{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261007-advance-20261008-r07-definite-integral-0100",
  "pilot_number": "R0064-32625",
  "number": 32625,
  "family_id": "qf-integral-affine-left-sum",
  "title": {
    "zh-Hant": "第 32625 題：仿射式的左端有限和",
    "en": "Question 32625: Finite Left-Endpoint Sum of an Affine Function"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261007-advance-20261008-r07-definite-integral-0100",
    "batch_id": "batch-20261007-advance-20261008-r07-definite-integral-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-definite-integral"
    ],
    "title_zh": "仿射式的左端有限和",
    "stem_zh": "全實數論域，以下函數／點均以明示實數模型解讀。f(x)=25x+(12)，區間 [-2,3] 分為 N=50 等份。求左端有限和 ΔΣ f(a+iΔ)，i=0至N−1、Δ=(b−a)/N。",
    "answer": {
      "kind": "expression",
      "value": "465/4",
      "explanation_zh": "左端有限和=465/4。產生端用等差級數閉式，答案是此有限和，不一般當成精確積分；常數式或零區間可恰與多項式矩積分相同。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 3,
      "skills": [
        "仿射式的左端有限和"
      ]
    },
    "variant": {
      "family_id": "qf-integral-affine-left-sum",
      "seed": 202610087100,
      "parameters": {
        "m": 25,
        "c": 12,
        "a": -2,
        "b": 3,
        "N": 50,
        "operation": "affine_left_sum",
        "domain": "reals",
        "semantic_template": "qf-integral-affine-left-sum/{\"inputs\":[\"25\",\"-38\",\"5\",\"50\"],\"operation\":\"affine_left_sum\"}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_integer_rational_arithmetic",
        "producer_finite_enumeration",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-07",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261007-advance-20261008-r07-definite-integral-0100",
    "batch_id": "batch-20261007-advance-20261008-r07-definite-integral-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "1664eca412eac2416d1eb84b1244342c91944c03ac1be08b95e49b7a3305df39",
      "family_id": "qf-integral-affine-left-sum",
      "seed": 202610087100,
      "semantic_template": "qf-integral-affine-left-sum/{\"inputs\":[\"25\",\"-38\",\"5\",\"50\"],\"operation\":\"affine_left_sum\"}"
    },
    "fields": {
      "title": "Finite Left-Endpoint Sum of an Affine Function",
      "stem": "On all R, f(x)=(25)x+(12). For a=-2≤b=3 and N=50 uniform cells, compute the finite left-endpoint sum with Δx=(b−a)/N and x_i=a+iΔx for i=0,…,N−1. Give a reduced fraction with positive denominator, retaining /1 for integers.",
      "answer": {
        "kind": "expression",
        "value": "465/4",
        "explanation": "Exact termwise rational accumulation gives the finite sum 465/4. A finite left sum generally differs from the integral, even though constant or zero-length special cases can agree; this is not a general Riemann-integrability proof."
      },
      "skills": [
        "Finite Left-Endpoint Sum of an Affine Function"
      ],
      "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."
    }
  },
  "givens": [
    {
      "id": "G001",
      "text": {
        "zh-Hant": "全實數論域，以下函數／點均以明示實數模型解讀。f(x)=25x+(12)，區間 [-2,3] 分為 N=50 等份。求左端有限和 ΔΣ f(a+iΔ)，i=0至N−1、Δ=(b−a)/N。",
        "en": "On all R, f(x)=(25)x+(12). For a=-2≤b=3 and N=50 uniform cells, compute the finite left-endpoint sum with Δx=(b−a)/N and x_i=a+iΔx for i=0,…,N−1. Give a reduced fraction with positive denominator, retaining /1 for integers."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [qf-integral-affine-left-sum, affine_left_sum, reals, []]",
      "after": "derived = [[finite_left_sum_fraction_RETAIN1]]",
      "relation": "implies",
      "rule_id": "R64_SOURCE_exact_legacy_interface_BASE_metadata_AND_output_roles",
      "conditions": [],
      "explanation": {
        "zh-Hant": "legacy核來源精確欄位及operation/domain；BASE整數保metadata但不當oracle。先分NUMBER、字串、fractionpair或vector等式，不強套同一格式；原中英／AST／status不變。",
        "en": "Legacy banks enforce their exact source keys/operation/domain.BASE integers retain metadata without treating it as an oracle.Distinguish NUMBER,string,fraction pair and vector equality formats;retain original bilingual data/AST/status."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "Left sum ORIGINAL upper minus lower endpoint := sub(3, -2)",
      "after": "Left sum ORIGINAL upper minus lower endpoint = 5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL upper minus lower endpoint，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL upper minus lower endpoint with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "3",
          "-2"
        ],
        "output": "5",
        "label": "Left sum ORIGINAL upper minus lower endpoint"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "Left sum ORIGINAL width over positive cell count := div(5, 50)",
      "after": "Left sum ORIGINAL width over positive cell count = 1/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL width over positive cell count，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL width over positive cell count with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "5",
          "50"
        ],
        "output": "1/10",
        "label": "Left sum ORIGINAL width over positive cell count"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "Left sum ORIGINAL sample0 index times width := mul(0, 1/10)",
      "after": "Left sum ORIGINAL sample0 index times width = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample0 index times width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample0 index times width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "1/10"
        ],
        "output": "0",
        "label": "Left sum ORIGINAL sample0 index times width"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "Left sum ORIGINAL sample0 lower plus offset := sum(-2, 0)",
      "after": "Left sum ORIGINAL sample0 lower plus offset = -2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample0 lower plus offset，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample0 lower plus offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "0"
        ],
        "output": "-2",
        "label": "Left sum ORIGINAL sample0 lower plus offset"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "Left sum ORIGINAL sample0 slope times x := mul(25, -2)",
      "after": "Left sum ORIGINAL sample0 slope times x = -50",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample0 slope times x，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample0 slope times x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-2"
        ],
        "output": "-50",
        "label": "Left sum ORIGINAL sample0 slope times x"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "Left sum ORIGINAL sample0 affine value plus c := sum(-50, 12)",
      "after": "Left sum ORIGINAL sample0 affine value plus c = -38",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample0 affine value plus c，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample0 affine value plus c with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-50",
          "12"
        ],
        "output": "-38",
        "label": "Left sum ORIGINAL sample0 affine value plus c"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "Left sum ORIGINAL sample0 value times cell width := mul(-38, 1/10)",
      "after": "Left sum ORIGINAL sample0 value times cell width = -19/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample0 value times cell width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample0 value times cell width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-38",
          "1/10"
        ],
        "output": "-19/5",
        "label": "Left sum ORIGINAL sample0 value times cell width"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "Left sum ORIGINAL accumulated term0 := sum(0, -19/5)",
      "after": "Left sum ORIGINAL accumulated term0 = -19/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term0，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL accumulated term0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "-19/5"
        ],
        "output": "-19/5",
        "label": "Left sum ORIGINAL accumulated term0"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "source roles = [0, 50, 25, 12, -2, 3, 1/10]",
      "after": "derived = [-2/1, -38/1, -19/5, -19/5]",
      "relation": "implies",
      "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。",
        "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "Left sum ORIGINAL sample1 index times width := mul(1, 1/10)",
      "after": "Left sum ORIGINAL sample1 index times width = 1/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample1 index times width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample1 index times width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1/10"
        ],
        "output": "1/10",
        "label": "Left sum ORIGINAL sample1 index times width"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "Left sum ORIGINAL sample1 lower plus offset := sum(-2, 1/10)",
      "after": "Left sum ORIGINAL sample1 lower plus offset = -19/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample1 lower plus offset，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample1 lower plus offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "1/10"
        ],
        "output": "-19/10",
        "label": "Left sum ORIGINAL sample1 lower plus offset"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "Left sum ORIGINAL sample1 slope times x := mul(25, -19/10)",
      "after": "Left sum ORIGINAL sample1 slope times x = -95/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample1 slope times x，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample1 slope times x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-19/10"
        ],
        "output": "-95/2",
        "label": "Left sum ORIGINAL sample1 slope times x"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "Left sum ORIGINAL sample1 affine value plus c := sum(-95/2, 12)",
      "after": "Left sum ORIGINAL sample1 affine value plus c = -71/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample1 affine value plus c，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample1 affine value plus c with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-95/2",
          "12"
        ],
        "output": "-71/2",
        "label": "Left sum ORIGINAL sample1 affine value plus c"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "Left sum ORIGINAL sample1 value times cell width := mul(-71/2, 1/10)",
      "after": "Left sum ORIGINAL sample1 value times cell width = -71/20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample1 value times cell width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample1 value times cell width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-71/2",
          "1/10"
        ],
        "output": "-71/20",
        "label": "Left sum ORIGINAL sample1 value times cell width"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "Left sum ORIGINAL accumulated term1 := sum(-19/5, -71/20)",
      "after": "Left sum ORIGINAL accumulated term1 = -147/20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term1，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL accumulated term1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-19/5",
          "-71/20"
        ],
        "output": "-147/20",
        "label": "Left sum ORIGINAL accumulated term1"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "source roles = [1, 50, 25, 12, -2, 3, 1/10]",
      "after": "derived = [-19/10, -71/2, -71/20, -147/20]",
      "relation": "implies",
      "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。",
        "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "Left sum ORIGINAL sample2 index times width := mul(2, 1/10)",
      "after": "Left sum ORIGINAL sample2 index times width = 1/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample2 index times width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample2 index times width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "1/10"
        ],
        "output": "1/5",
        "label": "Left sum ORIGINAL sample2 index times width"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "Left sum ORIGINAL sample2 lower plus offset := sum(-2, 1/5)",
      "after": "Left sum ORIGINAL sample2 lower plus offset = -9/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample2 lower plus offset，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample2 lower plus offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "1/5"
        ],
        "output": "-9/5",
        "label": "Left sum ORIGINAL sample2 lower plus offset"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "Left sum ORIGINAL sample2 slope times x := mul(25, -9/5)",
      "after": "Left sum ORIGINAL sample2 slope times x = -45",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample2 slope times x，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample2 slope times x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-9/5"
        ],
        "output": "-45",
        "label": "Left sum ORIGINAL sample2 slope times x"
      }
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "Left sum ORIGINAL sample2 affine value plus c := sum(-45, 12)",
      "after": "Left sum ORIGINAL sample2 affine value plus c = -33",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample2 affine value plus c，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample2 affine value plus c with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-45",
          "12"
        ],
        "output": "-33",
        "label": "Left sum ORIGINAL sample2 affine value plus c"
      }
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "Left sum ORIGINAL sample2 value times cell width := mul(-33, 1/10)",
      "after": "Left sum ORIGINAL sample2 value times cell width = -33/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample2 value times cell width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample2 value times cell width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-33",
          "1/10"
        ],
        "output": "-33/10",
        "label": "Left sum ORIGINAL sample2 value times cell width"
      }
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "Left sum ORIGINAL accumulated term2 := sum(-147/20, -33/10)",
      "after": "Left sum ORIGINAL accumulated term2 = -213/20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term2，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL accumulated term2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-147/20",
          "-33/10"
        ],
        "output": "-213/20",
        "label": "Left sum ORIGINAL accumulated term2"
      }
    },
    {
      "id": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "source roles = [2, 50, 25, 12, -2, 3, 1/10]",
      "after": "derived = [-9/5, -33/1, -33/10, -213/20]",
      "relation": "implies",
      "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。",
        "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "Left sum ORIGINAL sample3 index times width := mul(3, 1/10)",
      "after": "Left sum ORIGINAL sample3 index times width = 3/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample3 index times width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample3 index times width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3",
          "1/10"
        ],
        "output": "3/10",
        "label": "Left sum ORIGINAL sample3 index times width"
      }
    },
    {
      "id": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "Left sum ORIGINAL sample3 lower plus offset := sum(-2, 3/10)",
      "after": "Left sum ORIGINAL sample3 lower plus offset = -17/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample3 lower plus offset，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample3 lower plus offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "3/10"
        ],
        "output": "-17/10",
        "label": "Left sum ORIGINAL sample3 lower plus offset"
      }
    },
    {
      "id": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "Left sum ORIGINAL sample3 slope times x := mul(25, -17/10)",
      "after": "Left sum ORIGINAL sample3 slope times x = -85/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample3 slope times x，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample3 slope times x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-17/10"
        ],
        "output": "-85/2",
        "label": "Left sum ORIGINAL sample3 slope times x"
      }
    },
    {
      "id": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "Left sum ORIGINAL sample3 affine value plus c := sum(-85/2, 12)",
      "after": "Left sum ORIGINAL sample3 affine value plus c = -61/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample3 affine value plus c，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample3 affine value plus c with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-85/2",
          "12"
        ],
        "output": "-61/2",
        "label": "Left sum ORIGINAL sample3 affine value plus c"
      }
    },
    {
      "id": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "Left sum ORIGINAL sample3 value times cell width := mul(-61/2, 1/10)",
      "after": "Left sum ORIGINAL sample3 value times cell width = -61/20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample3 value times cell width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample3 value times cell width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-61/2",
          "1/10"
        ],
        "output": "-61/20",
        "label": "Left sum ORIGINAL sample3 value times cell width"
      }
    },
    {
      "id": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "Left sum ORIGINAL accumulated term3 := sum(-213/20, -61/20)",
      "after": "Left sum ORIGINAL accumulated term3 = -137/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term3，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL accumulated term3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-213/20",
          "-61/20"
        ],
        "output": "-137/10",
        "label": "Left sum ORIGINAL accumulated term3"
      }
    },
    {
      "id": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "source roles = [3, 50, 25, 12, -2, 3, 1/10]",
      "after": "derived = [-17/10, -61/2, -61/20, -137/10]",
      "relation": "implies",
      "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。",
        "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S032",
      "uses": [
        "G001",
        "S031"
      ],
      "before": "Left sum ORIGINAL sample4 index times width := mul(4, 1/10)",
      "after": "Left sum ORIGINAL sample4 index times width = 2/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample4 index times width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample4 index times width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "4",
          "1/10"
        ],
        "output": "2/5",
        "label": "Left sum ORIGINAL sample4 index times width"
      }
    },
    {
      "id": "S033",
      "uses": [
        "G001",
        "S032"
      ],
      "before": "Left sum ORIGINAL sample4 lower plus offset := sum(-2, 2/5)",
      "after": "Left sum ORIGINAL sample4 lower plus offset = -8/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample4 lower plus offset，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample4 lower plus offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "2/5"
        ],
        "output": "-8/5",
        "label": "Left sum ORIGINAL sample4 lower plus offset"
      }
    },
    {
      "id": "S034",
      "uses": [
        "G001",
        "S033"
      ],
      "before": "Left sum ORIGINAL sample4 slope times x := mul(25, -8/5)",
      "after": "Left sum ORIGINAL sample4 slope times x = -40",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample4 slope times x，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample4 slope times x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-8/5"
        ],
        "output": "-40",
        "label": "Left sum ORIGINAL sample4 slope times x"
      }
    },
    {
      "id": "S035",
      "uses": [
        "G001",
        "S034"
      ],
      "before": "Left sum ORIGINAL sample4 affine value plus c := sum(-40, 12)",
      "after": "Left sum ORIGINAL sample4 affine value plus c = -28",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample4 affine value plus c，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample4 affine value plus c with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-40",
          "12"
        ],
        "output": "-28",
        "label": "Left sum ORIGINAL sample4 affine value plus c"
      }
    },
    {
      "id": "S036",
      "uses": [
        "G001",
        "S035"
      ],
      "before": "Left sum ORIGINAL sample4 value times cell width := mul(-28, 1/10)",
      "after": "Left sum ORIGINAL sample4 value times cell width = -14/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample4 value times cell width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample4 value times cell width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-28",
          "1/10"
        ],
        "output": "-14/5",
        "label": "Left sum ORIGINAL sample4 value times cell width"
      }
    },
    {
      "id": "S037",
      "uses": [
        "G001",
        "S036"
      ],
      "before": "Left sum ORIGINAL accumulated term4 := sum(-137/10, -14/5)",
      "after": "Left sum ORIGINAL accumulated term4 = -33/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term4，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL accumulated term4 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-137/10",
          "-14/5"
        ],
        "output": "-33/2",
        "label": "Left sum ORIGINAL accumulated term4"
      }
    },
    {
      "id": "S038",
      "uses": [
        "G001",
        "S037"
      ],
      "before": "source roles = [4, 50, 25, 12, -2, 3, 1/10]",
      "after": "derived = [-8/5, -28/1, -14/5, -33/2]",
      "relation": "implies",
      "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。",
        "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S039",
      "uses": [
        "G001",
        "S038"
      ],
      "before": "Left sum ORIGINAL sample5 index times width := mul(5, 1/10)",
      "after": "Left sum ORIGINAL sample5 index times width = 1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample5 index times width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample5 index times width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "5",
          "1/10"
        ],
        "output": "1/2",
        "label": "Left sum ORIGINAL sample5 index times width"
      }
    },
    {
      "id": "S040",
      "uses": [
        "G001",
        "S039"
      ],
      "before": "Left sum ORIGINAL sample5 lower plus offset := sum(-2, 1/2)",
      "after": "Left sum ORIGINAL sample5 lower plus offset = -3/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample5 lower plus offset，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample5 lower plus offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "1/2"
        ],
        "output": "-3/2",
        "label": "Left sum ORIGINAL sample5 lower plus offset"
      }
    },
    {
      "id": "S041",
      "uses": [
        "G001",
        "S040"
      ],
      "before": "Left sum ORIGINAL sample5 slope times x := mul(25, -3/2)",
      "after": "Left sum ORIGINAL sample5 slope times x = -75/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample5 slope times x，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample5 slope times x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-3/2"
        ],
        "output": "-75/2",
        "label": "Left sum ORIGINAL sample5 slope times x"
      }
    },
    {
      "id": "S042",
      "uses": [
        "G001",
        "S041"
      ],
      "before": "Left sum ORIGINAL sample5 affine value plus c := sum(-75/2, 12)",
      "after": "Left sum ORIGINAL sample5 affine value plus c = -51/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample5 affine value plus c，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample5 affine value plus c with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-75/2",
