sum(-2, 18/25)",
      "after": "left sample36 x = -32/25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "起點加上第i個偏移得到採樣位置。",
        "en": "Add the indexed offset to the start."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample36 x，不以小數近似替代。",
          "en": "Recompute left sample36 x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "18/25"
        ],
        "output": "-32/25",
        "label": "left sample36 x"
      }
    },
    {
      "id": "S222",
      "uses": [
        "G001",
        "S221"
      ],
      "before": "left sample36 mx := mul(25, -32/25)",
      "after": "left sample36 mx = -32",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "代入仿射函數乘法部分。",
        "en": "Evaluate the product term of the affine integrand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample36 mx，不以小數近似替代。",
          "en": "Recompute left sample36 mx with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-32/25"
        ],
        "output": "-32",
        "label": "left sample36 mx"
      }
    },
    {
      "id": "S223",
      "uses": [
        "G001",
        "S222"
      ],
      "before": "left sample36 value := sum(-32, 12)",
      "after": "left sample36 value = -20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加常數c得到本採樣值。",
        "en": "Add c to obtain this sample value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample36 value，不以小數近似替代。",
          "en": "Recompute left sample36 value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-32",
          "12"
        ],
        "output": "-20",
        "label": "left sample36 value"
      }
    },
    {
      "id": "S224",
      "uses": [
        "G001",
        "S223"
      ],
      "before": "left sample36 rectangle := mul(-20, 1/50)",
      "after": "left sample36 rectangle = -2/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。",
        "en": "Multiply by the step to get the signed rectangle contribution,without absolute values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample36 rectangle，不以小數近似替代。",
          "en": "Recompute left sample36 rectangle with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-20",
          "1/50"
        ],
        "output": "-2/5",
        "label": "left sample36 rectangle"
      }
    },
    {
      "id": "S225",
      "uses": [
        "G001",
        "S224"
      ],
      "before": "[-32/25, -20, -2/5]",
      "after": "finite left-sum term index=36",
      "relation": "implies",
      "rule_id": "left_partition_term",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。",
        "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof."
      },
      "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": "S226",
      "uses": [
        "G001",
        "S225"
      ],
      "before": "left sample37 offset := mul(37, 1/50)",
      "after": "left sample37 offset = 37/50",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。",
        "en": "Left samples use i=0,...,N−1,not the final right endpoint."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample37 offset，不以小數近似替代。",
          "en": "Recompute left sample37 offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "37",
          "1/50"
        ],
        "output": "37/50",
        "label": "left sample37 offset"
      }
    },
    {
      "id": "S227",
      "uses": [
        "G001",
        "S226"
      ],
      "before": "left sample37 x := sum(-2, 37/50)",
      "after": "left sample37 x = -63/50",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "起點加上第i個偏移得到採樣位置。",
        "en": "Add the indexed offset to the start."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample37 x，不以小數近似替代。",
          "en": "Recompute left sample37 x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "37/50"
        ],
        "output": "-63/50",
        "label": "left sample37 x"
      }
    },
    {
      "id": "S228",
      "uses": [
        "G001",
        "S227"
      ],
      "before": "left sample37 mx := mul(25, -63/50)",
      "after": "left sample37 mx = -63/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "代入仿射函數乘法部分。",
        "en": "Evaluate the product term of the affine integrand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample37 mx，不以小數近似替代。",
          "en": "Recompute left sample37 mx with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-63/50"
        ],
        "output": "-63/2",
        "label": "left sample37 mx"
      }
    },
    {
      "id": "S229",
      "uses": [
        "G001",
        "S228"
      ],
      "before": "left sample37 value := sum(-63/2, 12)",
      "after": "left sample37 value = -39/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加常數c得到本採樣值。",
        "en": "Add c to obtain this sample value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample37 value，不以小數近似替代。",
          "en": "Recompute left sample37 value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-63/2",
          "12"
        ],
        "output": "-39/2",
        "label": "left sample37 value"
      }
    },
    {
      "id": "S230",
      "uses": [
        "G001",
        "S229"
      ],
      "before": "left sample37 rectangle := mul(-39/2, 1/50)",
      "after": "left sample37 rectangle = -39/100",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。",
        "en": "Multiply by the step to get the signed rectangle contribution,without absolute values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample37 rectangle，不以小數近似替代。",
          "en": "Recompute left sample37 rectangle with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-39/2",
          "1/50"
        ],
        "output": "-39/100",
        "label": "left sample37 rectangle"
      }
    },
    {
      "id": "S231",
      "uses": [
        "G001",
        "S230"
      ],
      "before": "[-63/50, -39/2, -39/100]",
      "after": "finite left-sum term index=37",
      "relation": "implies",
      "rule_id": "left_partition_term",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。",
        "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof."
      },
      "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": "S232",
      "uses": [
        "G001",
        "S231"
      ],
      "before": "left sample38 offset := mul(38, 1/50)",
      "after": "left sample38 offset = 19/25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。",
        "en": "Left samples use i=0,...,N−1,not the final right endpoint."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample38 offset，不以小數近似替代。",
          "en": "Recompute left sample38 offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "38",
          "1/50"
        ],
        "output": "19/25",
        "label": "left sample38 offset"
      }
    },
    {
      "id": "S233",
      "uses": [
        "G001",
        "S232"
      ],
      "before": "left sample38 x := sum(-2, 19/25)",
      "after": "left sample38 x = -31/25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "起點加上第i個偏移得到採樣位置。",
        "en": "Add the indexed offset to the start."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample38 x，不以小數近似替代。",
          "en": "Recompute left sample38 x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "19/25"
        ],
        "output": "-31/25",
        "label": "left sample38 x"
      }
    },
    {
      "id": "S234",
      "uses": [
        "G001",
        "S233"
      ],
      "before": "left sample38 mx := mul(25, -31/25)",
      "after": "left sample38 mx = -31",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "代入仿射函數乘法部分。",
        "en": "Evaluate the product term of the affine integrand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample38 mx，不以小數近似替代。",
          "en": "Recompute left sample38 mx with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-31/25"
        ],
        "output": "-31",
        "label": "left sample38 mx"
      }
    },
    {
      "id": "S235",
      "uses": [
        "G001",
        "S234"
      ],
      "before": "left sample38 value := sum(-31, 12)",
      "after": "left sample38 value = -19",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加常數c得到本採樣值。",
        "en": "Add c to obtain this sample value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample38 value，不以小數近似替代。",
          "en": "Recompute left sample38 value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-31",
          "12"
        ],
        "output": "-19",
        "label": "left sample38 value"
      }
    },
    {
      "id": "S236",
      "uses": [
        "G001",
        "S235"
      ],
      "before": "left sample38 rectangle := mul(-19, 1/50)",
      "after": "left sample38 rectangle = -19/50",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。",
        "en": "Multiply by the step to get the signed rectangle contribution,without absolute values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample38 rectangle，不以小數近似替代。",
          "en": "Recompute left sample38 rectangle with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-19",
          "1/50"
        ],
        "output": "-19/50",
        "label": "left sample38 rectangle"
      }
    },
    {
      "id": "S237",
      "uses": [
        "G001",
        "S236"
      ],
      "before": "[-31/25, -19, -19/50]",
      "after": "finite left-sum term index=38",
      "relation": "implies",
      "rule_id": "left_partition_term",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。",
        "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof."
      },
      "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": "S238",
      "uses": [
        "G001",
        "S237"
      ],
      "before": "left sample39 offset := mul(39, 1/50)",
      "after": "left sample39 offset = 39/50",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。",
        "en": "Left samples use i=0,...,N−1,not the final right endpoint."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample39 offset，不以小數近似替代。",
          "en": "Recompute left sample39 offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "39",
          "1/50"
        ],
        "output": "39/50",
        "label": "left sample39 offset"
      }
    },
    {
      "id": "S239",
      "uses": [
        "G001",
        "S238"
      ],
      "before": "left sample39 x := sum(-2, 39/50)",
      "after": "left sample39 x = -61/50",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "起點加上第i個偏移得到採樣位置。",
        "en": "Add the indexed offset to the start."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample39 x，不以小數近似替代。",
          "en": "Recompute left sample39 x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "39/50"
        ],
        "output": "-61/50",
        "label": "left sample39 x"
      }
    },
    {
      "id": "S240",
      "uses": [
        "G001",
        "S239"
      ],
      "before": "left sample39 mx := mul(25, -61/50)",
      "after": "left sample39 mx = -61/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "代入仿射函數乘法部分。",
        "en": "Evaluate the product term of the affine integrand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample39 mx，不以小數近似替代。",
          "en": "Recompute left sample39 mx with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-61/50"
        ],
        "output": "-61/2",
        "label": "left sample39 mx"
      }
    },
    {
      "id": "S241",
      "uses": [
        "G001",
        "S240"
      ],
      "before": "left sample39 value := sum(-61/2, 12)",
      "after": "left sample39 value = -37/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加常數c得到本採樣值。",
        "en": "Add c to obtain this sample value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample39 value，不以小數近似替代。",
          "en": "Recompute left sample39 value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-61/2",
          "12"
        ],
        "output": "-37/2",
        "label": "left sample39 value"
      }
    },
    {
      "id": "S242",
      "uses": [
        "G001",
        "S241"
      ],
      "before": "left sample39 rectangle := mul(-37/2, 1/50)",
      "after": "left sample39 rectangle = -37/100",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。",
        "en": "Multiply by the step to get the signed rectangle contribution,without absolute values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample39 rectangle，不以小數近似替代。",
          "en": "Recompute left sample39 rectangle with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-37/2",
          "1/50"
        ],
        "output": "-37/100",
        "label": "left sample39 rectangle"
      }
    },
    {
      "id": "S243",
      "uses": [
        "G001",
        "S242"
      ],
      "before": "[-61/50, -37/2, -37/100]",
      "after": "finite left-sum term index=39",
      "relation": "implies",
      "rule_id": "left_partition_term",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。",
        "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof."
      },
      "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": "S244",
      "uses": [
        "G001",
        "S243"
      ],
      "before": "left sample40 offset := mul(40, 1/50)",
      "after": "left sample40 offset = 4/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。",
        "en": "Left samples use i=0,...,N−1,not the final right endpoint."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample40 offset，不以小數近似替代。",
          "en": "Recompute left sample40 offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "40",
          "1/50"
        ],
        "output": "4/5",
        "label": "left sample40 offset"
      }
    },
    {
      "id": "S245",
      "uses": [
        "G001",
        "S244"
      ],
      "before": "left sample40 x := sum(-2, 4/5)",
      "after": "left sample40 x = -6/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "起點加上第i個偏移得到採樣位置。",
        "en": "Add the indexed offset to the start."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample40 x，不以小數近似替代。",
          "en": "Recompute left sample40 x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "4/5"
        ],
        "output": "-6/5",
        "label": "left sample40 x"
      }
    },
    {
      "id": "S246",
      "uses": [
        "G001",
        "S245"
      ],
      "before": "left sample40 mx := mul(25, -6/5)",
      "after": "left sample40 mx = -30",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "代入仿射函數乘法部分。",
        "en": "Evaluate the product term of the affine integrand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample40 mx，不以小數近似替代。",
          "en": "Recompute left sample40 mx with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-6/5"
        ],
        "output": "-30",
        "label": "left sample40 mx"
      }
    },
    {
      "id": "S247",
      "uses": [
        "G001",
        "S246"
      ],
      "before": "left sample40 value := sum(-30, 12)",
      "after": "left sample40 value = -18",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加常數c得到本採樣值。",
        "en": "Add c to obtain this sample value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample40 value，不以小數近似替代。",
          "en": "Recompute left sample40 value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-30",
          "12"
        ],
        "output": "-18",
        "label": "left sample40 value"
      }
    },
    {
      "id": "S248",
      "uses": [
        "G001",
        "S247"
      ],
      "before": "left sample40 rectangle := mul(-18, 1/50)",
      "after": "left sample40 rectangle = -9/25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。",
        "en": "Multiply by the step to get the signed rectangle contribution,without absolute values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample40 rectangle，不以小數近似替代。",
          "en": "Recompute left sample40 rectangle with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-18",
          "1/50"
        ],
        "output": "-9/25",
        "label": "left sample40 rectangle"
      }
    },
    {
      "id": "S249",
      "uses": [
        "G001",
        "S248"
      ],
      "before": "[-6/5, -18, -9/25]",
      "after": "finite left-sum term index=40",
      "relation": "implies",
      "rule_id": "left_partition_term",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。",
        "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof."