          "12"
        ],
        "output": "-51/2",
        "label": "Left sum ORIGINAL sample5 affine value plus c"
      }
    },
    {
      "id": "S043",
      "uses": [
        "G001",
        "S042"
      ],
      "before": "Left sum ORIGINAL sample5 value times cell width := mul(-51/2, 1/10)",
      "after": "Left sum ORIGINAL sample5 value times cell width = -51/20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample5 value times cell width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample5 value times cell width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-51/2",
          "1/10"
        ],
        "output": "-51/20",
        "label": "Left sum ORIGINAL sample5 value times cell width"
      }
    },
    {
      "id": "S044",
      "uses": [
        "G001",
        "S043"
      ],
      "before": "Left sum ORIGINAL accumulated term5 := sum(-33/2, -51/20)",
      "after": "Left sum ORIGINAL accumulated term5 = -381/20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term5，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL accumulated term5 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-33/2",
          "-51/20"
        ],
        "output": "-381/20",
        "label": "Left sum ORIGINAL accumulated term5"
      }
    },
    {
      "id": "S045",
      "uses": [
        "G001",
        "S044"
      ],
      "before": "source roles = [5, 50, 25, 12, -2, 3, 1/10]",
      "after": "derived = [-3/2, -51/2, -51/20, -381/20]",
      "relation": "implies",
      "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。",
        "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S046",
      "uses": [
        "G001",
        "S045"
      ],
      "before": "Left sum ORIGINAL sample6 index times width := mul(6, 1/10)",
      "after": "Left sum ORIGINAL sample6 index times width = 3/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample6 index times width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample6 index times width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "6",
          "1/10"
        ],
        "output": "3/5",
        "label": "Left sum ORIGINAL sample6 index times width"
      }
    },
    {
      "id": "S047",
      "uses": [
        "G001",
        "S046"
      ],
      "before": "Left sum ORIGINAL sample6 lower plus offset := sum(-2, 3/5)",
      "after": "Left sum ORIGINAL sample6 lower plus offset = -7/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample6 lower plus offset，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample6 lower plus offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "3/5"
        ],
        "output": "-7/5",
        "label": "Left sum ORIGINAL sample6 lower plus offset"
      }
    },
    {
      "id": "S048",
      "uses": [
        "G001",
        "S047"
      ],
      "before": "Left sum ORIGINAL sample6 slope times x := mul(25, -7/5)",
      "after": "Left sum ORIGINAL sample6 slope times x = -35",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample6 slope times x，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample6 slope times x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-7/5"
        ],
        "output": "-35",
        "label": "Left sum ORIGINAL sample6 slope times x"
      }
    },
    {
      "id": "S049",
      "uses": [
        "G001",
        "S048"
      ],
      "before": "Left sum ORIGINAL sample6 affine value plus c := sum(-35, 12)",
      "after": "Left sum ORIGINAL sample6 affine value plus c = -23",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample6 affine value plus c，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample6 affine value plus c with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-35",
          "12"
        ],
        "output": "-23",
        "label": "Left sum ORIGINAL sample6 affine value plus c"
      }
    },
    {
      "id": "S050",
      "uses": [
        "G001",
        "S049"
      ],
      "before": "Left sum ORIGINAL sample6 value times cell width := mul(-23, 1/10)",
      "after": "Left sum ORIGINAL sample6 value times cell width = -23/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample6 value times cell width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample6 value times cell width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-23",
          "1/10"
        ],
        "output": "-23/10",
        "label": "Left sum ORIGINAL sample6 value times cell width"
      }
    },
    {
      "id": "S051",
      "uses": [
        "G001",
        "S050"
      ],
      "before": "Left sum ORIGINAL accumulated term6 := sum(-381/20, -23/10)",
      "after": "Left sum ORIGINAL accumulated term6 = -427/20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term6，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL accumulated term6 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-381/20",
          "-23/10"
        ],
        "output": "-427/20",
        "label": "Left sum ORIGINAL accumulated term6"
      }
    },
    {
      "id": "S052",
      "uses": [
        "G001",
        "S051"
      ],
      "before": "source roles = [6, 50, 25, 12, -2, 3, 1/10]",
      "after": "derived = [-7/5, -23/1, -23/10, -427/20]",
      "relation": "implies",
      "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。",
        "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S053",
      "uses": [
        "G001",
        "S052"
      ],
      "before": "Left sum ORIGINAL sample7 index times width := mul(7, 1/10)",
      "after": "Left sum ORIGINAL sample7 index times width = 7/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample7 index times width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample7 index times width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "7",
          "1/10"
        ],
        "output": "7/10",
        "label": "Left sum ORIGINAL sample7 index times width"
      }
    },
    {
      "id": "S054",
      "uses": [
        "G001",
        "S053"
      ],
      "before": "Left sum ORIGINAL sample7 lower plus offset := sum(-2, 7/10)",
      "after": "Left sum ORIGINAL sample7 lower plus offset = -13/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample7 lower plus offset，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample7 lower plus offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "7/10"
        ],
        "output": "-13/10",
        "label": "Left sum ORIGINAL sample7 lower plus offset"
      }
    },
    {
      "id": "S055",
      "uses": [
        "G001",
        "S054"
      ],
      "before": "Left sum ORIGINAL sample7 slope times x := mul(25, -13/10)",
      "after": "Left sum ORIGINAL sample7 slope times x = -65/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample7 slope times x，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample7 slope times x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-13/10"
        ],
        "output": "-65/2",
        "label": "Left sum ORIGINAL sample7 slope times x"
      }
    },
    {
      "id": "S056",
      "uses": [
        "G001",
        "S055"
      ],
      "before": "Left sum ORIGINAL sample7 affine value plus c := sum(-65/2, 12)",
      "after": "Left sum ORIGINAL sample7 affine value plus c = -41/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample7 affine value plus c，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample7 affine value plus c with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-65/2",
          "12"
        ],
        "output": "-41/2",
        "label": "Left sum ORIGINAL sample7 affine value plus c"
      }
    },
    {
      "id": "S057",
      "uses": [
        "G001",
        "S056"
      ],
      "before": "Left sum ORIGINAL sample7 value times cell width := mul(-41/2, 1/10)",
      "after": "Left sum ORIGINAL sample7 value times cell width = -41/20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample7 value times cell width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample7 value times cell width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-41/2",
          "1/10"
        ],
        "output": "-41/20",
        "label": "Left sum ORIGINAL sample7 value times cell width"
      }
    },
    {
      "id": "S058",
      "uses": [
        "G001",
        "S057"
      ],
      "before": "Left sum ORIGINAL accumulated term7 := sum(-427/20, -41/20)",
      "after": "Left sum ORIGINAL accumulated term7 = -117/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term7，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL accumulated term7 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-427/20",
          "-41/20"
        ],
        "output": "-117/5",
        "label": "Left sum ORIGINAL accumulated term7"
      }
    },
    {
      "id": "S059",
      "uses": [
        "G001",
        "S058"
      ],
      "before": "source roles = [7, 50, 25, 12, -2, 3, 1/10]",
      "after": "derived = [-13/10, -41/2, -41/20, -117/5]",
      "relation": "implies",
      "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。",
        "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S060",
      "uses": [
        "G001",
        "S059"
      ],
      "before": "Left sum ORIGINAL sample8 index times width := mul(8, 1/10)",
      "after": "Left sum ORIGINAL sample8 index times width = 4/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample8 index times width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample8 index times width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "8",
          "1/10"
        ],
        "output": "4/5",
        "label": "Left sum ORIGINAL sample8 index times width"
      }
    },
    {
      "id": "S061",
      "uses": [
        "G001",
        "S060"
      ],
      "before": "Left sum ORIGINAL sample8 lower plus offset := sum(-2, 4/5)",
      "after": "Left sum ORIGINAL sample8 lower plus offset = -6/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample8 lower plus offset，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample8 lower plus offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "4/5"
        ],
        "output": "-6/5",
        "label": "Left sum ORIGINAL sample8 lower plus offset"
      }
    },
    {
      "id": "S062",
      "uses": [
        "G001",
        "S061"
      ],
      "before": "Left sum ORIGINAL sample8 slope times x := mul(25, -6/5)",
      "after": "Left sum ORIGINAL sample8 slope times x = -30",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample8 slope times x，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample8 slope times x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-6/5"
        ],
        "output": "-30",
        "label": "Left sum ORIGINAL sample8 slope times x"
      }
    },
    {
      "id": "S063",
      "uses": [
        "G001",
        "S062"
      ],
      "before": "Left sum ORIGINAL sample8 affine value plus c := sum(-30, 12)",
      "after": "Left sum ORIGINAL sample8 affine value plus c = -18",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample8 affine value plus c，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample8 affine value plus c with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-30",
          "12"
        ],
        "output": "-18",
        "label": "Left sum ORIGINAL sample8 affine value plus c"
      }
    },
    {
      "id": "S064",
      "uses": [
        "G001",
        "S063"
      ],
      "before": "Left sum ORIGINAL sample8 value times cell width := mul(-18, 1/10)",
      "after": "Left sum ORIGINAL sample8 value times cell width = -9/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample8 value times cell width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample8 value times cell width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-18",
          "1/10"
        ],
        "output": "-9/5",
        "label": "Left sum ORIGINAL sample8 value times cell width"
      }
    },
    {
      "id": "S065",
      "uses": [
        "G001",
        "S064"
      ],
      "before": "Left sum ORIGINAL accumulated term8 := sum(-117/5, -9/5)",
      "after": "Left sum ORIGINAL accumulated term8 = -126/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term8，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL accumulated term8 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-117/5",
          "-9/5"
        ],
        "output": "-126/5",
        "label": "Left sum ORIGINAL accumulated term8"
      }
    },
    {
      "id": "S066",
      "uses": [
        "G001",
        "S065"
      ],
      "before": "source roles = [8, 50, 25, 12, -2, 3, 1/10]",
      "after": "derived = [-6/5, -18/1, -9/5, -126/5]",
      "relation": "implies",
      "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。",
        "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S067",
      "uses": [
        "G001",
        "S066"
      ],
      "before": "Left sum ORIGINAL sample9 index times width := mul(9, 1/10)",
      "after": "Left sum ORIGINAL sample9 index times width = 9/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample9 index times width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample9 index times width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "9",
          "1/10"
        ],
        "output": "9/10",
        "label": "Left sum ORIGINAL sample9 index times width"
      }
    },
    {
      "id": "S068",
      "uses": [
        "G001",
        "S067"
      ],
      "before": "Left sum ORIGINAL sample9 lower plus offset := sum(-2, 9/10)",
      "after": "Left sum ORIGINAL sample9 lower plus offset = -11/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample9 lower plus offset，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample9 lower plus offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "9/10"
        ],
        "output": "-11/10",
        "label": "Left sum ORIGINAL sample9 lower plus offset"
      }
    },
    {
      "id": "S069",
      "uses": [
        "G001",
        "S068"
      ],
      "before": "Left sum ORIGINAL sample9 slope times x := mul(25, -11/10)",
      "after": "Left sum ORIGINAL sample9 slope times x = -55/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample9 slope times x，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample9 slope times x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-11/10"
        ],
        "output": "-55/2",
        "label": "Left sum ORIGINAL sample9 slope times x"
      }
    },
    {
      "id": "S070",
      "uses": [
        "G001",
        "S069"
      ],
      "before": "Left sum ORIGINAL sample9 affine value plus c := sum(-55/2, 12)",
      "after": "Left sum ORIGINAL sample9 affine value plus c = -31/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample9 affine value plus c，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample9 affine value plus c with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-55/2",
          "12"
        ],
        "output": "-31/2",
        "label": "Left sum ORIGINAL sample9 affine value plus c"
      }
    },
    {
      "id": "S071",
      "uses": [
        "G001",
        "S070"
      ],
      "before": "Left sum ORIGINAL sample9 value times cell width := mul(-31/2, 1/10)",
      "after": "Left sum ORIGINAL sample9 value times cell width = -31/20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample9 value times cell width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample9 value times cell width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-31/2",
          "1/10"
        ],
        "output": "-31/20",
        "label": "Left sum ORIGINAL sample9 value times cell width"
      }
    },
    {
      "id": "S072",
      "uses": [
        "G001",
        "S071"
      ],
      "before": "Left sum ORIGINAL accumulated term9 := sum(-126/5, -31/20)",
      "after": "Left sum ORIGINAL accumulated term9 = -107/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term9，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL accumulated term9 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-126/5",
          "-31/20"
        ],
        "output": "-107/4",
        "label": "Left sum ORIGINAL accumulated term9"
      }
    },
    {
      "id": "S073",
      "uses": [
        "G001",
        "S072"
      ],
      "before": "source roles = [9, 50, 25, 12, -2, 3, 1/10]",
      "after": "derived = [-11/10, -31/2, -31/20, -107/4]",
      "relation": "implies",
      "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。",
        "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S074",
      "uses": [
        "G001",
        "S073"
      ],
      "before": "Left sum ORIGINAL sample10 index times width := mul(10, 1/10)",
      "after": "Left sum ORIGINAL sample10 index times width = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample10 index times width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample10 index times width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "10",
          "1/10"
        ],
        "output": "1",
        "label": "Left sum ORIGINAL sample10 index times width"
      }
    },
    {
      "id": "S075",
      "uses": [
        "G001",
        "S074"
      ],
      "before": "Left sum ORIGINAL sample10 lower plus offset := sum(-2, 1)",
      "after": "Left sum ORIGINAL sample10 lower plus offset = -1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample10 lower plus offset，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample10 lower plus offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "1"
        ],
        "output": "-1",
        "label": "Left sum ORIGINAL sample10 lower plus offset"
      }
    },
    {
      "id": "S076",
      "uses": [
        "G001",
        "S075"
      ],
      "before": "Left sum ORIGINAL sample10 slope times x := mul(25, -1)",
      "after": "Left sum ORIGINAL sample10 slope times x = -25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample10 slope times x，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample10 slope times x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-1"
        ],
        "output": "-25",
        "label": "Left sum ORIGINAL sample10 slope times x"
      }
    },
    {
      "id": "S077",
      "uses": [
        "G001",
        "S076"
      ],
      "before": "Left sum ORIGINAL sample10 affine value plus c := sum(-25, 12)",
      "after": "Left sum ORIGINAL sample10 affine value plus c = -13",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample10 affine value plus c，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample10 affine value plus c with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-25",
          "12"
        ],
        "output": "-13",
        "label": "Left sum ORIGINAL sample10 affine value plus c"
      }
    },
    {
      "id": "S078",
      "uses": [
        "G001",
        "S077"
      ],
      "before": "Left sum ORIGINAL sample10 value times cell width := mul(-13, 1/10)",
      "after": "Left sum ORIGINAL sample10 value times cell width = -13/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample10 value times cell width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample10 value times cell width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-13",
          "1/10"
        ],
        "output": "-13/10",
        "label": "Left sum ORIGINAL sample10 value times cell width"
      }
    },
    {
      "id": "S079",
      "uses": [
        "G001",
        "S078"
      ],
      "before": "Left sum ORIGINAL accumulated term10 := sum(-107/4, -13/10)",
      "after": "Left sum ORIGINAL accumulated term10 = -561/20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term10，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL accumulated term10 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-107/4",
          "-13/10"
        ],
        "output": "-561/20",
        "label": "Left sum ORIGINAL accumulated term10"
      }
    },
    {
      "id": "S080",
      "uses": [
        "G001",
        "S079"
      ],
      "before": "source roles = [10, 50, 25, 12, -2, 3, 1/10]",
      "after": "derived = [-1/1, -13/1, -13/10, -561/20]",
      "relation": "implies",
      "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。",
        "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S081",
      "uses": [
        "G001",
        "S080"
      ],
      "before": "Left sum ORIGINAL sample11 index times width := mul(11, 1/10)",
      "after": "Left sum ORIGINAL sample11 index times width = 11/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample11 index times width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample11 index times width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "11",
          "1/10"
        ],
        "output": "11/10",
        "label": "Left sum ORIGINAL sample11 index times width"
      }
    },
    {
      "id": "S082",
      "uses": [
        "G001",
        "S081"
      ],
      "before": "Left sum ORIGINAL sample11 lower plus offset := sum(-2, 11/10)",
      "after": "Left sum ORIGINAL sample11 lower plus offset = -9/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample11 lower plus offset，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample11 lower plus offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "11/10"
        ],
        "output": "-9/10",
        "label": "Left sum ORIGINAL sample11 lower plus offset"
      }
    },
    {
      "id": "S083",
      "uses": [
        "G001",
        "S082"
      ],
      "before": "Left sum ORIGINAL sample11 slope times x := mul(25, -9/10)",
      "after": "Left sum ORIGINAL sample11 slope times x = -45/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample11 slope times x，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample11 slope times x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-9/10"
        ],
        "output": "-45/2",
        "label": "Left sum ORIGINAL sample11 slope times x"
      }
    },
    {
      "id": "S084",
      "uses": [
        "G001",
        "S083"
      ],
      "before": "Left sum ORIGINAL sample11 affine value plus c := sum(-45/2, 12)",
      "after": "Left sum ORIGINAL sample11 affine value plus c = -21/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample11 affine value plus c，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample11 affine value plus c with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-45/2",
          "12"
        ],
        "output": "-21/2",
        "label": "Left sum ORIGINAL sample11 affine value plus c"
      }
    },
    {
      "id": "S085",
      "uses": [
        "G001",
        "S084"
      ],
      "before": "Left sum ORIGINAL sample11 value times cell width := mul(-21/2, 1/10)",
      "after": "Left sum ORIGINAL sample11 value times cell width = -21/20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample11 value times cell width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample11 value times cell width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-21/2",
          "1/10"
        ],
        "output": "-21/20",
        "label": "Left sum ORIGINAL sample11 value times cell width"
      }
    },
    {
      "id": "S086",
      "uses": [
        "G001",
        "S085"
      ],
      "before": "Left sum ORIGINAL accumulated term11 := sum(-561/20, -21/20)",
      "after": "Left sum ORIGINAL accumulated term11 = -291/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term11，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL accumulated term11 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-561/20",
          "-21/20"
        ],
        "output": "-291/10",
        "label": "Left sum ORIGINAL accumulated term11"
      }
    },
    {
      "id": "S087",
      "uses": [
        "G001",
        "S086"
      ],
      "before": "source roles = [11, 50, 25, 12, -2, 3, 1/10]",
      "after": "derived = [-9/10, -21/2, -21/20, -291/10]",
      "relation": "implies",
      "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。",