      },
      "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": "S250",
      "uses": [
        "G001",
        "S249"
      ],
      "before": "left sample41 offset := mul(41, 1/50)",
      "after": "left sample41 offset = 41/50",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。",
        "en": "Left samples use i=0,...,N−1,not the final right endpoint."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample41 offset，不以小數近似替代。",
          "en": "Recompute left sample41 offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "41",
          "1/50"
        ],
        "output": "41/50",
        "label": "left sample41 offset"
      }
    },
    {
      "id": "S251",
      "uses": [
        "G001",
        "S250"
      ],
      "before": "left sample41 x := sum(-2, 41/50)",
      "after": "left sample41 x = -59/50",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "起點加上第i個偏移得到採樣位置。",
        "en": "Add the indexed offset to the start."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample41 x，不以小數近似替代。",
          "en": "Recompute left sample41 x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "41/50"
        ],
        "output": "-59/50",
        "label": "left sample41 x"
      }
    },
    {
      "id": "S252",
      "uses": [
        "G001",
        "S251"
      ],
      "before": "left sample41 mx := mul(25, -59/50)",
      "after": "left sample41 mx = -59/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "代入仿射函數乘法部分。",
        "en": "Evaluate the product term of the affine integrand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample41 mx，不以小數近似替代。",
          "en": "Recompute left sample41 mx with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-59/50"
        ],
        "output": "-59/2",
        "label": "left sample41 mx"
      }
    },
    {
      "id": "S253",
      "uses": [
        "G001",
        "S252"
      ],
      "before": "left sample41 value := sum(-59/2, 12)",
      "after": "left sample41 value = -35/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加常數c得到本採樣值。",
        "en": "Add c to obtain this sample value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample41 value，不以小數近似替代。",
          "en": "Recompute left sample41 value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-59/2",
          "12"
        ],
        "output": "-35/2",
        "label": "left sample41 value"
      }
    },
    {
      "id": "S254",
      "uses": [
        "G001",
        "S253"
      ],
      "before": "left sample41 rectangle := mul(-35/2, 1/50)",
      "after": "left sample41 rectangle = -7/20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。",
        "en": "Multiply by the step to get the signed rectangle contribution,without absolute values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample41 rectangle，不以小數近似替代。",
          "en": "Recompute left sample41 rectangle with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-35/2",
          "1/50"
        ],
        "output": "-7/20",
        "label": "left sample41 rectangle"
      }
    },
    {
      "id": "S255",
      "uses": [
        "G001",
        "S254"
      ],
      "before": "[-59/50, -35/2, -7/20]",
      "after": "finite left-sum term index=41",
      "relation": "implies",
      "rule_id": "left_partition_term",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。",
        "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof."
      },
      "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": "S256",
      "uses": [
        "G001",
        "S255"
      ],
      "before": "left sample42 offset := mul(42, 1/50)",
      "after": "left sample42 offset = 21/25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。",
        "en": "Left samples use i=0,...,N−1,not the final right endpoint."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample42 offset，不以小數近似替代。",
          "en": "Recompute left sample42 offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "42",
          "1/50"
        ],
        "output": "21/25",
        "label": "left sample42 offset"
      }
    },
    {
      "id": "S257",
      "uses": [
        "G001",
        "S256"
      ],
      "before": "left sample42 x := sum(-2, 21/25)",
      "after": "left sample42 x = -29/25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "起點加上第i個偏移得到採樣位置。",
        "en": "Add the indexed offset to the start."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample42 x，不以小數近似替代。",
          "en": "Recompute left sample42 x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "21/25"
        ],
        "output": "-29/25",
        "label": "left sample42 x"
      }
    },
    {
      "id": "S258",
      "uses": [
        "G001",
        "S257"
      ],
      "before": "left sample42 mx := mul(25, -29/25)",
      "after": "left sample42 mx = -29",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "代入仿射函數乘法部分。",
        "en": "Evaluate the product term of the affine integrand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample42 mx，不以小數近似替代。",
          "en": "Recompute left sample42 mx with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-29/25"
        ],
        "output": "-29",
        "label": "left sample42 mx"
      }
    },
    {
      "id": "S259",
      "uses": [
        "G001",
        "S258"
      ],
      "before": "left sample42 value := sum(-29, 12)",
      "after": "left sample42 value = -17",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加常數c得到本採樣值。",
        "en": "Add c to obtain this sample value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample42 value，不以小數近似替代。",
          "en": "Recompute left sample42 value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-29",
          "12"
        ],
        "output": "-17",
        "label": "left sample42 value"
      }
    },
    {
      "id": "S260",
      "uses": [
        "G001",
        "S259"
      ],
      "before": "left sample42 rectangle := mul(-17, 1/50)",
      "after": "left sample42 rectangle = -17/50",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。",
        "en": "Multiply by the step to get the signed rectangle contribution,without absolute values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample42 rectangle，不以小數近似替代。",
          "en": "Recompute left sample42 rectangle with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-17",
          "1/50"
        ],
        "output": "-17/50",
        "label": "left sample42 rectangle"
      }
    },
    {
      "id": "S261",
      "uses": [
        "G001",
        "S260"
      ],
      "before": "[-29/25, -17, -17/50]",
      "after": "finite left-sum term index=42",
      "relation": "implies",
      "rule_id": "left_partition_term",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。",
        "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof."
      },
      "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": "S262",
      "uses": [
        "G001",
        "S261"
      ],
      "before": "left sample43 offset := mul(43, 1/50)",
      "after": "left sample43 offset = 43/50",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。",
        "en": "Left samples use i=0,...,N−1,not the final right endpoint."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample43 offset，不以小數近似替代。",
          "en": "Recompute left sample43 offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "43",
          "1/50"
        ],
        "output": "43/50",
        "label": "left sample43 offset"
      }
    },
    {
      "id": "S263",
      "uses": [
        "G001",
        "S262"
      ],
      "before": "left sample43 x := sum(-2, 43/50)",
      "after": "left sample43 x = -57/50",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "起點加上第i個偏移得到採樣位置。",
        "en": "Add the indexed offset to the start."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample43 x，不以小數近似替代。",
          "en": "Recompute left sample43 x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "43/50"
        ],
        "output": "-57/50",
        "label": "left sample43 x"
      }
    },
    {
      "id": "S264",
      "uses": [
        "G001",
        "S263"
      ],
      "before": "left sample43 mx := mul(25, -57/50)",
      "after": "left sample43 mx = -57/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "代入仿射函數乘法部分。",
        "en": "Evaluate the product term of the affine integrand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample43 mx，不以小數近似替代。",
          "en": "Recompute left sample43 mx with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-57/50"
        ],
        "output": "-57/2",
        "label": "left sample43 mx"
      }
    },
    {
      "id": "S265",
      "uses": [
        "G001",
        "S264"
      ],
      "before": "left sample43 value := sum(-57/2, 12)",
      "after": "left sample43 value = -33/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加常數c得到本採樣值。",
        "en": "Add c to obtain this sample value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample43 value，不以小數近似替代。",
          "en": "Recompute left sample43 value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-57/2",
          "12"
        ],
        "output": "-33/2",
        "label": "left sample43 value"
      }
    },
    {
      "id": "S266",
      "uses": [
        "G001",
        "S265"
      ],
      "before": "left sample43 rectangle := mul(-33/2, 1/50)",
      "after": "left sample43 rectangle = -33/100",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。",
        "en": "Multiply by the step to get the signed rectangle contribution,without absolute values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample43 rectangle，不以小數近似替代。",
          "en": "Recompute left sample43 rectangle with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-33/2",
          "1/50"
        ],
        "output": "-33/100",
        "label": "left sample43 rectangle"
      }
    },
    {
      "id": "S267",
      "uses": [
        "G001",
        "S266"
      ],
      "before": "[-57/50, -33/2, -33/100]",
      "after": "finite left-sum term index=43",
      "relation": "implies",
      "rule_id": "left_partition_term",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。",
        "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof."
      },
      "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": "S268",
      "uses": [
        "G001",
        "S267"
      ],
      "before": "left sample44 offset := mul(44, 1/50)",
      "after": "left sample44 offset = 22/25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。",
        "en": "Left samples use i=0,...,N−1,not the final right endpoint."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample44 offset，不以小數近似替代。",
          "en": "Recompute left sample44 offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "44",
          "1/50"
        ],
        "output": "22/25",
        "label": "left sample44 offset"
      }
    },
    {
      "id": "S269",
      "uses": [
        "G001",
        "S268"
      ],
      "before": "left sample44 x := sum(-2, 22/25)",
      "after": "left sample44 x = -28/25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "起點加上第i個偏移得到採樣位置。",
        "en": "Add the indexed offset to the start."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample44 x，不以小數近似替代。",
          "en": "Recompute left sample44 x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "22/25"
        ],
        "output": "-28/25",
        "label": "left sample44 x"
      }
    },
    {
      "id": "S270",
      "uses": [
        "G001",
        "S269"
      ],
      "before": "left sample44 mx := mul(25, -28/25)",
      "after": "left sample44 mx = -28",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "代入仿射函數乘法部分。",
        "en": "Evaluate the product term of the affine integrand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample44 mx，不以小數近似替代。",
          "en": "Recompute left sample44 mx with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-28/25"
        ],
        "output": "-28",
        "label": "left sample44 mx"
      }
    },
    {
      "id": "S271",
      "uses": [
        "G001",
        "S270"
      ],
      "before": "left sample44 value := sum(-28, 12)",
      "after": "left sample44 value = -16",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加常數c得到本採樣值。",
        "en": "Add c to obtain this sample value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample44 value，不以小數近似替代。",
          "en": "Recompute left sample44 value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-28",
          "12"
        ],
        "output": "-16",
        "label": "left sample44 value"
      }
    },
    {
      "id": "S272",
      "uses": [
        "G001",
        "S271"
      ],
      "before": "left sample44 rectangle := mul(-16, 1/50)",
      "after": "left sample44 rectangle = -8/25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。",
        "en": "Multiply by the step to get the signed rectangle contribution,without absolute values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample44 rectangle，不以小數近似替代。",
          "en": "Recompute left sample44 rectangle with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-16",
          "1/50"
        ],
        "output": "-8/25",
        "label": "left sample44 rectangle"
      }
    },
    {
      "id": "S273",
      "uses": [
        "G001",
        "S272"
      ],
      "before": "[-28/25, -16, -8/25]",
      "after": "finite left-sum term index=44",
      "relation": "implies",
      "rule_id": "left_partition_term",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。",
        "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof."
      },
      "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": "S274",
      "uses": [
        "G001",
        "S273"
      ],
      "before": "left sample45 offset := mul(45, 1/50)",
      "after": "left sample45 offset = 9/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。",
        "en": "Left samples use i=0,...,N−1,not the final right endpoint."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample45 offset，不以小數近似替代。",
          "en": "Recompute left sample45 offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "45",
          "1/50"
        ],
        "output": "9/10",
        "label": "left sample45 offset"
      }
    },
    {
      "id": "S275",
      "uses": [
        "G001",
        "S274"
      ],
      "before": "left sample45 x := sum(-2, 9/10)",
      "after": "left sample45 x = -11/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "起點加上第i個偏移得到採樣位置。",
        "en": "Add the indexed offset to the start."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample45 x，不以小數近似替代。",
          "en": "Recompute left sample45 x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "9/10"
        ],
        "output": "-11/10",
        "label": "left sample45 x"
      }
    },
    {
      "id": "S276",
      "uses": [
        "G001",
        "S275"
      ],
      "before": "left sample45 mx := mul(25, -11/10)",
      "after": "left sample45 mx = -55/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "代入仿射函數乘法部分。",
        "en": "Evaluate the product term of the affine integrand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample45 mx，不以小數近似替代。",
          "en": "Recompute left sample45 mx with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-11/10"
        ],
        "output": "-55/2",
        "label": "left sample45 mx"
      }
    },
    {
      "id": "S277",
      "uses": [
        "G001",
        "S276"
      ],
      "before": "left sample45 value := sum(-55/2, 12)",
      "after": "left sample45 value = -31/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加常數c得到本採樣值。",
        "en": "Add c to obtain this sample value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample45 value，不以小數近似替代。",
          "en": "Recompute left sample45 value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-55/2",
          "12"
        ],
        "output": "-31/2",
        "label": "left sample45 value"
      }
    },
    {
      "id": "S278",
      "uses": [
        "G001",
        "S277"
      ],
      "before": "left sample45 rectangle := mul(-31/2, 1/50)",
      "after": "left sample45 rectangle = -31/100",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。",
        "en": "Multiply by the step to get the signed rectangle contribution,without absolute values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample45 rectangle，不以小數近似替代。",
          "en": "Recompute left sample45 rectangle with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-31/2",
          "1/50"
        ],
        "output": "-31/100",
        "label": "left sample45 rectangle"
      }
    },
    {
      "id": "S279",
      "uses": [
        "G001",
        "S278"
      ],
      "before": "[-11/10, -31/2, -31/100]",
      "after": "finite left-sum term index=45",
      "relation": "implies",
      "rule_id": "left_partition_term",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。",
        "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof."