        "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S088",
      "uses": [
        "G001",
        "S087"
      ],
      "before": "Left sum ORIGINAL sample12 index times width := mul(12, 1/10)",
      "after": "Left sum ORIGINAL sample12 index times width = 6/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample12 index times width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample12 index times width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "12",
          "1/10"
        ],
        "output": "6/5",
        "label": "Left sum ORIGINAL sample12 index times width"
      }
    },
    {
      "id": "S089",
      "uses": [
        "G001",
        "S088"
      ],
      "before": "Left sum ORIGINAL sample12 lower plus offset := sum(-2, 6/5)",
      "after": "Left sum ORIGINAL sample12 lower plus offset = -4/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample12 lower plus offset，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample12 lower plus offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "6/5"
        ],
        "output": "-4/5",
        "label": "Left sum ORIGINAL sample12 lower plus offset"
      }
    },
    {
      "id": "S090",
      "uses": [
        "G001",
        "S089"
      ],
      "before": "Left sum ORIGINAL sample12 slope times x := mul(25, -4/5)",
      "after": "Left sum ORIGINAL sample12 slope times x = -20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample12 slope times x，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample12 slope times x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-4/5"
        ],
        "output": "-20",
        "label": "Left sum ORIGINAL sample12 slope times x"
      }
    },
    {
      "id": "S091",
      "uses": [
        "G001",
        "S090"
      ],
      "before": "Left sum ORIGINAL sample12 affine value plus c := sum(-20, 12)",
      "after": "Left sum ORIGINAL sample12 affine value plus c = -8",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample12 affine value plus c，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample12 affine value plus c with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-20",
          "12"
        ],
        "output": "-8",
        "label": "Left sum ORIGINAL sample12 affine value plus c"
      }
    },
    {
      "id": "S092",
      "uses": [
        "G001",
        "S091"
      ],
      "before": "Left sum ORIGINAL sample12 value times cell width := mul(-8, 1/10)",
      "after": "Left sum ORIGINAL sample12 value times cell width = -4/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample12 value times cell width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample12 value times cell width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-8",
          "1/10"
        ],
        "output": "-4/5",
        "label": "Left sum ORIGINAL sample12 value times cell width"
      }
    },
    {
      "id": "S093",
      "uses": [
        "G001",
        "S092"
      ],
      "before": "Left sum ORIGINAL accumulated term12 := sum(-291/10, -4/5)",
      "after": "Left sum ORIGINAL accumulated term12 = -299/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term12，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL accumulated term12 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-291/10",
          "-4/5"
        ],
        "output": "-299/10",
        "label": "Left sum ORIGINAL accumulated term12"
      }
    },
    {
      "id": "S094",
      "uses": [
        "G001",
        "S093"
      ],
      "before": "source roles = [12, 50, 25, 12, -2, 3, 1/10]",
      "after": "derived = [-4/5, -8/1, -4/5, -299/10]",
      "relation": "implies",
      "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。",
        "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S095",
      "uses": [
        "G001",
        "S094"
      ],
      "before": "Left sum ORIGINAL sample13 index times width := mul(13, 1/10)",
      "after": "Left sum ORIGINAL sample13 index times width = 13/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample13 index times width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample13 index times width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "13",
          "1/10"
        ],
        "output": "13/10",
        "label": "Left sum ORIGINAL sample13 index times width"
      }
    },
    {
      "id": "S096",
      "uses": [
        "G001",
        "S095"
      ],
      "before": "Left sum ORIGINAL sample13 lower plus offset := sum(-2, 13/10)",
      "after": "Left sum ORIGINAL sample13 lower plus offset = -7/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample13 lower plus offset，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample13 lower plus offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "13/10"
        ],
        "output": "-7/10",
        "label": "Left sum ORIGINAL sample13 lower plus offset"
      }
    },
    {
      "id": "S097",
      "uses": [
        "G001",
        "S096"
      ],
      "before": "Left sum ORIGINAL sample13 slope times x := mul(25, -7/10)",
      "after": "Left sum ORIGINAL sample13 slope times x = -35/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample13 slope times x，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample13 slope times x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-7/10"
        ],
        "output": "-35/2",
        "label": "Left sum ORIGINAL sample13 slope times x"
      }
    },
    {
      "id": "S098",
      "uses": [
        "G001",
        "S097"
      ],
      "before": "Left sum ORIGINAL sample13 affine value plus c := sum(-35/2, 12)",
      "after": "Left sum ORIGINAL sample13 affine value plus c = -11/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample13 affine value plus c，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample13 affine value plus c with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-35/2",
          "12"
        ],
        "output": "-11/2",
        "label": "Left sum ORIGINAL sample13 affine value plus c"
      }
    },
    {
      "id": "S099",
      "uses": [
        "G001",
        "S098"
      ],
      "before": "Left sum ORIGINAL sample13 value times cell width := mul(-11/2, 1/10)",
      "after": "Left sum ORIGINAL sample13 value times cell width = -11/20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample13 value times cell width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample13 value times cell width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-11/2",
          "1/10"
        ],
        "output": "-11/20",
        "label": "Left sum ORIGINAL sample13 value times cell width"
      }
    },
    {
      "id": "S100",
      "uses": [
        "G001",
        "S099"
      ],
      "before": "Left sum ORIGINAL accumulated term13 := sum(-299/10, -11/20)",
      "after": "Left sum ORIGINAL accumulated term13 = -609/20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term13，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL accumulated term13 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-299/10",
          "-11/20"
        ],
        "output": "-609/20",
        "label": "Left sum ORIGINAL accumulated term13"
      }
    },
    {
      "id": "S101",
      "uses": [
        "G001",
        "S100"
      ],
      "before": "source roles = [13, 50, 25, 12, -2, 3, 1/10]",
      "after": "derived = [-7/10, -11/2, -11/20, -609/20]",
      "relation": "implies",
      "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。",
        "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S102",
      "uses": [
        "G001",
        "S101"
      ],
      "before": "Left sum ORIGINAL sample14 index times width := mul(14, 1/10)",
      "after": "Left sum ORIGINAL sample14 index times width = 7/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample14 index times width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample14 index times width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "14",
          "1/10"
        ],
        "output": "7/5",
        "label": "Left sum ORIGINAL sample14 index times width"
      }
    },
    {
      "id": "S103",
      "uses": [
        "G001",
        "S102"
      ],
      "before": "Left sum ORIGINAL sample14 lower plus offset := sum(-2, 7/5)",
      "after": "Left sum ORIGINAL sample14 lower plus offset = -3/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample14 lower plus offset，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample14 lower plus offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "7/5"
        ],
        "output": "-3/5",
        "label": "Left sum ORIGINAL sample14 lower plus offset"
      }
    },
    {
      "id": "S104",
      "uses": [
        "G001",
        "S103"
      ],
      "before": "Left sum ORIGINAL sample14 slope times x := mul(25, -3/5)",
      "after": "Left sum ORIGINAL sample14 slope times x = -15",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample14 slope times x，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample14 slope times x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-3/5"
        ],
        "output": "-15",
        "label": "Left sum ORIGINAL sample14 slope times x"
      }
    },
    {
      "id": "S105",
      "uses": [
        "G001",
        "S104"
      ],
      "before": "Left sum ORIGINAL sample14 affine value plus c := sum(-15, 12)",
      "after": "Left sum ORIGINAL sample14 affine value plus c = -3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample14 affine value plus c，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample14 affine value plus c with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-15",
          "12"
        ],
        "output": "-3",
        "label": "Left sum ORIGINAL sample14 affine value plus c"
      }
    },
    {
      "id": "S106",
      "uses": [
        "G001",
        "S105"
      ],
      "before": "Left sum ORIGINAL sample14 value times cell width := mul(-3, 1/10)",
      "after": "Left sum ORIGINAL sample14 value times cell width = -3/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample14 value times cell width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample14 value times cell width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-3",
          "1/10"
        ],
        "output": "-3/10",
        "label": "Left sum ORIGINAL sample14 value times cell width"
      }
    },
    {
      "id": "S107",
      "uses": [
        "G001",
        "S106"
      ],
      "before": "Left sum ORIGINAL accumulated term14 := sum(-609/20, -3/10)",
      "after": "Left sum ORIGINAL accumulated term14 = -123/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term14，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL accumulated term14 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-609/20",
          "-3/10"
        ],
        "output": "-123/4",
        "label": "Left sum ORIGINAL accumulated term14"
      }
    },
    {
      "id": "S108",
      "uses": [
        "G001",
        "S107"
      ],
      "before": "source roles = [14, 50, 25, 12, -2, 3, 1/10]",
      "after": "derived = [-3/5, -3/1, -3/10, -123/4]",
      "relation": "implies",
      "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。",
        "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S109",
      "uses": [
        "G001",
        "S108"
      ],
      "before": "Left sum ORIGINAL sample15 index times width := mul(15, 1/10)",
      "after": "Left sum ORIGINAL sample15 index times width = 3/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample15 index times width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample15 index times width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "15",
          "1/10"
        ],
        "output": "3/2",
        "label": "Left sum ORIGINAL sample15 index times width"
      }
    },
    {
      "id": "S110",
      "uses": [
        "G001",
        "S109"
      ],
      "before": "Left sum ORIGINAL sample15 lower plus offset := sum(-2, 3/2)",
      "after": "Left sum ORIGINAL sample15 lower plus offset = -1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample15 lower plus offset，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample15 lower plus offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "3/2"
        ],
        "output": "-1/2",
        "label": "Left sum ORIGINAL sample15 lower plus offset"
      }
    },
    {
      "id": "S111",
      "uses": [
        "G001",
        "S110"
      ],
      "before": "Left sum ORIGINAL sample15 slope times x := mul(25, -1/2)",
      "after": "Left sum ORIGINAL sample15 slope times x = -25/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample15 slope times x，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample15 slope times x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-1/2"
        ],
        "output": "-25/2",
        "label": "Left sum ORIGINAL sample15 slope times x"
      }
    },
    {
      "id": "S112",
      "uses": [
        "G001",
        "S111"
      ],
      "before": "Left sum ORIGINAL sample15 affine value plus c := sum(-25/2, 12)",
      "after": "Left sum ORIGINAL sample15 affine value plus c = -1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample15 affine value plus c，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample15 affine value plus c with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-25/2",
          "12"
        ],
        "output": "-1/2",
        "label": "Left sum ORIGINAL sample15 affine value plus c"
      }
    },
    {
      "id": "S113",
      "uses": [
        "G001",
        "S112"
      ],
      "before": "Left sum ORIGINAL sample15 value times cell width := mul(-1/2, 1/10)",
      "after": "Left sum ORIGINAL sample15 value times cell width = -1/20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample15 value times cell width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample15 value times cell width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1/2",
          "1/10"
        ],
        "output": "-1/20",
        "label": "Left sum ORIGINAL sample15 value times cell width"
      }
    },
    {
      "id": "S114",
      "uses": [
        "G001",
        "S113"
      ],
      "before": "Left sum ORIGINAL accumulated term15 := sum(-123/4, -1/20)",
      "after": "Left sum ORIGINAL accumulated term15 = -154/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term15，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL accumulated term15 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-123/4",
          "-1/20"
        ],
        "output": "-154/5",
        "label": "Left sum ORIGINAL accumulated term15"
      }
    },
    {
      "id": "S115",
      "uses": [
        "G001",
        "S114"
      ],
      "before": "source roles = [15, 50, 25, 12, -2, 3, 1/10]",
      "after": "derived = [-1/2, -1/2, -1/20, -154/5]",
      "relation": "implies",
      "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。",
        "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S116",
      "uses": [
        "G001",
        "S115"
      ],
      "before": "Left sum ORIGINAL sample16 index times width := mul(16, 1/10)",
      "after": "Left sum ORIGINAL sample16 index times width = 8/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample16 index times width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample16 index times width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "16",
          "1/10"
        ],
        "output": "8/5",
        "label": "Left sum ORIGINAL sample16 index times width"
      }
    },
    {
      "id": "S117",
      "uses": [
        "G001",
        "S116"
      ],
      "before": "Left sum ORIGINAL sample16 lower plus offset := sum(-2, 8/5)",
      "after": "Left sum ORIGINAL sample16 lower plus offset = -2/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample16 lower plus offset，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample16 lower plus offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "8/5"
        ],
        "output": "-2/5",
        "label": "Left sum ORIGINAL sample16 lower plus offset"
      }
    },
    {
      "id": "S118",
      "uses": [
        "G001",
        "S117"
      ],
      "before": "Left sum ORIGINAL sample16 slope times x := mul(25, -2/5)",
      "after": "Left sum ORIGINAL sample16 slope times x = -10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample16 slope times x，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample16 slope times x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-2/5"
        ],
        "output": "-10",
        "label": "Left sum ORIGINAL sample16 slope times x"
      }
    },
    {
      "id": "S119",
      "uses": [
        "G001",
        "S118"
      ],
      "before": "Left sum ORIGINAL sample16 affine value plus c := sum(-10, 12)",
      "after": "Left sum ORIGINAL sample16 affine value plus c = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample16 affine value plus c，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample16 affine value plus c with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-10",
          "12"
        ],
        "output": "2",
        "label": "Left sum ORIGINAL sample16 affine value plus c"
      }
    },
    {
      "id": "S120",
      "uses": [
        "G001",
        "S119"
      ],
      "before": "Left sum ORIGINAL sample16 value times cell width := mul(2, 1/10)",
      "after": "Left sum ORIGINAL sample16 value times cell width = 1/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample16 value times cell width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample16 value times cell width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "1/10"
        ],
        "output": "1/5",
        "label": "Left sum ORIGINAL sample16 value times cell width"
      }
    },
    {
      "id": "S121",
      "uses": [
        "G001",
        "S120"
      ],
      "before": "Left sum ORIGINAL accumulated term16 := sum(-154/5, 1/5)",
      "after": "Left sum ORIGINAL accumulated term16 = -153/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term16，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL accumulated term16 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-154/5",
          "1/5"
        ],
        "output": "-153/5",
        "label": "Left sum ORIGINAL accumulated term16"
      }
    },
    {
      "id": "S122",
      "uses": [
        "G001",
        "S121"
      ],
      "before": "source roles = [16, 50, 25, 12, -2, 3, 1/10]",
      "after": "derived = [-2/5, 2/1, 1/5, -153/5]",
      "relation": "implies",
      "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。",
        "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S123",
      "uses": [
        "G001",
        "S122"
      ],
      "before": "Left sum ORIGINAL sample17 index times width := mul(17, 1/10)",
      "after": "Left sum ORIGINAL sample17 index times width = 17/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample17 index times width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample17 index times width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "17",
          "1/10"
        ],
        "output": "17/10",
        "label": "Left sum ORIGINAL sample17 index times width"
      }
    },
    {
      "id": "S124",
      "uses": [
        "G001",
        "S123"
      ],
      "before": "Left sum ORIGINAL sample17 lower plus offset := sum(-2, 17/10)",
      "after": "Left sum ORIGINAL sample17 lower plus offset = -3/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample17 lower plus offset，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample17 lower plus offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "17/10"
        ],
        "output": "-3/10",
        "label": "Left sum ORIGINAL sample17 lower plus offset"
      }
    },
    {
      "id": "S125",
      "uses": [
        "G001",
        "S124"
      ],
      "before": "Left sum ORIGINAL sample17 slope times x := mul(25, -3/10)",
      "after": "Left sum ORIGINAL sample17 slope times x = -15/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample17 slope times x，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample17 slope times x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-3/10"
        ],
        "output": "-15/2",
        "label": "Left sum ORIGINAL sample17 slope times x"
      }
    },
    {
      "id": "S126",
      "uses": [
        "G001",
        "S125"
      ],
      "before": "Left sum ORIGINAL sample17 affine value plus c := sum(-15/2, 12)",
      "after": "Left sum ORIGINAL sample17 affine value plus c = 9/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample17 affine value plus c，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample17 affine value plus c with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-15/2",
          "12"
        ],
        "output": "9/2",
        "label": "Left sum ORIGINAL sample17 affine value plus c"
      }
    },
    {
      "id": "S127",
      "uses": [
        "G001",
        "S126"
      ],
      "before": "Left sum ORIGINAL sample17 value times cell width := mul(9/2, 1/10)",
      "after": "Left sum ORIGINAL sample17 value times cell width = 9/20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample17 value times cell width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample17 value times cell width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "9/2",
          "1/10"
        ],
        "output": "9/20",
        "label": "Left sum ORIGINAL sample17 value times cell width"
      }
    },
    {
      "id": "S128",
      "uses": [
        "G001",
        "S127"
      ],
      "before": "Left sum ORIGINAL accumulated term17 := sum(-153/5, 9/20)",
      "after": "Left sum ORIGINAL accumulated term17 = -603/20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term17，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL accumulated term17 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-153/5",
          "9/20"
        ],
        "output": "-603/20",
        "label": "Left sum ORIGINAL accumulated term17"
      }
    },
    {
      "id": "S129",
      "uses": [
        "G001",
        "S128"
      ],
      "before": "source roles = [17, 50, 25, 12, -2, 3, 1/10]",
      "after": "derived = [-3/10, 9/2, 9/20, -603/20]",
      "relation": "implies",
      "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。",
        "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S130",
      "uses": [
        "G001",
        "S129"
      ],
      "before": "Left sum ORIGINAL sample18 index times width := mul(18, 1/10)",
      "after": "Left sum ORIGINAL sample18 index times width = 9/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample18 index times width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample18 index times width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "18",
          "1/10"
        ],
        "output": "9/5",
        "label": "Left sum ORIGINAL sample18 index times width"
      }
    },
    {
      "id": "S131",
      "uses": [
        "G001",
        "S130"
      ],
      "before": "Left sum ORIGINAL sample18 lower plus offset := sum(-2, 9/5)",
      "after": "Left sum ORIGINAL sample18 lower plus offset = -1/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample18 lower plus offset，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample18 lower plus offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "9/5"
        ],
        "output": "-1/5",
        "label": "Left sum ORIGINAL sample18 lower plus offset"
      }
    },
    {
      "id": "S132",
      "uses": [
        "G001",
        "S131"
      ],
      "before": "Left sum ORIGINAL sample18 slope times x := mul(25, -1/5)",
      "after": "Left sum ORIGINAL sample18 slope times x = -5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample18 slope times x，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample18 slope times x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-1/5"
        ],
        "output": "-5",
        "label": "Left sum ORIGINAL sample18 slope times x"
      }