      },
      "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": "S280",
      "uses": [
        "G001",
        "S279"
      ],
      "before": "left sample46 offset := mul(46, 1/50)",
      "after": "left sample46 offset = 23/25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。",
        "en": "Left samples use i=0,...,N−1,not the final right endpoint."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample46 offset，不以小數近似替代。",
          "en": "Recompute left sample46 offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "46",
          "1/50"
        ],
        "output": "23/25",
        "label": "left sample46 offset"
      }
    },
    {
      "id": "S281",
      "uses": [
        "G001",
        "S280"
      ],
      "before": "left sample46 x := sum(-2, 23/25)",
      "after": "left sample46 x = -27/25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "起點加上第i個偏移得到採樣位置。",
        "en": "Add the indexed offset to the start."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample46 x，不以小數近似替代。",
          "en": "Recompute left sample46 x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "23/25"
        ],
        "output": "-27/25",
        "label": "left sample46 x"
      }
    },
    {
      "id": "S282",
      "uses": [
        "G001",
        "S281"
      ],
      "before": "left sample46 mx := mul(25, -27/25)",
      "after": "left sample46 mx = -27",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "代入仿射函數乘法部分。",
        "en": "Evaluate the product term of the affine integrand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample46 mx，不以小數近似替代。",
          "en": "Recompute left sample46 mx with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-27/25"
        ],
        "output": "-27",
        "label": "left sample46 mx"
      }
    },
    {
      "id": "S283",
      "uses": [
        "G001",
        "S282"
      ],
      "before": "left sample46 value := sum(-27, 12)",
      "after": "left sample46 value = -15",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加常數c得到本採樣值。",
        "en": "Add c to obtain this sample value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample46 value，不以小數近似替代。",
          "en": "Recompute left sample46 value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-27",
          "12"
        ],
        "output": "-15",
        "label": "left sample46 value"
      }
    },
    {
      "id": "S284",
      "uses": [
        "G001",
        "S283"
      ],
      "before": "left sample46 rectangle := mul(-15, 1/50)",
      "after": "left sample46 rectangle = -3/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。",
        "en": "Multiply by the step to get the signed rectangle contribution,without absolute values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample46 rectangle，不以小數近似替代。",
          "en": "Recompute left sample46 rectangle with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-15",
          "1/50"
        ],
        "output": "-3/10",
        "label": "left sample46 rectangle"
      }
    },
    {
      "id": "S285",
      "uses": [
        "G001",
        "S284"
      ],
      "before": "[-27/25, -15, -3/10]",
      "after": "finite left-sum term index=46",
      "relation": "implies",
      "rule_id": "left_partition_term",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。",
        "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof."
      },
      "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": "S286",
      "uses": [
        "G001",
        "S285"
      ],
      "before": "left sample47 offset := mul(47, 1/50)",
      "after": "left sample47 offset = 47/50",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。",
        "en": "Left samples use i=0,...,N−1,not the final right endpoint."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample47 offset，不以小數近似替代。",
          "en": "Recompute left sample47 offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "47",
          "1/50"
        ],
        "output": "47/50",
        "label": "left sample47 offset"
      }
    },
    {
      "id": "S287",
      "uses": [
        "G001",
        "S286"
      ],
      "before": "left sample47 x := sum(-2, 47/50)",
      "after": "left sample47 x = -53/50",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "起點加上第i個偏移得到採樣位置。",
        "en": "Add the indexed offset to the start."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample47 x，不以小數近似替代。",
          "en": "Recompute left sample47 x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "47/50"
        ],
        "output": "-53/50",
        "label": "left sample47 x"
      }
    },
    {
      "id": "S288",
      "uses": [
        "G001",
        "S287"
      ],
      "before": "left sample47 mx := mul(25, -53/50)",
      "after": "left sample47 mx = -53/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "代入仿射函數乘法部分。",
        "en": "Evaluate the product term of the affine integrand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample47 mx，不以小數近似替代。",
          "en": "Recompute left sample47 mx with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-53/50"
        ],
        "output": "-53/2",
        "label": "left sample47 mx"
      }
    },
    {
      "id": "S289",
      "uses": [
        "G001",
        "S288"
      ],
      "before": "left sample47 value := sum(-53/2, 12)",
      "after": "left sample47 value = -29/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加常數c得到本採樣值。",
        "en": "Add c to obtain this sample value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample47 value，不以小數近似替代。",
          "en": "Recompute left sample47 value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-53/2",
          "12"
        ],
        "output": "-29/2",
        "label": "left sample47 value"
      }
    },
    {
      "id": "S290",
      "uses": [
        "G001",
        "S289"
      ],
      "before": "left sample47 rectangle := mul(-29/2, 1/50)",
      "after": "left sample47 rectangle = -29/100",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。",
        "en": "Multiply by the step to get the signed rectangle contribution,without absolute values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample47 rectangle，不以小數近似替代。",
          "en": "Recompute left sample47 rectangle with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-29/2",
          "1/50"
        ],
        "output": "-29/100",
        "label": "left sample47 rectangle"
      }
    },
    {
      "id": "S291",
      "uses": [
        "G001",
        "S290"
      ],
      "before": "[-53/50, -29/2, -29/100]",
      "after": "finite left-sum term index=47",
      "relation": "implies",
      "rule_id": "left_partition_term",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。",
        "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof."
      },
      "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": "S292",
      "uses": [
        "G001",
        "S291"
      ],
      "before": "left sample48 offset := mul(48, 1/50)",
      "after": "left sample48 offset = 24/25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。",
        "en": "Left samples use i=0,...,N−1,not the final right endpoint."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample48 offset，不以小數近似替代。",
          "en": "Recompute left sample48 offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "48",
          "1/50"
        ],
        "output": "24/25",
        "label": "left sample48 offset"
      }
    },
    {
      "id": "S293",
      "uses": [
        "G001",
        "S292"
      ],
      "before": "left sample48 x := sum(-2, 24/25)",
      "after": "left sample48 x = -26/25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "起點加上第i個偏移得到採樣位置。",
        "en": "Add the indexed offset to the start."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample48 x，不以小數近似替代。",
          "en": "Recompute left sample48 x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "24/25"
        ],
        "output": "-26/25",
        "label": "left sample48 x"
      }
    },
    {
      "id": "S294",
      "uses": [
        "G001",
        "S293"
      ],
      "before": "left sample48 mx := mul(25, -26/25)",
      "after": "left sample48 mx = -26",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "代入仿射函數乘法部分。",
        "en": "Evaluate the product term of the affine integrand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample48 mx，不以小數近似替代。",
          "en": "Recompute left sample48 mx with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-26/25"
        ],
        "output": "-26",
        "label": "left sample48 mx"
      }
    },
    {
      "id": "S295",
      "uses": [
        "G001",
        "S294"
      ],
      "before": "left sample48 value := sum(-26, 12)",
      "after": "left sample48 value = -14",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加常數c得到本採樣值。",
        "en": "Add c to obtain this sample value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample48 value，不以小數近似替代。",
          "en": "Recompute left sample48 value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-26",
          "12"
        ],
        "output": "-14",
        "label": "left sample48 value"
      }
    },
    {
      "id": "S296",
      "uses": [
        "G001",
        "S295"
      ],
      "before": "left sample48 rectangle := mul(-14, 1/50)",
      "after": "left sample48 rectangle = -7/25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。",
        "en": "Multiply by the step to get the signed rectangle contribution,without absolute values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample48 rectangle，不以小數近似替代。",
          "en": "Recompute left sample48 rectangle with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-14",
          "1/50"
        ],
        "output": "-7/25",
        "label": "left sample48 rectangle"
      }
    },
    {
      "id": "S297",
      "uses": [
        "G001",
        "S296"
      ],
      "before": "[-26/25, -14, -7/25]",
      "after": "finite left-sum term index=48",
      "relation": "implies",
      "rule_id": "left_partition_term",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。",
        "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof."