    },
    {
      "id": "S133",
      "uses": [
        "G001",
        "S132"
      ],
      "before": "Left sum ORIGINAL sample18 affine value plus c := sum(-5, 12)",
      "after": "Left sum ORIGINAL sample18 affine value plus c = 7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample18 affine value plus c，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample18 affine value plus c with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-5",
          "12"
        ],
        "output": "7",
        "label": "Left sum ORIGINAL sample18 affine value plus c"
      }
    },
    {
      "id": "S134",
      "uses": [
        "G001",
        "S133"
      ],
      "before": "Left sum ORIGINAL sample18 value times cell width := mul(7, 1/10)",
      "after": "Left sum ORIGINAL sample18 value times cell width = 7/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample18 value times cell width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample18 value times cell width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "7",
          "1/10"
        ],
        "output": "7/10",
        "label": "Left sum ORIGINAL sample18 value times cell width"
      }
    },
    {
      "id": "S135",
      "uses": [
        "G001",
        "S134"
      ],
      "before": "Left sum ORIGINAL accumulated term18 := sum(-603/20, 7/10)",
      "after": "Left sum ORIGINAL accumulated term18 = -589/20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term18，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL accumulated term18 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-603/20",
          "7/10"
        ],
        "output": "-589/20",
        "label": "Left sum ORIGINAL accumulated term18"
      }
    },
    {
      "id": "S136",
      "uses": [
        "G001",
        "S135"
      ],
      "before": "source roles = [18, 50, 25, 12, -2, 3, 1/10]",
      "after": "derived = [-1/5, 7/1, 7/10, -589/20]",
      "relation": "implies",
      "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。",
        "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S137",
      "uses": [
        "G001",
        "S136"
      ],
      "before": "Left sum ORIGINAL sample19 index times width := mul(19, 1/10)",
      "after": "Left sum ORIGINAL sample19 index times width = 19/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample19 index times width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample19 index times width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "19",
          "1/10"
        ],
        "output": "19/10",
        "label": "Left sum ORIGINAL sample19 index times width"
      }
    },
    {
      "id": "S138",
      "uses": [
        "G001",
        "S137"
      ],
      "before": "Left sum ORIGINAL sample19 lower plus offset := sum(-2, 19/10)",
      "after": "Left sum ORIGINAL sample19 lower plus offset = -1/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample19 lower plus offset，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample19 lower plus offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "19/10"
        ],
        "output": "-1/10",
        "label": "Left sum ORIGINAL sample19 lower plus offset"
      }
    },
    {
      "id": "S139",
      "uses": [
        "G001",
        "S138"
      ],
      "before": "Left sum ORIGINAL sample19 slope times x := mul(25, -1/10)",
      "after": "Left sum ORIGINAL sample19 slope times x = -5/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample19 slope times x，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample19 slope times x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "-1/10"
        ],
        "output": "-5/2",
        "label": "Left sum ORIGINAL sample19 slope times x"
      }
    },
    {
      "id": "S140",
      "uses": [
        "G001",
        "S139"
      ],
      "before": "Left sum ORIGINAL sample19 affine value plus c := sum(-5/2, 12)",
      "after": "Left sum ORIGINAL sample19 affine value plus c = 19/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample19 affine value plus c，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample19 affine value plus c with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-5/2",
          "12"
        ],
        "output": "19/2",
        "label": "Left sum ORIGINAL sample19 affine value plus c"
      }
    },
    {
      "id": "S141",
      "uses": [
        "G001",
        "S140"
      ],
      "before": "Left sum ORIGINAL sample19 value times cell width := mul(19/2, 1/10)",
      "after": "Left sum ORIGINAL sample19 value times cell width = 19/20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample19 value times cell width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample19 value times cell width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "19/2",
          "1/10"
        ],
        "output": "19/20",
        "label": "Left sum ORIGINAL sample19 value times cell width"
      }
    },
    {
      "id": "S142",
      "uses": [
        "G001",
        "S141"
      ],
      "before": "Left sum ORIGINAL accumulated term19 := sum(-589/20, 19/20)",
      "after": "Left sum ORIGINAL accumulated term19 = -57/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term19，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL accumulated term19 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-589/20",
          "19/20"
        ],
        "output": "-57/2",
        "label": "Left sum ORIGINAL accumulated term19"
      }
    },
    {
      "id": "S143",
      "uses": [
        "G001",
        "S142"
      ],
      "before": "source roles = [19, 50, 25, 12, -2, 3, 1/10]",
      "after": "derived = [-1/10, 19/2, 19/20, -57/2]",
      "relation": "implies",
      "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。",
        "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S144",
      "uses": [
        "G001",
        "S143"
      ],
      "before": "Left sum ORIGINAL sample20 index times width := mul(20, 1/10)",
      "after": "Left sum ORIGINAL sample20 index times width = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample20 index times width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample20 index times width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "1/10"
        ],
        "output": "2",
        "label": "Left sum ORIGINAL sample20 index times width"
      }
    },
    {
      "id": "S145",
      "uses": [
        "G001",
        "S144"
      ],
      "before": "Left sum ORIGINAL sample20 lower plus offset := sum(-2, 2)",
      "after": "Left sum ORIGINAL sample20 lower plus offset = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample20 lower plus offset，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample20 lower plus offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "2"
        ],
        "output": "0",
        "label": "Left sum ORIGINAL sample20 lower plus offset"
      }
    },
    {
      "id": "S146",
      "uses": [
        "G001",
        "S145"
      ],
      "before": "Left sum ORIGINAL sample20 slope times x := mul(25, 0)",
      "after": "Left sum ORIGINAL sample20 slope times x = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample20 slope times x，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample20 slope times x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "0"
        ],
        "output": "0",
        "label": "Left sum ORIGINAL sample20 slope times x"
      }
    },
    {
      "id": "S147",
      "uses": [
        "G001",
        "S146"
      ],
      "before": "Left sum ORIGINAL sample20 affine value plus c := sum(0, 12)",
      "after": "Left sum ORIGINAL sample20 affine value plus c = 12",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample20 affine value plus c，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample20 affine value plus c with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "12"
        ],
        "output": "12",
        "label": "Left sum ORIGINAL sample20 affine value plus c"
      }
    },
    {
      "id": "S148",
      "uses": [
        "G001",
        "S147"
      ],
      "before": "Left sum ORIGINAL sample20 value times cell width := mul(12, 1/10)",
      "after": "Left sum ORIGINAL sample20 value times cell width = 6/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample20 value times cell width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample20 value times cell width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "12",
          "1/10"
        ],
        "output": "6/5",
        "label": "Left sum ORIGINAL sample20 value times cell width"
      }
    },
    {
      "id": "S149",
      "uses": [
        "G001",
        "S148"
      ],
      "before": "Left sum ORIGINAL accumulated term20 := sum(-57/2, 6/5)",
      "after": "Left sum ORIGINAL accumulated term20 = -273/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term20，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL accumulated term20 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-57/2",
          "6/5"
        ],
        "output": "-273/10",
        "label": "Left sum ORIGINAL accumulated term20"
      }
    },
    {
      "id": "S150",
      "uses": [
        "G001",
        "S149"
      ],
      "before": "source roles = [20, 50, 25, 12, -2, 3, 1/10]",
      "after": "derived = [0/1, 12/1, 6/5, -273/10]",
      "relation": "implies",
      "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。",
        "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S151",
      "uses": [
        "G001",
        "S150"
      ],
      "before": "Left sum ORIGINAL sample21 index times width := mul(21, 1/10)",
      "after": "Left sum ORIGINAL sample21 index times width = 21/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample21 index times width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample21 index times width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "21",
          "1/10"
        ],
        "output": "21/10",
        "label": "Left sum ORIGINAL sample21 index times width"
      }
    },
    {
      "id": "S152",
      "uses": [
        "G001",
        "S151"
      ],
      "before": "Left sum ORIGINAL sample21 lower plus offset := sum(-2, 21/10)",
      "after": "Left sum ORIGINAL sample21 lower plus offset = 1/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample21 lower plus offset，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample21 lower plus offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "21/10"
        ],
        "output": "1/10",
        "label": "Left sum ORIGINAL sample21 lower plus offset"
      }
    },
    {
      "id": "S153",
      "uses": [
        "G001",
        "S152"
      ],
      "before": "Left sum ORIGINAL sample21 slope times x := mul(25, 1/10)",
      "after": "Left sum ORIGINAL sample21 slope times x = 5/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample21 slope times x，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample21 slope times x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "1/10"
        ],
        "output": "5/2",
        "label": "Left sum ORIGINAL sample21 slope times x"
      }
    },
    {
      "id": "S154",
      "uses": [
        "G001",
        "S153"
      ],
      "before": "Left sum ORIGINAL sample21 affine value plus c := sum(5/2, 12)",
      "after": "Left sum ORIGINAL sample21 affine value plus c = 29/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample21 affine value plus c，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample21 affine value plus c with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "5/2",
          "12"
        ],
        "output": "29/2",
        "label": "Left sum ORIGINAL sample21 affine value plus c"
      }
    },
    {
      "id": "S155",
      "uses": [
        "G001",
        "S154"
      ],
      "before": "Left sum ORIGINAL sample21 value times cell width := mul(29/2, 1/10)",
      "after": "Left sum ORIGINAL sample21 value times cell width = 29/20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample21 value times cell width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample21 value times cell width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "29/2",
          "1/10"
        ],
        "output": "29/20",
        "label": "Left sum ORIGINAL sample21 value times cell width"
      }
    },
    {
      "id": "S156",
      "uses": [
        "G001",
        "S155"
      ],
      "before": "Left sum ORIGINAL accumulated term21 := sum(-273/10, 29/20)",
      "after": "Left sum ORIGINAL accumulated term21 = -517/20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term21，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL accumulated term21 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-273/10",
          "29/20"
        ],
        "output": "-517/20",
        "label": "Left sum ORIGINAL accumulated term21"
      }
    },
    {
      "id": "S157",
      "uses": [
        "G001",
        "S156"
      ],
      "before": "source roles = [21, 50, 25, 12, -2, 3, 1/10]",
      "after": "derived = [1/10, 29/2, 29/20, -517/20]",
      "relation": "implies",
      "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。",
        "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S158",
      "uses": [
        "G001",
        "S157"
      ],
      "before": "Left sum ORIGINAL sample22 index times width := mul(22, 1/10)",
      "after": "Left sum ORIGINAL sample22 index times width = 11/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample22 index times width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample22 index times width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "22",
          "1/10"
        ],
        "output": "11/5",
        "label": "Left sum ORIGINAL sample22 index times width"
      }
    },
    {
      "id": "S159",
      "uses": [
        "G001",
        "S158"
      ],
      "before": "Left sum ORIGINAL sample22 lower plus offset := sum(-2, 11/5)",
      "after": "Left sum ORIGINAL sample22 lower plus offset = 1/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample22 lower plus offset，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample22 lower plus offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "11/5"
        ],
        "output": "1/5",
        "label": "Left sum ORIGINAL sample22 lower plus offset"
      }
    },
    {
      "id": "S160",
      "uses": [
        "G001",
        "S159"
      ],
      "before": "Left sum ORIGINAL sample22 slope times x := mul(25, 1/5)",
      "after": "Left sum ORIGINAL sample22 slope times x = 5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample22 slope times x，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample22 slope times x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "1/5"
        ],
        "output": "5",
        "label": "Left sum ORIGINAL sample22 slope times x"
      }
    },
    {
      "id": "S161",
      "uses": [
        "G001",
        "S160"
      ],
      "before": "Left sum ORIGINAL sample22 affine value plus c := sum(5, 12)",
      "after": "Left sum ORIGINAL sample22 affine value plus c = 17",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample22 affine value plus c，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample22 affine value plus c with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "5",
          "12"
        ],
        "output": "17",
        "label": "Left sum ORIGINAL sample22 affine value plus c"
      }
    },
    {
      "id": "S162",
      "uses": [
        "G001",
        "S161"
      ],
      "before": "Left sum ORIGINAL sample22 value times cell width := mul(17, 1/10)",
      "after": "Left sum ORIGINAL sample22 value times cell width = 17/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample22 value times cell width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample22 value times cell width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "17",
          "1/10"
        ],
        "output": "17/10",
        "label": "Left sum ORIGINAL sample22 value times cell width"
      }
    },
    {
      "id": "S163",
      "uses": [
        "G001",
        "S162"
      ],
      "before": "Left sum ORIGINAL accumulated term22 := sum(-517/20, 17/10)",
      "after": "Left sum ORIGINAL accumulated term22 = -483/20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term22，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL accumulated term22 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-517/20",
          "17/10"
        ],
        "output": "-483/20",
        "label": "Left sum ORIGINAL accumulated term22"
      }
    },
    {
      "id": "S164",
      "uses": [
        "G001",
        "S163"
      ],
      "before": "source roles = [22, 50, 25, 12, -2, 3, 1/10]",
      "after": "derived = [1/5, 17/1, 17/10, -483/20]",
      "relation": "implies",
      "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。",
        "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S165",
      "uses": [
        "G001",
        "S164"
      ],
      "before": "Left sum ORIGINAL sample23 index times width := mul(23, 1/10)",
      "after": "Left sum ORIGINAL sample23 index times width = 23/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample23 index times width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample23 index times width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "23",
          "1/10"
        ],
        "output": "23/10",
        "label": "Left sum ORIGINAL sample23 index times width"
      }
    },
    {
      "id": "S166",
      "uses": [
        "G001",
        "S165"
      ],
      "before": "Left sum ORIGINAL sample23 lower plus offset := sum(-2, 23/10)",
      "after": "Left sum ORIGINAL sample23 lower plus offset = 3/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample23 lower plus offset，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample23 lower plus offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "23/10"
        ],
        "output": "3/10",
        "label": "Left sum ORIGINAL sample23 lower plus offset"
      }
    },
    {
      "id": "S167",
      "uses": [
        "G001",
        "S166"
      ],
      "before": "Left sum ORIGINAL sample23 slope times x := mul(25, 3/10)",
      "after": "Left sum ORIGINAL sample23 slope times x = 15/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample23 slope times x，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample23 slope times x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "3/10"
        ],
        "output": "15/2",
        "label": "Left sum ORIGINAL sample23 slope times x"
      }
    },
    {
      "id": "S168",
      "uses": [
        "G001",
        "S167"
      ],
      "before": "Left sum ORIGINAL sample23 affine value plus c := sum(15/2, 12)",
      "after": "Left sum ORIGINAL sample23 affine value plus c = 39/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample23 affine value plus c，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample23 affine value plus c with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "15/2",
          "12"
        ],
        "output": "39/2",
        "label": "Left sum ORIGINAL sample23 affine value plus c"
      }
    },
    {
      "id": "S169",
      "uses": [
        "G001",
        "S168"
      ],
      "before": "Left sum ORIGINAL sample23 value times cell width := mul(39/2, 1/10)",
      "after": "Left sum ORIGINAL sample23 value times cell width = 39/20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample23 value times cell width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample23 value times cell width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "39/2",
          "1/10"
        ],
        "output": "39/20",
        "label": "Left sum ORIGINAL sample23 value times cell width"
      }
    },
    {
      "id": "S170",
      "uses": [
        "G001",
        "S169"
      ],
      "before": "Left sum ORIGINAL accumulated term23 := sum(-483/20, 39/20)",
      "after": "Left sum ORIGINAL accumulated term23 = -111/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term23，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL accumulated term23 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-483/20",
          "39/20"
        ],
        "output": "-111/5",
        "label": "Left sum ORIGINAL accumulated term23"
      }
    },
    {
      "id": "S171",
      "uses": [
        "G001",
        "S170"
      ],
      "before": "source roles = [23, 50, 25, 12, -2, 3, 1/10]",
      "after": "derived = [3/10, 39/2, 39/20, -111/5]",
      "relation": "implies",
      "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。",
        "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S172",
      "uses": [
        "G001",
        "S171"
      ],
      "before": "Left sum ORIGINAL sample24 index times width := mul(24, 1/10)",
      "after": "Left sum ORIGINAL sample24 index times width = 12/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample24 index times width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample24 index times width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "24",
          "1/10"
        ],
        "output": "12/5",
        "label": "Left sum ORIGINAL sample24 index times width"
      }
    },
    {
      "id": "S173",
      "uses": [
        "G001",
        "S172"
      ],
      "before": "Left sum ORIGINAL sample24 lower plus offset := sum(-2, 12/5)",
      "after": "Left sum ORIGINAL sample24 lower plus offset = 2/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample24 lower plus offset，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample24 lower plus offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "12/5"
        ],
        "output": "2/5",
        "label": "Left sum ORIGINAL sample24 lower plus offset"
      }
    },
    {
      "id": "S174",
      "uses": [
        "G001",
        "S173"
      ],
      "before": "Left sum ORIGINAL sample24 slope times x := mul(25, 2/5)",
      "after": "Left sum ORIGINAL sample24 slope times x = 10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample24 slope times x，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample24 slope times x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "2/5"
        ],
        "output": "10",
        "label": "Left sum ORIGINAL sample24 slope times x"
      }
    },
    {
      "id": "S175",
      "uses": [
        "G001",
        "S174"
      ],
      "before": "Left sum ORIGINAL sample24 affine value plus c := sum(10, 12)",
      "after": "Left sum ORIGINAL sample24 affine value plus c = 22",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample24 affine value plus c，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample24 affine value plus c with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "10",
          "12"
        ],
        "output": "22",
        "label": "Left sum ORIGINAL sample24 affine value plus c"
      }
    },
    {
      "id": "S176",
      "uses": [
        "G001",
        "S175"
      ],
      "before": "Left sum ORIGINAL sample24 value times cell width := mul(22, 1/10)",
      "after": "Left sum ORIGINAL sample24 value times cell width = 11/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample24 value times cell width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample24 value times cell width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "22",
          "1/10"
        ],
        "output": "11/5",
        "label": "Left sum ORIGINAL sample24 value times cell width"
      }
    },
    {
      "id": "S177",
      "uses": [
        "G001",
        "S176"
      ],
      "before": "Left sum ORIGINAL accumulated term24 := sum(-111/5, 11/5)",
      "after": "Left sum ORIGINAL accumulated term24 = -20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term24，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL accumulated term24 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-111/5",
          "11/5"
        ],
        "output": "-20",
        "label": "Left sum ORIGINAL accumulated term24"
      }
    },