      },
      "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": "S298",
      "uses": [
        "G001",
        "S297"
      ],
      "before": "left sample49 offset := mul(49, 1/50)",
      "after": "left sample49 offset = 49/50",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。",
        "en": "Left samples use i=0,...,N−1,not the final right endpoint."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample49 offset，不以小數近似替代。",
          "en": "Recompute left sample49 offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "49",
          "1/50"
        ],
        "output": "49/50",
        "label": "left sample49 offset"
      }
    },
    {
      "id": "S299",
      "uses": [
        "G001",
        "S298"
      ],
      "before": "left sample49 x := sum(-2, 49/50)",
      "after": "left sample49 x = -51/50",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "起點加上第i個偏移得到採樣位置。",
        "en": "Add the indexed offset to the start."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample49 x，不以小數近似替代。",
          "en": "Recompute left sample49 x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "49/50"
        ],
        "output": "-51/50",
        "label": "left sample49 x"
      }
    },
    {
      "id": "S300",
      "uses": [
        "G001",
        "S299"
      ],
      "before": "left sample49 mx := mul(25, -51/50)",
      "after": "left sample49 mx = -51/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "代入仿射函數乘法部分。",
        "en": "Evaluate the product term of the affine integrand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample49 mx，不以小數近似替代。",
          "en": "Recompute left sample49 mx with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-51/50"
        ],
        "output": "-51/2",
        "label": "left sample49 mx"
      }
    },
    {
      "id": "S301",
      "uses": [
        "G001",
        "S300"
      ],
      "before": "left sample49 value := sum(-51/2, 12)",
      "after": "left sample49 value = -27/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加常數c得到本採樣值。",
        "en": "Add c to obtain this sample value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample49 value，不以小數近似替代。",
          "en": "Recompute left sample49 value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-51/2",
          "12"
        ],
        "output": "-27/2",
        "label": "left sample49 value"
      }
    },
    {
      "id": "S302",
      "uses": [
        "G001",
        "S301"
      ],
      "before": "left sample49 rectangle := mul(-27/2, 1/50)",
      "after": "left sample49 rectangle = -27/100",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。",
        "en": "Multiply by the step to get the signed rectangle contribution,without absolute values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left sample49 rectangle，不以小數近似替代。",
          "en": "Recompute left sample49 rectangle with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-27/2",
          "1/50"
        ],
        "output": "-27/100",
        "label": "left sample49 rectangle"
      }
    },
    {
      "id": "S303",
      "uses": [
        "G001",
        "S302"
      ],
      "before": "[-51/50, -27/2, -27/100]",
      "after": "finite left-sum term index=49",
      "relation": "implies",
      "rule_id": "left_partition_term",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。",
        "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof."
      },
      "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": "S304",
      "uses": [
        "G001",
        "S303"
      ],
      "before": "finite left sum := sum(-19/25, -3/4, -37/50, -73/100, -18/25, -71/100, -7/10, -69/100, -17/25, -67/100, -33/50, -13/20, -16/25, -63/100, -31/50, -61/100, -3/5, -59/100, -29/50, -57/100, -14/25, -11/20, -27/50, -53/100, -13/25, -51/100, -1/2, -49/100, -12/25, -47/100, -23/50, -9/20, -11/25, -43/100, -21/50, -41/100, -2/5, -39/100, -19/50, -37/100, -9/25, -7/20, -17/50, -33/100, -8/25, -31/100, -3/10, -29/100, -7/25, -27/100)",
      "after": "finite left sum = -103/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把N項完整相加。",
        "en": "Add all N contributions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite left sum，不以小數近似替代。",
          "en": "Recompute finite left sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-19/25",
          "-3/4",
          "-37/50",
          "-73/100",
          "-18/25",
          "-71/100",
          "-7/10",
          "-69/100",
          "-17/25",
          "-67/100",
          "-33/50",
          "-13/20",
          "-16/25",
          "-63/100",
          "-31/50",
          "-61/100",
          "-3/5",
          "-59/100",
          "-29/50",
          "-57/100",
          "-14/25",
          "-11/20",
          "-27/50",
          "-53/100",
          "-13/25",
          "-51/100",
          "-1/2",
          "-49/100",
          "-12/25",
          "-47/100",
          "-23/50",
          "-9/20",
          "-11/25",
          "-43/100",
          "-21/50",
          "-41/100",
          "-2/5",
          "-39/100",
          "-19/50",
          "-37/100",
          "-9/25",
          "-7/20",
          "-17/50",
          "-33/100",
          "-8/25",
          "-31/100",
          "-3/10",
          "-29/100",
          "-7/25",
          "-27/100"
        ],
        "output": "-103/4",
        "label": "finite left sum"
      }
    },
    {
      "id": "S305",
      "uses": [
        "G001",
        "S304"
      ],
      "before": "affine integral primitive coefficient1 := div(12, 1)",
      "after": "affine integral primitive coefficient1 = 12",
      "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 affine integral primitive coefficient1，不以小數近似替代。",
          "en": "Recompute affine integral primitive coefficient1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "12",
          "1"
        ],
        "output": "12",
        "label": "affine integral primitive coefficient1"
      }
    },
    {
      "id": "S306",
      "uses": [
        "G001",
        "S305"
      ],
      "before": "affine integral primitive coefficient2 := div(25, 2)",
      "after": "affine integral primitive coefficient2 = 25/2",
      "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 affine integral primitive coefficient2，不以小數近似替代。",
          "en": "Recompute affine integral primitive coefficient2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "25",
          "2"
        ],
        "output": "25/2",
        "label": "affine integral primitive coefficient2"
      }
    },
    {
      "id": "S307",
      "uses": [
        "G001",
        "S306"
      ],
      "before": "[0, 12, 25/2]",
      "after": "H′=the supplied polynomial;integrate from -2 to -1",
      "relation": "implies",
      "rule_id": "polynomial_primitive_identity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐項微分回原係數確定原函數；積分常數在端點差相消。",
        "en": "Termwise differentiation recovers the original coefficients;the integration constant cancels in the endpoint difference."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S308",
      "uses": [
        "G001",
        "S307"
      ],
      "before": "affine integral H(b) ascending coefficients=[0, 12, 25/2]",