    {
      "id": "S178",
      "uses": [
        "G001",
        "S177"
      ],
      "before": "source roles = [24, 50, 25, 12, -2, 3, 1/10]",
      "after": "derived = [2/5, 22/1, 11/5, -20/1]",
      "relation": "implies",
      "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。",
        "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S179",
      "uses": [
        "G001",
        "S178"
      ],
      "before": "Left sum ORIGINAL sample25 index times width := mul(25, 1/10)",
      "after": "Left sum ORIGINAL sample25 index times width = 5/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample25 index times width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample25 index times width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "1/10"
        ],
        "output": "5/2",
        "label": "Left sum ORIGINAL sample25 index times width"
      }
    },
    {
      "id": "S180",
      "uses": [
        "G001",
        "S179"
      ],
      "before": "Left sum ORIGINAL sample25 lower plus offset := sum(-2, 5/2)",
      "after": "Left sum ORIGINAL sample25 lower plus offset = 1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample25 lower plus offset，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample25 lower plus offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "5/2"
        ],
        "output": "1/2",
        "label": "Left sum ORIGINAL sample25 lower plus offset"
      }
    },
    {
      "id": "S181",
      "uses": [
        "G001",
        "S180"
      ],
      "before": "Left sum ORIGINAL sample25 slope times x := mul(25, 1/2)",
      "after": "Left sum ORIGINAL sample25 slope times x = 25/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample25 slope times x，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample25 slope times x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "1/2"
        ],
        "output": "25/2",
        "label": "Left sum ORIGINAL sample25 slope times x"
      }
    },
    {
      "id": "S182",
      "uses": [
        "G001",
        "S181"
      ],
      "before": "Left sum ORIGINAL sample25 affine value plus c := sum(25/2, 12)",
      "after": "Left sum ORIGINAL sample25 affine value plus c = 49/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample25 affine value plus c，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample25 affine value plus c with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "25/2",
          "12"
        ],
        "output": "49/2",
        "label": "Left sum ORIGINAL sample25 affine value plus c"
      }
    },
    {
      "id": "S183",
      "uses": [
        "G001",
        "S182"
      ],
      "before": "Left sum ORIGINAL sample25 value times cell width := mul(49/2, 1/10)",
      "after": "Left sum ORIGINAL sample25 value times cell width = 49/20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample25 value times cell width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample25 value times cell width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "49/2",
          "1/10"
        ],
        "output": "49/20",
        "label": "Left sum ORIGINAL sample25 value times cell width"
      }
    },
    {
      "id": "S184",
      "uses": [
        "G001",
        "S183"
      ],
      "before": "Left sum ORIGINAL accumulated term25 := sum(-20, 49/20)",
      "after": "Left sum ORIGINAL accumulated term25 = -351/20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term25，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL accumulated term25 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-20",
          "49/20"
        ],
        "output": "-351/20",
        "label": "Left sum ORIGINAL accumulated term25"
      }
    },
    {
      "id": "S185",
      "uses": [
        "G001",
        "S184"
      ],
      "before": "source roles = [25, 50, 25, 12, -2, 3, 1/10]",
      "after": "derived = [1/2, 49/2, 49/20, -351/20]",
      "relation": "implies",
      "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。",
        "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S186",
      "uses": [
        "G001",
        "S185"
      ],
      "before": "Left sum ORIGINAL sample26 index times width := mul(26, 1/10)",
      "after": "Left sum ORIGINAL sample26 index times width = 13/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample26 index times width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample26 index times width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "26",
          "1/10"
        ],
        "output": "13/5",
        "label": "Left sum ORIGINAL sample26 index times width"
      }
    },
    {
      "id": "S187",
      "uses": [
        "G001",
        "S186"
      ],
      "before": "Left sum ORIGINAL sample26 lower plus offset := sum(-2, 13/5)",
      "after": "Left sum ORIGINAL sample26 lower plus offset = 3/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample26 lower plus offset，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample26 lower plus offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "13/5"
        ],
        "output": "3/5",
        "label": "Left sum ORIGINAL sample26 lower plus offset"
      }
    },
    {
      "id": "S188",
      "uses": [
        "G001",
        "S187"
      ],
      "before": "Left sum ORIGINAL sample26 slope times x := mul(25, 3/5)",
      "after": "Left sum ORIGINAL sample26 slope times x = 15",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample26 slope times x，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample26 slope times x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "3/5"
        ],
        "output": "15",
        "label": "Left sum ORIGINAL sample26 slope times x"
      }
    },
    {
      "id": "S189",
      "uses": [
        "G001",
        "S188"
      ],
      "before": "Left sum ORIGINAL sample26 affine value plus c := sum(15, 12)",
      "after": "Left sum ORIGINAL sample26 affine value plus c = 27",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample26 affine value plus c，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample26 affine value plus c with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "15",
          "12"
        ],
        "output": "27",
        "label": "Left sum ORIGINAL sample26 affine value plus c"
      }
    },
    {
      "id": "S190",
      "uses": [
        "G001",
        "S189"
      ],
      "before": "Left sum ORIGINAL sample26 value times cell width := mul(27, 1/10)",
      "after": "Left sum ORIGINAL sample26 value times cell width = 27/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample26 value times cell width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample26 value times cell width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "27",
          "1/10"
        ],
        "output": "27/10",
        "label": "Left sum ORIGINAL sample26 value times cell width"
      }
    },
    {
      "id": "S191",
      "uses": [
        "G001",
        "S190"
      ],
      "before": "Left sum ORIGINAL accumulated term26 := sum(-351/20, 27/10)",
      "after": "Left sum ORIGINAL accumulated term26 = -297/20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term26，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL accumulated term26 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-351/20",
          "27/10"
        ],
        "output": "-297/20",
        "label": "Left sum ORIGINAL accumulated term26"
      }
    },
    {
      "id": "S192",
      "uses": [
        "G001",
        "S191"
      ],
      "before": "source roles = [26, 50, 25, 12, -2, 3, 1/10]",
      "after": "derived = [3/5, 27/1, 27/10, -297/20]",
      "relation": "implies",
      "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。",
        "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S193",
      "uses": [
        "G001",
        "S192"
      ],
      "before": "Left sum ORIGINAL sample27 index times width := mul(27, 1/10)",
      "after": "Left sum ORIGINAL sample27 index times width = 27/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample27 index times width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample27 index times width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "27",
          "1/10"
        ],
        "output": "27/10",
        "label": "Left sum ORIGINAL sample27 index times width"
      }
    },
    {
      "id": "S194",
      "uses": [
        "G001",
        "S193"
      ],
      "before": "Left sum ORIGINAL sample27 lower plus offset := sum(-2, 27/10)",
      "after": "Left sum ORIGINAL sample27 lower plus offset = 7/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample27 lower plus offset，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample27 lower plus offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "27/10"
        ],
        "output": "7/10",
        "label": "Left sum ORIGINAL sample27 lower plus offset"
      }
    },
    {
      "id": "S195",
      "uses": [
        "G001",
        "S194"
      ],
      "before": "Left sum ORIGINAL sample27 slope times x := mul(25, 7/10)",
      "after": "Left sum ORIGINAL sample27 slope times x = 35/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample27 slope times x，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample27 slope times x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "7/10"
        ],
        "output": "35/2",
        "label": "Left sum ORIGINAL sample27 slope times x"
      }
    },
    {
      "id": "S196",
      "uses": [
        "G001",
        "S195"
      ],
      "before": "Left sum ORIGINAL sample27 affine value plus c := sum(35/2, 12)",
      "after": "Left sum ORIGINAL sample27 affine value plus c = 59/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample27 affine value plus c，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample27 affine value plus c with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "35/2",
          "12"
        ],
        "output": "59/2",
        "label": "Left sum ORIGINAL sample27 affine value plus c"
      }
    },
    {
      "id": "S197",
      "uses": [
        "G001",
        "S196"
      ],
      "before": "Left sum ORIGINAL sample27 value times cell width := mul(59/2, 1/10)",
      "after": "Left sum ORIGINAL sample27 value times cell width = 59/20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample27 value times cell width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample27 value times cell width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "59/2",
          "1/10"
        ],
        "output": "59/20",
        "label": "Left sum ORIGINAL sample27 value times cell width"
      }
    },
    {
      "id": "S198",
      "uses": [
        "G001",
        "S197"
      ],
      "before": "Left sum ORIGINAL accumulated term27 := sum(-297/20, 59/20)",
      "after": "Left sum ORIGINAL accumulated term27 = -119/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term27，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL accumulated term27 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-297/20",
          "59/20"
        ],
        "output": "-119/10",
        "label": "Left sum ORIGINAL accumulated term27"
      }
    },
    {
      "id": "S199",
      "uses": [
        "G001",
        "S198"
      ],
      "before": "source roles = [27, 50, 25, 12, -2, 3, 1/10]",
      "after": "derived = [7/10, 59/2, 59/20, -119/10]",
      "relation": "implies",
      "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。",
        "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S200",
      "uses": [
        "G001",
        "S199"
      ],
      "before": "Left sum ORIGINAL sample28 index times width := mul(28, 1/10)",
      "after": "Left sum ORIGINAL sample28 index times width = 14/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample28 index times width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample28 index times width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "28",
          "1/10"
        ],
        "output": "14/5",
        "label": "Left sum ORIGINAL sample28 index times width"
      }
    },
    {
      "id": "S201",
      "uses": [
        "G001",
        "S200"
      ],
      "before": "Left sum ORIGINAL sample28 lower plus offset := sum(-2, 14/5)",
      "after": "Left sum ORIGINAL sample28 lower plus offset = 4/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample28 lower plus offset，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample28 lower plus offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "14/5"
        ],
        "output": "4/5",
        "label": "Left sum ORIGINAL sample28 lower plus offset"
      }
    },
    {
      "id": "S202",
      "uses": [
        "G001",
        "S201"
      ],
      "before": "Left sum ORIGINAL sample28 slope times x := mul(25, 4/5)",
      "after": "Left sum ORIGINAL sample28 slope times x = 20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample28 slope times x，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample28 slope times x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "4/5"
        ],
        "output": "20",
        "label": "Left sum ORIGINAL sample28 slope times x"
      }
    },
    {
      "id": "S203",
      "uses": [
        "G001",
        "S202"
      ],
      "before": "Left sum ORIGINAL sample28 affine value plus c := sum(20, 12)",
      "after": "Left sum ORIGINAL sample28 affine value plus c = 32",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample28 affine value plus c，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample28 affine value plus c with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "20",
          "12"
        ],
        "output": "32",
        "label": "Left sum ORIGINAL sample28 affine value plus c"
      }
    },
    {
      "id": "S204",
      "uses": [
        "G001",
        "S203"
      ],
      "before": "Left sum ORIGINAL sample28 value times cell width := mul(32, 1/10)",
      "after": "Left sum ORIGINAL sample28 value times cell width = 16/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample28 value times cell width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample28 value times cell width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "32",
          "1/10"
        ],
        "output": "16/5",
        "label": "Left sum ORIGINAL sample28 value times cell width"
      }
    },
    {
      "id": "S205",
      "uses": [
        "G001",
        "S204"
      ],
      "before": "Left sum ORIGINAL accumulated term28 := sum(-119/10, 16/5)",
      "after": "Left sum ORIGINAL accumulated term28 = -87/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term28，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL accumulated term28 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-119/10",
          "16/5"
        ],
        "output": "-87/10",
        "label": "Left sum ORIGINAL accumulated term28"
      }
    },
    {
      "id": "S206",
      "uses": [
        "G001",
        "S205"
      ],
      "before": "source roles = [28, 50, 25, 12, -2, 3, 1/10]",
      "after": "derived = [4/5, 32/1, 16/5, -87/10]",
      "relation": "implies",
      "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。",
        "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S207",
      "uses": [
        "G001",
        "S206"
      ],
      "before": "Left sum ORIGINAL sample29 index times width := mul(29, 1/10)",
      "after": "Left sum ORIGINAL sample29 index times width = 29/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample29 index times width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample29 index times width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "29",
          "1/10"
        ],
        "output": "29/10",
        "label": "Left sum ORIGINAL sample29 index times width"
      }
    },
    {
      "id": "S208",
      "uses": [
        "G001",
        "S207"
      ],
      "before": "Left sum ORIGINAL sample29 lower plus offset := sum(-2, 29/10)",
      "after": "Left sum ORIGINAL sample29 lower plus offset = 9/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample29 lower plus offset，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample29 lower plus offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "29/10"
        ],
        "output": "9/10",
        "label": "Left sum ORIGINAL sample29 lower plus offset"
      }
    },
    {
      "id": "S209",
      "uses": [
        "G001",
        "S208"
      ],
      "before": "Left sum ORIGINAL sample29 slope times x := mul(25, 9/10)",
      "after": "Left sum ORIGINAL sample29 slope times x = 45/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample29 slope times x，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample29 slope times x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "9/10"
        ],
        "output": "45/2",
        "label": "Left sum ORIGINAL sample29 slope times x"
      }
    },
    {
      "id": "S210",
      "uses": [
        "G001",
        "S209"
      ],
      "before": "Left sum ORIGINAL sample29 affine value plus c := sum(45/2, 12)",
      "after": "Left sum ORIGINAL sample29 affine value plus c = 69/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample29 affine value plus c，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample29 affine value plus c with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "45/2",
          "12"
        ],
        "output": "69/2",
        "label": "Left sum ORIGINAL sample29 affine value plus c"
      }
    },
    {
      "id": "S211",
      "uses": [
        "G001",
        "S210"
      ],
      "before": "Left sum ORIGINAL sample29 value times cell width := mul(69/2, 1/10)",
      "after": "Left sum ORIGINAL sample29 value times cell width = 69/20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample29 value times cell width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample29 value times cell width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "69/2",
          "1/10"
        ],
        "output": "69/20",
        "label": "Left sum ORIGINAL sample29 value times cell width"
      }
    },
    {
      "id": "S212",
      "uses": [
        "G001",
        "S211"
      ],
      "before": "Left sum ORIGINAL accumulated term29 := sum(-87/10, 69/20)",
      "after": "Left sum ORIGINAL accumulated term29 = -21/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term29，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL accumulated term29 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-87/10",
          "69/20"
        ],
        "output": "-21/4",
        "label": "Left sum ORIGINAL accumulated term29"
      }
    },
    {
      "id": "S213",
      "uses": [
        "G001",
        "S212"
      ],
      "before": "source roles = [29, 50, 25, 12, -2, 3, 1/10]",
      "after": "derived = [9/10, 69/2, 69/20, -21/4]",
      "relation": "implies",
      "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。",
        "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S214",
      "uses": [
        "G001",
        "S213"
      ],
      "before": "Left sum ORIGINAL sample30 index times width := mul(30, 1/10)",
      "after": "Left sum ORIGINAL sample30 index times width = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample30 index times width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample30 index times width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "30",
          "1/10"
        ],
        "output": "3",
        "label": "Left sum ORIGINAL sample30 index times width"
      }
    },
    {
      "id": "S215",
      "uses": [
        "G001",
        "S214"
      ],
      "before": "Left sum ORIGINAL sample30 lower plus offset := sum(-2, 3)",
      "after": "Left sum ORIGINAL sample30 lower plus offset = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample30 lower plus offset，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample30 lower plus offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "3"
        ],
        "output": "1",
        "label": "Left sum ORIGINAL sample30 lower plus offset"
      }
    },
    {
      "id": "S216",
      "uses": [
        "G001",
        "S215"
      ],
      "before": "Left sum ORIGINAL sample30 slope times x := mul(25, 1)",
      "after": "Left sum ORIGINAL sample30 slope times x = 25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample30 slope times x，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample30 slope times x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "1"
        ],
        "output": "25",
        "label": "Left sum ORIGINAL sample30 slope times x"
      }
    },
    {
      "id": "S217",
      "uses": [
        "G001",
        "S216"
      ],
      "before": "Left sum ORIGINAL sample30 affine value plus c := sum(25, 12)",
      "after": "Left sum ORIGINAL sample30 affine value plus c = 37",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample30 affine value plus c，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample30 affine value plus c with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "25",
          "12"
        ],
        "output": "37",
        "label": "Left sum ORIGINAL sample30 affine value plus c"
      }
    },
    {
      "id": "S218",
      "uses": [
        "G001",
        "S217"
      ],
      "before": "Left sum ORIGINAL sample30 value times cell width := mul(37, 1/10)",
      "after": "Left sum ORIGINAL sample30 value times cell width = 37/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample30 value times cell width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample30 value times cell width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "37",
          "1/10"
        ],
        "output": "37/10",
        "label": "Left sum ORIGINAL sample30 value times cell width"
      }
    },
    {
      "id": "S219",
      "uses": [
        "G001",
        "S218"
      ],
      "before": "Left sum ORIGINAL accumulated term30 := sum(-21/4, 37/10)",
      "after": "Left sum ORIGINAL accumulated term30 = -31/20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term30，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL accumulated term30 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-21/4",
          "37/10"
        ],
        "output": "-31/20",
        "label": "Left sum ORIGINAL accumulated term30"
      }
    },
    {
      "id": "S220",
      "uses": [
        "G001",
        "S219"
      ],
      "before": "source roles = [30, 50, 25, 12, -2, 3, 1/10]",
      "after": "derived = [1/1, 37/1, 37/10, -31/20]",
      "relation": "implies",
      "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。",
        "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S221",
      "uses": [
        "G001",
        "S220"
      ],
      "before": "Left sum ORIGINAL sample31 index times width := mul(31, 1/10)",
      "after": "Left sum ORIGINAL sample31 index times width = 31/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample31 index times width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample31 index times width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "31",
          "1/10"
        ],
        "output": "31/10",
        "label": "Left sum ORIGINAL sample31 index times width"
      }
    },
    {
      "id": "S222",
      "uses": [
        "G001",
        "S221"
      ],
      "before": "Left sum ORIGINAL sample31 lower plus offset := sum(-2, 31/10)",
      "after": "Left sum ORIGINAL sample31 lower plus offset = 11/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample31 lower plus offset，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample31 lower plus offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "31/10"
        ],
        "output": "11/10",
        "label": "Left sum ORIGINAL sample31 lower plus offset"
      }
    },
    {
      "id": "S223",
      "uses": [
        "G001",
        "S222"
      ],
      "before": "Left sum ORIGINAL sample31 slope times x := mul(25, 11/10)",