      "after": "evaluate at x=-1",
      "relation": "implies",
      "rule_id": "polynomial_evaluation_order",
      "conditions": [],
      "explanation": {
        "zh-Hant": "升冪係數用Horner由最高非零項向常數項求值；略去零最高項不改函數，也不改原始係數清單。",
        "en": "Evaluate ascending coefficients by Horner from the highest nonzero term down;ignoring redundant top zero terms changes neither the function nor the retained original list."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S309",
      "uses": [
        "G001",
        "S308"
      ],
      "before": "affine integral H(b) Horner multiply before degree1 := mul(25/2, -1)",
      "after": "affine integral H(b) Horner multiply before degree1 = -25/2",
      "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 affine integral H(b) Horner multiply before degree1，不以小數近似替代。",
          "en": "Recompute affine integral H(b) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25/2",
          "-1"
        ],
        "output": "-25/2",
        "label": "affine integral H(b) Horner multiply before degree1"
      }
    },
    {
      "id": "S310",
      "uses": [
        "G001",
        "S309"
      ],
      "before": "affine integral H(b) Horner add degree1 := sum(-25/2, 12)",
      "after": "affine integral H(b) Horner add degree1 = -1/2",
      "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 affine integral H(b) Horner add degree1，不以小數近似替代。",
          "en": "Recompute affine integral H(b) Horner add degree1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-25/2",
          "12"
        ],
        "output": "-1/2",
        "label": "affine integral H(b) Horner add degree1"
      }
    },
    {
      "id": "S311",
      "uses": [
        "G001",
        "S310"
      ],
      "before": "affine integral H(b) Horner multiply before degree0 := mul(-1/2, -1)",
      "after": "affine integral H(b) Horner multiply before degree0 = 1/2",
      "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 affine integral H(b) Horner multiply before degree0，不以小數近似替代。",
          "en": "Recompute affine integral H(b) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1/2",
          "-1"
        ],
        "output": "1/2",
        "label": "affine integral H(b) Horner multiply before degree0"
      }
    },
    {
      "id": "S312",
      "uses": [
        "G001",
        "S311"
      ],
      "before": "affine integral H(b) Horner add degree0 := sum(1/2, 0)",
      "after": "affine integral H(b) Horner add degree0 = 1/2",
      "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 affine integral H(b) Horner add degree0，不以小數近似替代。",
          "en": "Recompute affine integral H(b) Horner add degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1/2",
          "0"
        ],
        "output": "1/2",
        "label": "affine integral H(b) Horner add degree0"
      }
    },
    {
      "id": "S313",
      "uses": [
        "G001",
        "S312"
      ],
      "before": "affine integral H(a) ascending coefficients=[0, 12, 25/2]",
      "after": "evaluate at x=-2",
      "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": "S314",
      "uses": [
        "G001",
        "S313"
      ],
      "before": "affine integral H(a) Horner multiply before degree1 := mul(25/2, -2)",
      "after": "affine integral H(a) Horner multiply before degree1 = -25",
      "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 affine integral H(a) Horner multiply before degree1，不以小數近似替代。",
          "en": "Recompute affine integral H(a) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25/2",
          "-2"
        ],
        "output": "-25",
        "label": "affine integral H(a) Horner multiply before degree1"
      }
    },
    {
      "id": "S315",
      "uses": [
        "G001",
        "S314"
      ],
      "before": "affine integral H(a) Horner add degree1 := sum(-25, 12)",
      "after": "affine integral H(a) Horner add degree1 = -13",
      "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 affine integral H(a) Horner add degree1，不以小數近似替代。",
          "en": "Recompute affine integral H(a) Horner add degree1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-25",
          "12"
        ],
        "output": "-13",
        "label": "affine integral H(a) Horner add degree1"
      }
    },
    {
      "id": "S316",
      "uses": [
        "G001",
        "S315"
      ],
      "before": "affine integral H(a) Horner multiply before degree0 := mul(-13, -2)",
      "after": "affine integral H(a) Horner multiply before degree0 = 26",
      "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 affine integral H(a) Horner multiply before degree0，不以小數近似替代。",
          "en": "Recompute affine integral H(a) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-13",
          "-2"
        ],
        "output": "26",
        "label": "affine integral H(a) Horner multiply before degree0"
      }
    },
    {
      "id": "S317",
      "uses": [
        "G001",
        "S316"
      ],
      "before": "affine integral H(a) Horner add degree0 := sum(26, 0)",
      "after": "affine integral H(a) Horner add degree0 = 26",
      "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 affine integral H(a) Horner add degree0，不以小數近似替代。",
          "en": "Recompute affine integral H(a) Horner add degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "26",
          "0"
        ],
        "output": "26",
        "label": "affine integral H(a) Horner add degree0"
      }
    },
    {
      "id": "S318",
      "uses": [
        "G001",
        "S317"
      ],
      "before": "affine integral oriented moment := sub(1/2, 26)",
      "after": "affine integral oriented moment = -51/2",
      "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 affine integral oriented moment，不以小數近似替代。",
          "en": "Recompute affine integral oriented moment with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "1/2",
          "26"
        ],
        "output": "-51/2",
        "label": "affine integral oriented moment"
      }
    },
    {
      "id": "S319",
      "uses": [
        "G001",
        "S318"
      ],
      "before": "integral minus left sum := sub(-51/2, -103/4)",
      "after": "integral minus left sum = 1/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "方向固定為I−S_N；負斜率可產生負差，不是絕對誤差。",
        "en": "The requested direction is I−S_N;a negative slope may give a negative difference,not absolute error."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 integral minus left sum，不以小數近似替代。",
          "en": "Recompute integral minus left sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-51/2",
          "-103/4"
        ],
        "output": "1/4",
        "label": "integral minus left sum"
      }
    },
    {
      "id": "S320",
      "uses": [
        "G001",
        "S319"
      ],
      "before": "I=-51/2;S_N=-103/4;N=50",
      "after": "signed finite-sum difference=1/4",
      "relation": "implies",
      "rule_id": "finite_sum_vs_integral",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這是精確有限N的差；不把一次數值符合當作一般收斂速度或任意函數的積分證明。",
        "en": "This is the exact difference for finite N,not evidence of a general convergence rate or integrability of arbitrary functions."