      "after": "Left sum ORIGINAL sample31 slope times x = 55/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample31 slope times x，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample31 slope times x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "11/10"
        ],
        "output": "55/2",
        "label": "Left sum ORIGINAL sample31 slope times x"
      }
    },
    {
      "id": "S224",
      "uses": [
        "G001",
        "S223"
      ],
      "before": "Left sum ORIGINAL sample31 affine value plus c := sum(55/2, 12)",
      "after": "Left sum ORIGINAL sample31 affine value plus c = 79/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample31 affine value plus c，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample31 affine value plus c with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "55/2",
          "12"
        ],
        "output": "79/2",
        "label": "Left sum ORIGINAL sample31 affine value plus c"
      }
    },
    {
      "id": "S225",
      "uses": [
        "G001",
        "S224"
      ],
      "before": "Left sum ORIGINAL sample31 value times cell width := mul(79/2, 1/10)",
      "after": "Left sum ORIGINAL sample31 value times cell width = 79/20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample31 value times cell width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample31 value times cell width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "79/2",
          "1/10"
        ],
        "output": "79/20",
        "label": "Left sum ORIGINAL sample31 value times cell width"
      }
    },
    {
      "id": "S226",
      "uses": [
        "G001",
        "S225"
      ],
      "before": "Left sum ORIGINAL accumulated term31 := sum(-31/20, 79/20)",
      "after": "Left sum ORIGINAL accumulated term31 = 12/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term31，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL accumulated term31 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-31/20",
          "79/20"
        ],
        "output": "12/5",
        "label": "Left sum ORIGINAL accumulated term31"
      }
    },
    {
      "id": "S227",
      "uses": [
        "G001",
        "S226"
      ],
      "before": "source roles = [31, 50, 25, 12, -2, 3, 1/10]",
      "after": "derived = [11/10, 79/2, 79/20, 12/5]",
      "relation": "implies",
      "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。",
        "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S228",
      "uses": [
        "G001",
        "S227"
      ],
      "before": "Left sum ORIGINAL sample32 index times width := mul(32, 1/10)",
      "after": "Left sum ORIGINAL sample32 index times width = 16/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample32 index times width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample32 index times width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "32",
          "1/10"
        ],
        "output": "16/5",
        "label": "Left sum ORIGINAL sample32 index times width"
      }
    },
    {
      "id": "S229",
      "uses": [
        "G001",
        "S228"
      ],
      "before": "Left sum ORIGINAL sample32 lower plus offset := sum(-2, 16/5)",
      "after": "Left sum ORIGINAL sample32 lower plus offset = 6/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample32 lower plus offset，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample32 lower plus offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "16/5"
        ],
        "output": "6/5",
        "label": "Left sum ORIGINAL sample32 lower plus offset"
      }
    },
    {
      "id": "S230",
      "uses": [
        "G001",
        "S229"
      ],
      "before": "Left sum ORIGINAL sample32 slope times x := mul(25, 6/5)",
      "after": "Left sum ORIGINAL sample32 slope times x = 30",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample32 slope times x，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample32 slope times x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "6/5"
        ],
        "output": "30",
        "label": "Left sum ORIGINAL sample32 slope times x"
      }
    },
    {
      "id": "S231",
      "uses": [
        "G001",
        "S230"
      ],
      "before": "Left sum ORIGINAL sample32 affine value plus c := sum(30, 12)",
      "after": "Left sum ORIGINAL sample32 affine value plus c = 42",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample32 affine value plus c，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample32 affine value plus c with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "30",
          "12"
        ],
        "output": "42",
        "label": "Left sum ORIGINAL sample32 affine value plus c"
      }
    },
    {
      "id": "S232",
      "uses": [
        "G001",
        "S231"
      ],
      "before": "Left sum ORIGINAL sample32 value times cell width := mul(42, 1/10)",
      "after": "Left sum ORIGINAL sample32 value times cell width = 21/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample32 value times cell width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample32 value times cell width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "42",
          "1/10"
        ],
        "output": "21/5",
        "label": "Left sum ORIGINAL sample32 value times cell width"
      }
    },
    {
      "id": "S233",
      "uses": [
        "G001",
        "S232"
      ],
      "before": "Left sum ORIGINAL accumulated term32 := sum(12/5, 21/5)",
      "after": "Left sum ORIGINAL accumulated term32 = 33/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term32，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL accumulated term32 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "12/5",
          "21/5"
        ],
        "output": "33/5",
        "label": "Left sum ORIGINAL accumulated term32"
      }
    },
    {
      "id": "S234",
      "uses": [
        "G001",
        "S233"
      ],
      "before": "source roles = [32, 50, 25, 12, -2, 3, 1/10]",
      "after": "derived = [6/5, 42/1, 21/5, 33/5]",
      "relation": "implies",
      "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。",
        "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S235",
      "uses": [
        "G001",
        "S234"
      ],
      "before": "Left sum ORIGINAL sample33 index times width := mul(33, 1/10)",
      "after": "Left sum ORIGINAL sample33 index times width = 33/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample33 index times width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample33 index times width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "33",
          "1/10"
        ],
        "output": "33/10",
        "label": "Left sum ORIGINAL sample33 index times width"
      }
    },
    {
      "id": "S236",
      "uses": [
        "G001",
        "S235"
      ],
      "before": "Left sum ORIGINAL sample33 lower plus offset := sum(-2, 33/10)",
      "after": "Left sum ORIGINAL sample33 lower plus offset = 13/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample33 lower plus offset，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample33 lower plus offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "33/10"
        ],
        "output": "13/10",
        "label": "Left sum ORIGINAL sample33 lower plus offset"
      }
    },
    {
      "id": "S237",
      "uses": [
        "G001",
        "S236"
      ],
      "before": "Left sum ORIGINAL sample33 slope times x := mul(25, 13/10)",
      "after": "Left sum ORIGINAL sample33 slope times x = 65/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample33 slope times x，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample33 slope times x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "13/10"
        ],
        "output": "65/2",
        "label": "Left sum ORIGINAL sample33 slope times x"
      }
    },
    {
      "id": "S238",
      "uses": [
        "G001",
        "S237"
      ],
      "before": "Left sum ORIGINAL sample33 affine value plus c := sum(65/2, 12)",
      "after": "Left sum ORIGINAL sample33 affine value plus c = 89/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample33 affine value plus c，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample33 affine value plus c with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "65/2",
          "12"
        ],
        "output": "89/2",
        "label": "Left sum ORIGINAL sample33 affine value plus c"
      }
    },
    {
      "id": "S239",
      "uses": [
        "G001",
        "S238"
      ],
      "before": "Left sum ORIGINAL sample33 value times cell width := mul(89/2, 1/10)",
      "after": "Left sum ORIGINAL sample33 value times cell width = 89/20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample33 value times cell width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample33 value times cell width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "89/2",
          "1/10"
        ],
        "output": "89/20",
        "label": "Left sum ORIGINAL sample33 value times cell width"
      }
    },
    {
      "id": "S240",
      "uses": [
        "G001",
        "S239"
      ],
      "before": "Left sum ORIGINAL accumulated term33 := sum(33/5, 89/20)",
      "after": "Left sum ORIGINAL accumulated term33 = 221/20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term33，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL accumulated term33 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "33/5",
          "89/20"
        ],
        "output": "221/20",
        "label": "Left sum ORIGINAL accumulated term33"
      }
    },
    {
      "id": "S241",
      "uses": [
        "G001",
        "S240"
      ],
      "before": "source roles = [33, 50, 25, 12, -2, 3, 1/10]",
      "after": "derived = [13/10, 89/2, 89/20, 221/20]",
      "relation": "implies",
      "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。",
        "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S242",
      "uses": [
        "G001",
        "S241"
      ],
      "before": "Left sum ORIGINAL sample34 index times width := mul(34, 1/10)",
      "after": "Left sum ORIGINAL sample34 index times width = 17/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample34 index times width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample34 index times width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "34",
          "1/10"
        ],
        "output": "17/5",
        "label": "Left sum ORIGINAL sample34 index times width"
      }
    },
    {
      "id": "S243",
      "uses": [
        "G001",
        "S242"
      ],
      "before": "Left sum ORIGINAL sample34 lower plus offset := sum(-2, 17/5)",
      "after": "Left sum ORIGINAL sample34 lower plus offset = 7/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample34 lower plus offset，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample34 lower plus offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "17/5"
        ],
        "output": "7/5",
        "label": "Left sum ORIGINAL sample34 lower plus offset"
      }
    },
    {
      "id": "S244",
      "uses": [
        "G001",
        "S243"
      ],
      "before": "Left sum ORIGINAL sample34 slope times x := mul(25, 7/5)",
      "after": "Left sum ORIGINAL sample34 slope times x = 35",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample34 slope times x，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample34 slope times x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "7/5"
        ],
        "output": "35",
        "label": "Left sum ORIGINAL sample34 slope times x"
      }
    },
    {
      "id": "S245",
      "uses": [
        "G001",
        "S244"
      ],
      "before": "Left sum ORIGINAL sample34 affine value plus c := sum(35, 12)",
      "after": "Left sum ORIGINAL sample34 affine value plus c = 47",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample34 affine value plus c，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample34 affine value plus c with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "35",
          "12"
        ],
        "output": "47",
        "label": "Left sum ORIGINAL sample34 affine value plus c"
      }
    },
    {
      "id": "S246",
      "uses": [
        "G001",
        "S245"
      ],
      "before": "Left sum ORIGINAL sample34 value times cell width := mul(47, 1/10)",
      "after": "Left sum ORIGINAL sample34 value times cell width = 47/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample34 value times cell width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample34 value times cell width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "47",
          "1/10"
        ],
        "output": "47/10",
        "label": "Left sum ORIGINAL sample34 value times cell width"
      }
    },
    {
      "id": "S247",
      "uses": [
        "G001",
        "S246"
      ],
      "before": "Left sum ORIGINAL accumulated term34 := sum(221/20, 47/10)",
      "after": "Left sum ORIGINAL accumulated term34 = 63/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term34，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL accumulated term34 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "221/20",
          "47/10"
        ],
        "output": "63/4",
        "label": "Left sum ORIGINAL accumulated term34"
      }
    },
    {
      "id": "S248",
      "uses": [
        "G001",
        "S247"
      ],
      "before": "source roles = [34, 50, 25, 12, -2, 3, 1/10]",
      "after": "derived = [7/5, 47/1, 47/10, 63/4]",
      "relation": "implies",
      "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。",
        "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S249",
      "uses": [
        "G001",
        "S248"
      ],
      "before": "Left sum ORIGINAL sample35 index times width := mul(35, 1/10)",
      "after": "Left sum ORIGINAL sample35 index times width = 7/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample35 index times width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample35 index times width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "35",
          "1/10"
        ],
        "output": "7/2",
        "label": "Left sum ORIGINAL sample35 index times width"
      }
    },
    {
      "id": "S250",
      "uses": [
        "G001",
        "S249"
      ],
      "before": "Left sum ORIGINAL sample35 lower plus offset := sum(-2, 7/2)",
      "after": "Left sum ORIGINAL sample35 lower plus offset = 3/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample35 lower plus offset，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample35 lower plus offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "7/2"
        ],
        "output": "3/2",
        "label": "Left sum ORIGINAL sample35 lower plus offset"
      }
    },
    {
      "id": "S251",
      "uses": [
        "G001",
        "S250"
      ],
      "before": "Left sum ORIGINAL sample35 slope times x := mul(25, 3/2)",
      "after": "Left sum ORIGINAL sample35 slope times x = 75/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample35 slope times x，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample35 slope times x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "3/2"
        ],
        "output": "75/2",
        "label": "Left sum ORIGINAL sample35 slope times x"
      }
    },
    {
      "id": "S252",
      "uses": [
        "G001",
        "S251"
      ],
      "before": "Left sum ORIGINAL sample35 affine value plus c := sum(75/2, 12)",
      "after": "Left sum ORIGINAL sample35 affine value plus c = 99/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample35 affine value plus c，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample35 affine value plus c with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "75/2",
          "12"
        ],
        "output": "99/2",
        "label": "Left sum ORIGINAL sample35 affine value plus c"
      }
    },
    {
      "id": "S253",
      "uses": [
        "G001",
        "S252"
      ],
      "before": "Left sum ORIGINAL sample35 value times cell width := mul(99/2, 1/10)",
      "after": "Left sum ORIGINAL sample35 value times cell width = 99/20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample35 value times cell width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample35 value times cell width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "99/2",
          "1/10"
        ],
        "output": "99/20",
        "label": "Left sum ORIGINAL sample35 value times cell width"
      }
    },
    {
      "id": "S254",
      "uses": [
        "G001",
        "S253"
      ],
      "before": "Left sum ORIGINAL accumulated term35 := sum(63/4, 99/20)",
      "after": "Left sum ORIGINAL accumulated term35 = 207/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term35，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL accumulated term35 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "63/4",
          "99/20"
        ],
        "output": "207/10",
        "label": "Left sum ORIGINAL accumulated term35"
      }
    },
    {
      "id": "S255",
      "uses": [
        "G001",
        "S254"
      ],
      "before": "source roles = [35, 50, 25, 12, -2, 3, 1/10]",
      "after": "derived = [3/2, 99/2, 99/20, 207/10]",
      "relation": "implies",
      "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。",
        "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S256",
      "uses": [
        "G001",
        "S255"
      ],
      "before": "Left sum ORIGINAL sample36 index times width := mul(36, 1/10)",
      "after": "Left sum ORIGINAL sample36 index times width = 18/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample36 index times width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample36 index times width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "36",
          "1/10"
        ],
        "output": "18/5",
        "label": "Left sum ORIGINAL sample36 index times width"
      }
    },
    {
      "id": "S257",
      "uses": [
        "G001",
        "S256"
      ],
      "before": "Left sum ORIGINAL sample36 lower plus offset := sum(-2, 18/5)",
      "after": "Left sum ORIGINAL sample36 lower plus offset = 8/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample36 lower plus offset，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample36 lower plus offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "18/5"
        ],
        "output": "8/5",
        "label": "Left sum ORIGINAL sample36 lower plus offset"
      }
    },
    {
      "id": "S258",
      "uses": [
        "G001",
        "S257"
      ],
      "before": "Left sum ORIGINAL sample36 slope times x := mul(25, 8/5)",
      "after": "Left sum ORIGINAL sample36 slope times x = 40",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample36 slope times x，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample36 slope times x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "8/5"
        ],
        "output": "40",
        "label": "Left sum ORIGINAL sample36 slope times x"
      }
    },
    {
      "id": "S259",
      "uses": [
        "G001",
        "S258"
      ],
      "before": "Left sum ORIGINAL sample36 affine value plus c := sum(40, 12)",
      "after": "Left sum ORIGINAL sample36 affine value plus c = 52",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample36 affine value plus c，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample36 affine value plus c with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "40",
          "12"
        ],
        "output": "52",
        "label": "Left sum ORIGINAL sample36 affine value plus c"
      }
    },
    {
      "id": "S260",
      "uses": [
        "G001",
        "S259"
      ],
      "before": "Left sum ORIGINAL sample36 value times cell width := mul(52, 1/10)",
      "after": "Left sum ORIGINAL sample36 value times cell width = 26/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample36 value times cell width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample36 value times cell width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "52",
          "1/10"
        ],
        "output": "26/5",
        "label": "Left sum ORIGINAL sample36 value times cell width"
      }
    },
    {
      "id": "S261",
      "uses": [
        "G001",
        "S260"
      ],
      "before": "Left sum ORIGINAL accumulated term36 := sum(207/10, 26/5)",
      "after": "Left sum ORIGINAL accumulated term36 = 259/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term36，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL accumulated term36 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "207/10",
          "26/5"
        ],
        "output": "259/10",
        "label": "Left sum ORIGINAL accumulated term36"
      }
    },
    {
      "id": "S262",
      "uses": [
        "G001",
        "S261"
      ],
      "before": "source roles = [36, 50, 25, 12, -2, 3, 1/10]",
      "after": "derived = [8/5, 52/1, 26/5, 259/10]",
      "relation": "implies",
      "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。",
        "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S263",
      "uses": [
        "G001",
        "S262"
      ],
      "before": "Left sum ORIGINAL sample37 index times width := mul(37, 1/10)",
      "after": "Left sum ORIGINAL sample37 index times width = 37/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample37 index times width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample37 index times width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "37",
          "1/10"
        ],
        "output": "37/10",
        "label": "Left sum ORIGINAL sample37 index times width"
      }
    },
    {
      "id": "S264",
      "uses": [
        "G001",
        "S263"
      ],
      "before": "Left sum ORIGINAL sample37 lower plus offset := sum(-2, 37/10)",
      "after": "Left sum ORIGINAL sample37 lower plus offset = 17/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample37 lower plus offset，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample37 lower plus offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "37/10"
        ],
        "output": "17/10",
        "label": "Left sum ORIGINAL sample37 lower plus offset"
      }
    },
    {
      "id": "S265",
      "uses": [
        "G001",
        "S264"
      ],
      "before": "Left sum ORIGINAL sample37 slope times x := mul(25, 17/10)",
      "after": "Left sum ORIGINAL sample37 slope times x = 85/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample37 slope times x，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample37 slope times x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "17/10"
        ],
        "output": "85/2",
        "label": "Left sum ORIGINAL sample37 slope times x"
      }
    },
    {
      "id": "S266",
      "uses": [
        "G001",
        "S265"
      ],
      "before": "Left sum ORIGINAL sample37 affine value plus c := sum(85/2, 12)",
      "after": "Left sum ORIGINAL sample37 affine value plus c = 109/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample37 affine value plus c，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample37 affine value plus c with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "85/2",
          "12"
        ],
        "output": "109/2",
        "label": "Left sum ORIGINAL sample37 affine value plus c"
      }
    },
    {
      "id": "S267",
      "uses": [
        "G001",
        "S266"
      ],
      "before": "Left sum ORIGINAL sample37 value times cell width := mul(109/2, 1/10)",
      "after": "Left sum ORIGINAL sample37 value times cell width = 109/20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample37 value times cell width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample37 value times cell width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "109/2",
          "1/10"
        ],
        "output": "109/20",
        "label": "Left sum ORIGINAL sample37 value times cell width"
      }
    },
    {
      "id": "S268",
      "uses": [
        "G001",
        "S267"
      ],
      "before": "Left sum ORIGINAL accumulated term37 := sum(259/10, 109/20)",
      "after": "Left sum ORIGINAL accumulated term37 = 627/20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term37，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL accumulated term37 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "259/10",
          "109/20"
        ],
        "output": "627/20",
        "label": "Left sum ORIGINAL accumulated term37"
      }
    },
    {
      "id": "S269",
      "uses": [
        "G001",
        "S268"
      ],
      "before": "source roles = [37, 50, 25, 12, -2, 3, 1/10]",
      "after": "derived = [17/10, 109/2, 109/20, 627/20]",
      "relation": "implies",
      "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。",
        "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S270",
      "uses": [
        "G001",
        "S269"
      ],
      "before": "Left sum ORIGINAL sample38 index times width := mul(38, 1/10)",
      "after": "Left sum ORIGINAL sample38 index times width = 19/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample38 index times width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample38 index times width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "38",
          "1/10"
        ],
        "output": "19/5",
        "label": "Left sum ORIGINAL sample38 index times width"
      }
    },
    {
      "id": "S271",
      "uses": [
        "G001",
        "S270"
      ],
      "before": "Left sum ORIGINAL sample38 lower plus offset := sum(-2, 19/5)",
      "after": "Left sum ORIGINAL sample38 lower plus offset = 9/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample38 lower plus offset，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample38 lower plus offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "19/5"
        ],
        "output": "9/5",
        "label": "Left sum ORIGINAL sample38 lower plus offset"
      }
    },
    {
      "id": "S272",
      "uses": [
        "G001",
        "S271"
      ],
      "before": "Left sum ORIGINAL sample38 slope times x := mul(25, 9/5)",
      "after": "Left sum ORIGINAL sample38 slope times x = 45",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample38 slope times x，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample38 slope times x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "9/5"
        ],
        "output": "45",
        "label": "Left sum ORIGINAL sample38 slope times x"
      }
    },
    {
      "id": "S273",
      "uses": [
        "G001",
        "S272"
      ],
      "before": "Left sum ORIGINAL sample38 affine value plus c := sum(45, 12)",
      "after": "Left sum ORIGINAL sample38 affine value plus c = 57",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample38 affine value plus c，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample38 affine value plus c with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "45",
          "12"
        ],
        "output": "57",
        "label": "Left sum ORIGINAL sample38 affine value plus c"
      }
    },
    {
      "id": "S274",
      "uses": [
        "G001",
        "S273"
      ],
      "before": "Left sum ORIGINAL sample38 value times cell width := mul(57, 1/10)",
      "after": "Left sum ORIGINAL sample38 value times cell width = 57/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample38 value times cell width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample38 value times cell width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "57",
          "1/10"
        ],
        "output": "57/10",
        "label": "Left sum ORIGINAL sample38 value times cell width"
      }
    },
    {
      "id": "S275",
      "uses": [
        "G001",
        "S274"
      ],
      "before": "Left sum ORIGINAL accumulated term38 := sum(627/20, 57/10)",
      "after": "Left sum ORIGINAL accumulated term38 = 741/20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term38，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL accumulated term38 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "627/20",
          "57/10"
        ],
        "output": "741/20",
        "label": "Left sum ORIGINAL accumulated term38"
      }
    },
    {
      "id": "S276",
      "uses": [
        "G001",
        "S275"
      ],
      "before": "source roles = [38, 50, 25, 12, -2, 3, 1/10]",
      "after": "derived = [9/5, 57/1, 57/10, 741/20]",
      "relation": "implies",
      "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。",
        "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S277",
      "uses": [
        "G001",
        "S276"
      ],
      "before": "Left sum ORIGINAL sample39 index times width := mul(39, 1/10)",
      "after": "Left sum ORIGINAL sample39 index times width = 39/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample39 index times width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample39 index times width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "39",
          "1/10"
        ],
        "output": "39/10",
        "label": "Left sum ORIGINAL sample39 index times width"
      }
    },
    {
      "id": "S278",
      "uses": [
        "G001",
        "S277"
      ],
      "before": "Left sum ORIGINAL sample39 lower plus offset := sum(-2, 39/10)",
      "after": "Left sum ORIGINAL sample39 lower plus offset = 19/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample39 lower plus offset，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample39 lower plus offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "39/10"
        ],
        "output": "19/10",
        "label": "Left sum ORIGINAL sample39 lower plus offset"
      }
    },
    {
      "id": "S279",
      "uses": [
        "G001",
        "S278"
      ],
      "before": "Left sum ORIGINAL sample39 slope times x := mul(25, 19/10)",
      "after": "Left sum ORIGINAL sample39 slope times x = 95/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample39 slope times x，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample39 slope times x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "19/10"
        ],
        "output": "95/2",
        "label": "Left sum ORIGINAL sample39 slope times x"
      }
    },
    {
      "id": "S280",
      "uses": [
        "G001",
        "S279"
      ],
      "before": "Left sum ORIGINAL sample39 affine value plus c := sum(95/2, 12)",
      "after": "Left sum ORIGINAL sample39 affine value plus c = 119/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample39 affine value plus c，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample39 affine value plus c with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "95/2",
          "12"
        ],
        "output": "119/2",
        "label": "Left sum ORIGINAL sample39 affine value plus c"
      }
    },
    {
      "id": "S281",
      "uses": [
        "G001",
        "S280"
      ],
      "before": "Left sum ORIGINAL sample39 value times cell width := mul(119/2, 1/10)",
      "after": "Left sum ORIGINAL sample39 value times cell width = 119/20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample39 value times cell width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample39 value times cell width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "119/2",
          "1/10"
        ],
        "output": "119/20",
        "label": "Left sum ORIGINAL sample39 value times cell width"
      }
    },
    {
      "id": "S282",
      "uses": [
        "G001",
        "S281"
      ],
      "before": "Left sum ORIGINAL accumulated term39 := sum(741/20, 119/20)",
      "after": "Left sum ORIGINAL accumulated term39 = 43",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term39，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL accumulated term39 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "741/20",
          "119/20"
        ],
        "output": "43",
        "label": "Left sum ORIGINAL accumulated term39"
      }
    },
    {
      "id": "S283",
      "uses": [
        "G001",
        "S282"
      ],
      "before": "source roles = [39, 50, 25, 12, -2, 3, 1/10]",
      "after": "derived = [19/10, 119/2, 119/20, 43/1]",
      "relation": "implies",
      "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。",
        "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S284",
      "uses": [
        "G001",
        "S283"
      ],
      "before": "Left sum ORIGINAL sample40 index times width := mul(40, 1/10)",
      "after": "Left sum ORIGINAL sample40 index times width = 4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample40 index times width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample40 index times width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "40",
          "1/10"
        ],
        "output": "4",
        "label": "Left sum ORIGINAL sample40 index times width"
      }
    },
    {
      "id": "S285",
      "uses": [
        "G001",
        "S284"
      ],
      "before": "Left sum ORIGINAL sample40 lower plus offset := sum(-2, 4)",
      "after": "Left sum ORIGINAL sample40 lower plus offset = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample40 lower plus offset，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample40 lower plus offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "4"
        ],
        "output": "2",
        "label": "Left sum ORIGINAL sample40 lower plus offset"
      }
    },
    {
      "id": "S286",
      "uses": [
        "G001",
        "S285"
      ],
      "before": "Left sum ORIGINAL sample40 slope times x := mul(25, 2)",
      "after": "Left sum ORIGINAL sample40 slope times x = 50",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample40 slope times x，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample40 slope times x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "2"
        ],
        "output": "50",
        "label": "Left sum ORIGINAL sample40 slope times x"
      }
    },
    {
      "id": "S287",
      "uses": [
        "G001",
        "S286"
      ],
      "before": "Left sum ORIGINAL sample40 affine value plus c := sum(50, 12)",
      "after": "Left sum ORIGINAL sample40 affine value plus c = 62",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample40 affine value plus c，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample40 affine value plus c with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "50",
          "12"
        ],
        "output": "62",
        "label": "Left sum ORIGINAL sample40 affine value plus c"
      }
    },
    {
      "id": "S288",
      "uses": [
        "G001",
        "S287"
      ],
      "before": "Left sum ORIGINAL sample40 value times cell width := mul(62, 1/10)",
      "after": "Left sum ORIGINAL sample40 value times cell width = 31/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample40 value times cell width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample40 value times cell width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "62",
          "1/10"
        ],
        "output": "31/5",
        "label": "Left sum ORIGINAL sample40 value times cell width"
      }
    },
    {
      "id": "S289",
      "uses": [
        "G001",
        "S288"
      ],
      "before": "Left sum ORIGINAL accumulated term40 := sum(43, 31/5)",
      "after": "Left sum ORIGINAL accumulated term40 = 246/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term40，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL accumulated term40 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "43",
          "31/5"
        ],
        "output": "246/5",
        "label": "Left sum ORIGINAL accumulated term40"
      }
    },
    {
      "id": "S290",
      "uses": [
        "G001",
        "S289"
      ],
      "before": "source roles = [40, 50, 25, 12, -2, 3, 1/10]",
      "after": "derived = [2/1, 62/1, 31/5, 246/5]",
      "relation": "implies",
      "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。",
        "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S291",
      "uses": [
        "G001",
        "S290"
      ],
      "before": "Left sum ORIGINAL sample41 index times width := mul(41, 1/10)",
      "after": "Left sum ORIGINAL sample41 index times width = 41/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample41 index times width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample41 index times width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "41",
          "1/10"
        ],
        "output": "41/10",
        "label": "Left sum ORIGINAL sample41 index times width"
      }
    },
    {
      "id": "S292",
      "uses": [
        "G001",
        "S291"
      ],
      "before": "Left sum ORIGINAL sample41 lower plus offset := sum(-2, 41/10)",
      "after": "Left sum ORIGINAL sample41 lower plus offset = 21/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample41 lower plus offset，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample41 lower plus offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "41/10"
        ],
        "output": "21/10",
        "label": "Left sum ORIGINAL sample41 lower plus offset"
      }
    },
    {
      "id": "S293",
      "uses": [
        "G001",
        "S292"
      ],
      "before": "Left sum ORIGINAL sample41 slope times x := mul(25, 21/10)",
      "after": "Left sum ORIGINAL sample41 slope times x = 105/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample41 slope times x，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample41 slope times x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "21/10"
        ],
        "output": "105/2",
        "label": "Left sum ORIGINAL sample41 slope times x"
      }
    },
    {
      "id": "S294",
      "uses": [
        "G001",
        "S293"
      ],
      "before": "Left sum ORIGINAL sample41 affine value plus c := sum(105/2, 12)",
      "after": "Left sum ORIGINAL sample41 affine value plus c = 129/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample41 affine value plus c，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample41 affine value plus c with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "105/2",
          "12"
        ],
        "output": "129/2",
        "label": "Left sum ORIGINAL sample41 affine value plus c"
      }
    },
    {
      "id": "S295",
      "uses": [
        "G001",
        "S294"
      ],
      "before": "Left sum ORIGINAL sample41 value times cell width := mul(129/2, 1/10)",
      "after": "Left sum ORIGINAL sample41 value times cell width = 129/20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample41 value times cell width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample41 value times cell width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "129/2",
          "1/10"
        ],
        "output": "129/20",
        "label": "Left sum ORIGINAL sample41 value times cell width"
      }
    },
    {
      "id": "S296",
      "uses": [
        "G001",
        "S295"
      ],
      "before": "Left sum ORIGINAL accumulated term41 := sum(246/5, 129/20)",
      "after": "Left sum ORIGINAL accumulated term41 = 1113/20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term41，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL accumulated term41 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "246/5",
          "129/20"
        ],
        "output": "1113/20",
        "label": "Left sum ORIGINAL accumulated term41"
      }
    },
    {
      "id": "S297",
      "uses": [
        "G001",
        "S296"
      ],
      "before": "source roles = [41, 50, 25, 12, -2, 3, 1/10]",
      "after": "derived = [21/10, 129/2, 129/20, 1113/20]",
      "relation": "implies",
      "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。",
        "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S298",
      "uses": [
        "G001",
        "S297"
      ],
      "before": "Left sum ORIGINAL sample42 index times width := mul(42, 1/10)",
      "after": "Left sum ORIGINAL sample42 index times width = 21/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample42 index times width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample42 index times width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "42",
          "1/10"
        ],
        "output": "21/5",
        "label": "Left sum ORIGINAL sample42 index times width"
      }
    },
    {
      "id": "S299",
      "uses": [
        "G001",
        "S298"
      ],
      "before": "Left sum ORIGINAL sample42 lower plus offset := sum(-2, 21/5)",
      "after": "Left sum ORIGINAL sample42 lower plus offset = 11/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample42 lower plus offset，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample42 lower plus offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "21/5"
        ],
        "output": "11/5",
        "label": "Left sum ORIGINAL sample42 lower plus offset"
      }
    },
    {
      "id": "S300",
      "uses": [
        "G001",
        "S299"
      ],
      "before": "Left sum ORIGINAL sample42 slope times x := mul(25, 11/5)",
      "after": "Left sum ORIGINAL sample42 slope times x = 55",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample42 slope times x，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample42 slope times x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "11/5"
        ],
        "output": "55",
        "label": "Left sum ORIGINAL sample42 slope times x"
      }
    },
    {
      "id": "S301",
      "uses": [
        "G001",
        "S300"
      ],
      "before": "Left sum ORIGINAL sample42 affine value plus c := sum(55, 12)",
      "after": "Left sum ORIGINAL sample42 affine value plus c = 67",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample42 affine value plus c，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample42 affine value plus c with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "55",
          "12"
        ],
        "output": "67",
        "label": "Left sum ORIGINAL sample42 affine value plus c"
      }
    },
    {
      "id": "S302",
      "uses": [
        "G001",
        "S301"
      ],
      "before": "Left sum ORIGINAL sample42 value times cell width := mul(67, 1/10)",
      "after": "Left sum ORIGINAL sample42 value times cell width = 67/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample42 value times cell width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample42 value times cell width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "67",
          "1/10"
        ],
        "output": "67/10",
        "label": "Left sum ORIGINAL sample42 value times cell width"
      }
    },
    {
      "id": "S303",
      "uses": [
        "G001",
        "S302"
      ],
      "before": "Left sum ORIGINAL accumulated term42 := sum(1113/20, 67/10)",
      "after": "Left sum ORIGINAL accumulated term42 = 1247/20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term42，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL accumulated term42 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1113/20",
          "67/10"
        ],
        "output": "1247/20",
        "label": "Left sum ORIGINAL accumulated term42"
      }
    },
    {
      "id": "S304",
      "uses": [
        "G001",
        "S303"
      ],
      "before": "source roles = [42, 50, 25, 12, -2, 3, 1/10]",
      "after": "derived = [11/5, 67/1, 67/10, 1247/20]",
      "relation": "implies",
      "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。",
        "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S305",
      "uses": [
        "G001",
        "S304"
      ],
      "before": "Left sum ORIGINAL sample43 index times width := mul(43, 1/10)",
      "after": "Left sum ORIGINAL sample43 index times width = 43/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample43 index times width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample43 index times width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "43",
          "1/10"
        ],
        "output": "43/10",
        "label": "Left sum ORIGINAL sample43 index times width"
      }
    },
    {
      "id": "S306",
      "uses": [
        "G001",
        "S305"
      ],
      "before": "Left sum ORIGINAL sample43 lower plus offset := sum(-2, 43/10)",
      "after": "Left sum ORIGINAL sample43 lower plus offset = 23/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample43 lower plus offset，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample43 lower plus offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "43/10"
        ],
        "output": "23/10",
        "label": "Left sum ORIGINAL sample43 lower plus offset"
      }
    },
    {
      "id": "S307",
      "uses": [
        "G001",
        "S306"
      ],
      "before": "Left sum ORIGINAL sample43 slope times x := mul(25, 23/10)",
      "after": "Left sum ORIGINAL sample43 slope times x = 115/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample43 slope times x，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample43 slope times x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "23/10"
        ],
        "output": "115/2",
        "label": "Left sum ORIGINAL sample43 slope times x"
      }
    },
    {
      "id": "S308",
      "uses": [
        "G001",
        "S307"
      ],
      "before": "Left sum ORIGINAL sample43 affine value plus c := sum(115/2, 12)",
      "after": "Left sum ORIGINAL sample43 affine value plus c = 139/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample43 affine value plus c，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample43 affine value plus c with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "115/2",
          "12"
        ],
        "output": "139/2",
        "label": "Left sum ORIGINAL sample43 affine value plus c"
      }
    },
    {
      "id": "S309",
      "uses": [
        "G001",
        "S308"
      ],
      "before": "Left sum ORIGINAL sample43 value times cell width := mul(139/2, 1/10)",
      "after": "Left sum ORIGINAL sample43 value times cell width = 139/20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample43 value times cell width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample43 value times cell width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "139/2",
          "1/10"
        ],
        "output": "139/20",
        "label": "Left sum ORIGINAL sample43 value times cell width"
      }
    },
    {
      "id": "S310",
      "uses": [
        "G001",
        "S309"
      ],
      "before": "Left sum ORIGINAL accumulated term43 := sum(1247/20, 139/20)",
      "after": "Left sum ORIGINAL accumulated term43 = 693/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term43，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL accumulated term43 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1247/20",
          "139/20"
        ],
        "output": "693/10",
        "label": "Left sum ORIGINAL accumulated term43"
      }
    },
    {
      "id": "S311",
      "uses": [
        "G001",
        "S310"
      ],
      "before": "source roles = [43, 50, 25, 12, -2, 3, 1/10]",
      "after": "derived = [23/10, 139/2, 139/20, 693/10]",
      "relation": "implies",
      "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。",
        "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S312",
      "uses": [
        "G001",
        "S311"
      ],
      "before": "Left sum ORIGINAL sample44 index times width := mul(44, 1/10)",
      "after": "Left sum ORIGINAL sample44 index times width = 22/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample44 index times width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample44 index times width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "44",
          "1/10"
        ],
        "output": "22/5",
        "label": "Left sum ORIGINAL sample44 index times width"
      }
    },
    {
      "id": "S313",
      "uses": [
        "G001",
        "S312"
      ],
      "before": "Left sum ORIGINAL sample44 lower plus offset := sum(-2, 22/5)",
      "after": "Left sum ORIGINAL sample44 lower plus offset = 12/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample44 lower plus offset，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample44 lower plus offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "22/5"