      },
      "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": "1/4",
    "integral": "-51/2",
    "left_sum": "-103/4",
    "step": "1/50",
    "partition_values": [
      [
        "-2",
        "-38",
        "-19/25"
      ],
      [
        "-99/50",
        "-75/2",
        "-3/4"
      ],
      [
        "-49/25",
        "-37",
        "-37/50"
      ],
      [
        "-97/50",
        "-73/2",
        "-73/100"
      ],
      [
        "-48/25",
        "-36",
        "-18/25"
      ],
      [
        "-19/10",
        "-71/2",
        "-71/100"
      ],
      [
        "-47/25",
        "-35",
        "-7/10"
      ],
      [
        "-93/50",
        "-69/2",
        "-69/100"
      ],
      [
        "-46/25",
        "-34",
        "-17/25"
      ],
      [
        "-91/50",
        "-67/2",
        "-67/100"
      ],
      [
        "-9/5",
        "-33",
        "-33/50"
      ],
      [
        "-89/50",
        "-65/2",
        "-13/20"
      ],
      [
        "-44/25",
        "-32",
        "-16/25"
      ],
      [
        "-87/50",
        "-63/2",
        "-63/100"
      ],
      [
        "-43/25",
        "-31",
        "-31/50"
      ],
      [
        "-17/10",
        "-61/2",
        "-61/100"
      ],
      [
        "-42/25",
        "-30",
        "-3/5"
      ],
      [
        "-83/50",
        "-59/2",
        "-59/100"
      ],
      [
        "-41/25",
        "-29",
        "-29/50"
      ],
      [
        "-81/50",
        "-57/2",
        "-57/100"
      ],
      [
        "-8/5",
        "-28",
        "-14/25"
      ],
      [
        "-79/50",
        "-55/2",
        "-11/20"
      ],
      [
        "-39/25",
        "-27",
        "-27/50"
      ],
      [
        "-77/50",
        "-53/2",
        "-53/100"
      ],
      [
        "-38/25",
        "-26",
        "-13/25"
      ],
      [
        "-3/2",
        "-51/2",
        "-51/100"
      ],
      [
        "-37/25",
        "-25",
        "-1/2"
      ],
      [
        "-73/50",
        "-49/2",
        "-49/100"
      ],
      [
        "-36/25",
        "-24",
        "-12/25"
      ],
      [
        "-71/50",
        "-47/2",
        "-47/100"
      ],
      [
        "-7/5",
        "-23",
        "-23/50"
      ],
      [
        "-69/50",
        "-45/2",
        "-9/20"
      ],
      [
        "-34/25",
        "-22",
        "-11/25"
      ],
      [
        "-67/50",
        "-43/2",
        "-43/100"
      ],
      [
        "-33/25",
        "-21",
        "-21/50"
      ],
      [
        "-13/10",
        "-41/2",
        "-41/100"
      ],
      [
        "-32/25",
        "-20",
        "-2/5"
      ],
      [
        "-63/50",
        "-39/2",
        "-39/100"
      ],
      [
        "-31/25",
        "-19",
        "-19/50"
      ],
      [
        "-61/50",
        "-37/2",
        "-37/100"
      ],
      [
        "-6/5",
        "-18",
        "-9/25"
      ],
      [
        "-59/50",
        "-35/2",
        "-7/20"
      ],
      [
        "-29/25",
        "-17",
        "-17/50"
      ],
      [
        "-57/50",
        "-33/2",
        "-33/100"
      ],
      [
        "-28/25",
        "-16",
        "-8/25"
      ],
      [
        "-11/10",
        "-31/2",
        "-31/100"
      ],
      [
        "-27/25",
        "-15",
        "-3/10"
      ],
      [
        "-53/50",
        "-29/2",
        "-29/100"
      ],
      [
        "-26/25",
        "-14",
        "-7/25"
      ],
      [
        "-51/50",
        "-27/2",
        "-27/100"
      ]
    ]
  },
  "conclusion": {
    "canonical_value": "1/4",
    "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": 75,
      "original_line_sha256": "9a9af4b8e61e8b7f244b930f8f7f50a164526a9c1231a7778dfe85fa293a9eca",
      "question_compact_sha256": "f9d5dd056a03791cfe650b3b96d8afcde27f070e51be8c93db8ef551efbc068e",
      "english_row_compact_sha256": "101ccf5d0bce3101cb8c298aac4812cfac303616f7e32edaf2bd621a05ba19b1",
      "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": {
        "sha256": "f1fc56fc2ac48870dcaa53e6269bb2e365c1529c5540683ba6b6cd103ddcc5f7",
        "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": "bb39d070e83df78cd4dead1790a22f00ecb90bfc93f34a0b3cc7bd68dc00922e",
    "original_candidate_line_sha256": "bb39d070e83df78cd4dead1790a22f00ecb90bfc93f34a0b3cc7bd68dc00922e"
  },
  "historical_source_state": {
    "review_state": {
      "derivation": "candidate_authored_and_locally_checked",
      "source_question_state": "candidate",
      "independent_AI_or_human_review": "NOT_PERFORMED",
      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
    },
    "scope_note": {
      "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。",
      "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged."
    },
    "label": "Frozen producer metadata; current per-question review is separate below."
  },
  "revision": {
    "id": "review-20261026-004-p7",
    "review_date": "2026-10-26",
    "owner_authorized_reader_publication": true,
    "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.",
    "review_receipts": {
      "semantic_review_sha256": "568eefd71709ac3dc16a0d5b3b4ad3e83376a8ef300c9e5493eee0721500b29d",
      "question_first_witness_sha256": "d964b91bedb9d4918bbc055f567762a09f5a58af6c876268e719d5100332d855"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 320,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "4ae243022e50692b960e31309a2a52a3885163ba1da42fd2a45985ec6a91f44f",
      "arithmetic_steps": 265
    },
    "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."
  }
}