        ],
        "output": "12/5",
        "label": "Left sum ORIGINAL sample44 lower plus offset"
      }
    },
    {
      "id": "S314",
      "uses": [
        "G001",
        "S313"
      ],
      "before": "Left sum ORIGINAL sample44 slope times x := mul(25, 12/5)",
      "after": "Left sum ORIGINAL sample44 slope times x = 60",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample44 slope times x，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample44 slope times x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "12/5"
        ],
        "output": "60",
        "label": "Left sum ORIGINAL sample44 slope times x"
      }
    },
    {
      "id": "S315",
      "uses": [
        "G001",
        "S314"
      ],
      "before": "Left sum ORIGINAL sample44 affine value plus c := sum(60, 12)",
      "after": "Left sum ORIGINAL sample44 affine value plus c = 72",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample44 affine value plus c，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample44 affine value plus c with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "60",
          "12"
        ],
        "output": "72",
        "label": "Left sum ORIGINAL sample44 affine value plus c"
      }
    },
    {
      "id": "S316",
      "uses": [
        "G001",
        "S315"
      ],
      "before": "Left sum ORIGINAL sample44 value times cell width := mul(72, 1/10)",
      "after": "Left sum ORIGINAL sample44 value times cell width = 36/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample44 value times cell width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample44 value times cell width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "72",
          "1/10"
        ],
        "output": "36/5",
        "label": "Left sum ORIGINAL sample44 value times cell width"
      }
    },
    {
      "id": "S317",
      "uses": [
        "G001",
        "S316"
      ],
      "before": "Left sum ORIGINAL accumulated term44 := sum(693/10, 36/5)",
      "after": "Left sum ORIGINAL accumulated term44 = 153/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term44，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL accumulated term44 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "693/10",
          "36/5"
        ],
        "output": "153/2",
        "label": "Left sum ORIGINAL accumulated term44"
      }
    },
    {
      "id": "S318",
      "uses": [
        "G001",
        "S317"
      ],
      "before": "source roles = [44, 50, 25, 12, -2, 3, 1/10]",
      "after": "derived = [12/5, 72/1, 36/5, 153/2]",
      "relation": "implies",
      "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。",
        "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S319",
      "uses": [
        "G001",
        "S318"
      ],
      "before": "Left sum ORIGINAL sample45 index times width := mul(45, 1/10)",
      "after": "Left sum ORIGINAL sample45 index times width = 9/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample45 index times width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample45 index times width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "45",
          "1/10"
        ],
        "output": "9/2",
        "label": "Left sum ORIGINAL sample45 index times width"
      }
    },
    {
      "id": "S320",
      "uses": [
        "G001",
        "S319"
      ],
      "before": "Left sum ORIGINAL sample45 lower plus offset := sum(-2, 9/2)",
      "after": "Left sum ORIGINAL sample45 lower plus offset = 5/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample45 lower plus offset，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample45 lower plus offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "9/2"
        ],
        "output": "5/2",
        "label": "Left sum ORIGINAL sample45 lower plus offset"
      }
    },
    {
      "id": "S321",
      "uses": [
        "G001",
        "S320"
      ],
      "before": "Left sum ORIGINAL sample45 slope times x := mul(25, 5/2)",
      "after": "Left sum ORIGINAL sample45 slope times x = 125/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample45 slope times x，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample45 slope times x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "5/2"
        ],
        "output": "125/2",
        "label": "Left sum ORIGINAL sample45 slope times x"
      }
    },
    {
      "id": "S322",
      "uses": [
        "G001",
        "S321"
      ],
      "before": "Left sum ORIGINAL sample45 affine value plus c := sum(125/2, 12)",
      "after": "Left sum ORIGINAL sample45 affine value plus c = 149/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample45 affine value plus c，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample45 affine value plus c with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "125/2",
          "12"
        ],
        "output": "149/2",
        "label": "Left sum ORIGINAL sample45 affine value plus c"
      }
    },
    {
      "id": "S323",
      "uses": [
        "G001",
        "S322"
      ],
      "before": "Left sum ORIGINAL sample45 value times cell width := mul(149/2, 1/10)",
      "after": "Left sum ORIGINAL sample45 value times cell width = 149/20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample45 value times cell width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample45 value times cell width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "149/2",
          "1/10"
        ],
        "output": "149/20",
        "label": "Left sum ORIGINAL sample45 value times cell width"
      }
    },
    {
      "id": "S324",
      "uses": [
        "G001",
        "S323"
      ],
      "before": "Left sum ORIGINAL accumulated term45 := sum(153/2, 149/20)",
      "after": "Left sum ORIGINAL accumulated term45 = 1679/20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term45，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL accumulated term45 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "153/2",
          "149/20"
        ],
        "output": "1679/20",
        "label": "Left sum ORIGINAL accumulated term45"
      }
    },
    {
      "id": "S325",
      "uses": [
        "G001",
        "S324"
      ],
      "before": "source roles = [45, 50, 25, 12, -2, 3, 1/10]",
      "after": "derived = [5/2, 149/2, 149/20, 1679/20]",
      "relation": "implies",
      "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。",
        "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S326",
      "uses": [
        "G001",
        "S325"
      ],
      "before": "Left sum ORIGINAL sample46 index times width := mul(46, 1/10)",
      "after": "Left sum ORIGINAL sample46 index times width = 23/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample46 index times width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample46 index times width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "46",
          "1/10"
        ],
        "output": "23/5",
        "label": "Left sum ORIGINAL sample46 index times width"
      }
    },
    {
      "id": "S327",
      "uses": [
        "G001",
        "S326"
      ],
      "before": "Left sum ORIGINAL sample46 lower plus offset := sum(-2, 23/5)",
      "after": "Left sum ORIGINAL sample46 lower plus offset = 13/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample46 lower plus offset，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample46 lower plus offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "23/5"
        ],
        "output": "13/5",
        "label": "Left sum ORIGINAL sample46 lower plus offset"
      }
    },
    {
      "id": "S328",
      "uses": [
        "G001",
        "S327"
      ],
      "before": "Left sum ORIGINAL sample46 slope times x := mul(25, 13/5)",
      "after": "Left sum ORIGINAL sample46 slope times x = 65",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample46 slope times x，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample46 slope times x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "13/5"
        ],
        "output": "65",
        "label": "Left sum ORIGINAL sample46 slope times x"
      }
    },
    {
      "id": "S329",
      "uses": [
        "G001",
        "S328"
      ],
      "before": "Left sum ORIGINAL sample46 affine value plus c := sum(65, 12)",
      "after": "Left sum ORIGINAL sample46 affine value plus c = 77",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample46 affine value plus c，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample46 affine value plus c with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "65",
          "12"
        ],
        "output": "77",
        "label": "Left sum ORIGINAL sample46 affine value plus c"
      }
    },
    {
      "id": "S330",
      "uses": [
        "G001",
        "S329"
      ],
      "before": "Left sum ORIGINAL sample46 value times cell width := mul(77, 1/10)",
      "after": "Left sum ORIGINAL sample46 value times cell width = 77/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample46 value times cell width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample46 value times cell width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "77",
          "1/10"
        ],
        "output": "77/10",
        "label": "Left sum ORIGINAL sample46 value times cell width"
      }
    },
    {
      "id": "S331",
      "uses": [
        "G001",
        "S330"
      ],
      "before": "Left sum ORIGINAL accumulated term46 := sum(1679/20, 77/10)",
      "after": "Left sum ORIGINAL accumulated term46 = 1833/20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term46，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL accumulated term46 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1679/20",
          "77/10"
        ],
        "output": "1833/20",
        "label": "Left sum ORIGINAL accumulated term46"
      }
    },
    {
      "id": "S332",
      "uses": [
        "G001",
        "S331"
      ],
      "before": "source roles = [46, 50, 25, 12, -2, 3, 1/10]",
      "after": "derived = [13/5, 77/1, 77/10, 1833/20]",
      "relation": "implies",
      "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。",
        "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S333",
      "uses": [
        "G001",
        "S332"
      ],
      "before": "Left sum ORIGINAL sample47 index times width := mul(47, 1/10)",
      "after": "Left sum ORIGINAL sample47 index times width = 47/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample47 index times width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample47 index times width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "47",
          "1/10"
        ],
        "output": "47/10",
        "label": "Left sum ORIGINAL sample47 index times width"
      }
    },
    {
      "id": "S334",
      "uses": [
        "G001",
        "S333"
      ],
      "before": "Left sum ORIGINAL sample47 lower plus offset := sum(-2, 47/10)",
      "after": "Left sum ORIGINAL sample47 lower plus offset = 27/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample47 lower plus offset，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample47 lower plus offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "47/10"
        ],
        "output": "27/10",
        "label": "Left sum ORIGINAL sample47 lower plus offset"
      }
    },
    {
      "id": "S335",
      "uses": [
        "G001",
        "S334"
      ],
      "before": "Left sum ORIGINAL sample47 slope times x := mul(25, 27/10)",
      "after": "Left sum ORIGINAL sample47 slope times x = 135/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample47 slope times x，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample47 slope times x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "27/10"
        ],
        "output": "135/2",
        "label": "Left sum ORIGINAL sample47 slope times x"
      }
    },
    {
      "id": "S336",
      "uses": [
        "G001",
        "S335"
      ],
      "before": "Left sum ORIGINAL sample47 affine value plus c := sum(135/2, 12)",
      "after": "Left sum ORIGINAL sample47 affine value plus c = 159/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample47 affine value plus c，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample47 affine value plus c with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "135/2",
          "12"
        ],
        "output": "159/2",
        "label": "Left sum ORIGINAL sample47 affine value plus c"
      }
    },
    {
      "id": "S337",
      "uses": [
        "G001",
        "S336"
      ],
      "before": "Left sum ORIGINAL sample47 value times cell width := mul(159/2, 1/10)",
      "after": "Left sum ORIGINAL sample47 value times cell width = 159/20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample47 value times cell width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample47 value times cell width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "159/2",
          "1/10"
        ],
        "output": "159/20",
        "label": "Left sum ORIGINAL sample47 value times cell width"
      }
    },
    {
      "id": "S338",
      "uses": [
        "G001",
        "S337"
      ],
      "before": "Left sum ORIGINAL accumulated term47 := sum(1833/20, 159/20)",
      "after": "Left sum ORIGINAL accumulated term47 = 498/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term47，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL accumulated term47 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1833/20",
          "159/20"
        ],
        "output": "498/5",
        "label": "Left sum ORIGINAL accumulated term47"
      }
    },
    {
      "id": "S339",
      "uses": [
        "G001",
        "S338"
      ],
      "before": "source roles = [47, 50, 25, 12, -2, 3, 1/10]",
      "after": "derived = [27/10, 159/2, 159/20, 498/5]",
      "relation": "implies",
      "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。",
        "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S340",
      "uses": [
        "G001",
        "S339"
      ],
      "before": "Left sum ORIGINAL sample48 index times width := mul(48, 1/10)",
      "after": "Left sum ORIGINAL sample48 index times width = 24/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample48 index times width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample48 index times width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "48",
          "1/10"
        ],
        "output": "24/5",
        "label": "Left sum ORIGINAL sample48 index times width"
      }
    },
    {
      "id": "S341",
      "uses": [
        "G001",
        "S340"
      ],
      "before": "Left sum ORIGINAL sample48 lower plus offset := sum(-2, 24/5)",
      "after": "Left sum ORIGINAL sample48 lower plus offset = 14/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample48 lower plus offset，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample48 lower plus offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "24/5"
        ],
        "output": "14/5",
        "label": "Left sum ORIGINAL sample48 lower plus offset"
      }
    },
    {
      "id": "S342",
      "uses": [
        "G001",
        "S341"
      ],
      "before": "Left sum ORIGINAL sample48 slope times x := mul(25, 14/5)",
      "after": "Left sum ORIGINAL sample48 slope times x = 70",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample48 slope times x，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample48 slope times x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "14/5"
        ],
        "output": "70",
        "label": "Left sum ORIGINAL sample48 slope times x"
      }
    },
    {
      "id": "S343",
      "uses": [
        "G001",
        "S342"
      ],
      "before": "Left sum ORIGINAL sample48 affine value plus c := sum(70, 12)",
      "after": "Left sum ORIGINAL sample48 affine value plus c = 82",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample48 affine value plus c，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample48 affine value plus c with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "70",
          "12"
        ],
        "output": "82",
        "label": "Left sum ORIGINAL sample48 affine value plus c"
      }
    },
    {
      "id": "S344",
      "uses": [
        "G001",
        "S343"
      ],
      "before": "Left sum ORIGINAL sample48 value times cell width := mul(82, 1/10)",
      "after": "Left sum ORIGINAL sample48 value times cell width = 41/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample48 value times cell width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample48 value times cell width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "82",
          "1/10"
        ],
        "output": "41/5",
        "label": "Left sum ORIGINAL sample48 value times cell width"
      }
    },
    {
      "id": "S345",
      "uses": [
        "G001",
        "S344"
      ],
      "before": "Left sum ORIGINAL accumulated term48 := sum(498/5, 41/5)",
      "after": "Left sum ORIGINAL accumulated term48 = 539/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term48，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL accumulated term48 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "498/5",
          "41/5"
        ],
        "output": "539/5",
        "label": "Left sum ORIGINAL accumulated term48"
      }
    },
    {
      "id": "S346",
      "uses": [
        "G001",
        "S345"
      ],
      "before": "source roles = [48, 50, 25, 12, -2, 3, 1/10]",
      "after": "derived = [14/5, 82/1, 41/5, 539/5]",
      "relation": "implies",
      "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。",
        "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S347",
      "uses": [
        "G001",
        "S346"
      ],
      "before": "Left sum ORIGINAL sample49 index times width := mul(49, 1/10)",
      "after": "Left sum ORIGINAL sample49 index times width = 49/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample49 index times width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample49 index times width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "49",
          "1/10"
        ],
        "output": "49/10",
        "label": "Left sum ORIGINAL sample49 index times width"
      }
    },
    {
      "id": "S348",
      "uses": [
        "G001",
        "S347"
      ],
      "before": "Left sum ORIGINAL sample49 lower plus offset := sum(-2, 49/10)",
      "after": "Left sum ORIGINAL sample49 lower plus offset = 29/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample49 lower plus offset，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample49 lower plus offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "49/10"
        ],
        "output": "29/10",
        "label": "Left sum ORIGINAL sample49 lower plus offset"
      }
    },
    {
      "id": "S349",
      "uses": [
        "G001",
        "S348"
      ],
      "before": "Left sum ORIGINAL sample49 slope times x := mul(25, 29/10)",
      "after": "Left sum ORIGINAL sample49 slope times x = 145/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample49 slope times x，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample49 slope times x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "29/10"
        ],
        "output": "145/2",
        "label": "Left sum ORIGINAL sample49 slope times x"
      }
    },
    {
      "id": "S350",
      "uses": [
        "G001",
        "S349"
      ],
      "before": "Left sum ORIGINAL sample49 affine value plus c := sum(145/2, 12)",
      "after": "Left sum ORIGINAL sample49 affine value plus c = 169/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample49 affine value plus c，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample49 affine value plus c with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "145/2",
          "12"
        ],
        "output": "169/2",
        "label": "Left sum ORIGINAL sample49 affine value plus c"
      }
    },
    {
      "id": "S351",
      "uses": [
        "G001",
        "S350"
      ],
      "before": "Left sum ORIGINAL sample49 value times cell width := mul(169/2, 1/10)",
      "after": "Left sum ORIGINAL sample49 value times cell width = 169/20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample49 value times cell width，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL sample49 value times cell width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "169/2",
          "1/10"
        ],
        "output": "169/20",
        "label": "Left sum ORIGINAL sample49 value times cell width"
      }
    },
    {
      "id": "S352",
      "uses": [
        "G001",
        "S351"
      ],
      "before": "Left sum ORIGINAL accumulated term49 := sum(539/5, 169/20)",
      "after": "Left sum ORIGINAL accumulated term49 = 465/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term49，不以小數近似替代。",
          "en": "Recompute Left sum ORIGINAL accumulated term49 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "539/5",
          "169/20"
        ],
        "output": "465/4",
        "label": "Left sum ORIGINAL accumulated term49"
      }
    },
    {
      "id": "S353",
      "uses": [
        "G001",
        "S352"
      ],
      "before": "source roles = [49, 50, 25, 12, -2, 3, 1/10]",
      "after": "derived = [29/10, 169/2, 169/20, 465/4]",
      "relation": "implies",
      "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。",
        "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S354",
      "uses": [
        "G001",
        "S353"
      ],
      "before": "source roles = [50, 25, 12, -2, 3]",
      "after": "derived = [1/10, 465/4]",
      "relation": "implies",
      "rule_id": "LeftSum_FIXED_finite_partition_NOT_integral_or_limit_claim",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這是固定N的有限左端和，一般不等於積分；常數／零寬特例可相等，不代表任意函數可積性或N→∞證明。",
        "en": "This is a FIXED-N finite left sum,generally not the integral.Constants/zero-width cases may agree,without proving arbitrary integrability or a limit as N→∞."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": "465/4"
  },
  "conclusion": {
    "canonical_value": "465/4",
    "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-20261008-r07-definite-integral-foundations",
      "batch_path": "public/data/questions/batches/batch-20261007-advance-20261008-r07-definite-integral-foundations/questions.jsonl",
      "batch_sha256": "f08b46f392acfe2749b49fcea9773a4b95227d50a72cafa6f06b32903b00deef",
      "original_line": 100,
      "original_line_sha256": "394432a11ca6756635186fb492dca80725ca58fd28f99836623163ca217bbeb3",
      "question_compact_sha256": "1664eca412eac2416d1eb84b1244342c91944c03ac1be08b95e49b7a3305df39",
      "english_row_compact_sha256": "4a56882ff0ee622a66b40035336fd70bb861ef0e015145d6af457792e933611f",
      "english_manifest_path": "public/data/i18n/editions/batch-20261007-advance-20261008-r07-definite-integral-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261007-advance-20261008-r07-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": "528186aa825c3054db15f4ce0a03ad58f72be7e7eb5bb236c4cc7ef34ff5a470",
    "original_candidate_sha256": "7f47d196eb7bb914d89f194581bbf7ecdf720bb1c06ea65c8174c35704960d59",
    "original_candidate_line_sha256": "7f47d196eb7bb914d89f194581bbf7ecdf720bb1c06ea65c8174c35704960d59"
  },
  "historical_source_state": {
    "review_state": {
      "derivation": "candidate_authored_and_locally_checked",
      "source_question_state": "candidate",
      "independent_AI_or_human_review": "NOT_PERFORMED",
      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
    },
    "scope_note": {
      "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。",
      "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged."
    },
    "label": "Frozen producer metadata; current per-question review is separate below."
  },
  "revision": {
    "id": "review-20261026-011-p65",
    "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": "8b9ea3ddfab468e9e3f0a0fc0f7682987d5e030be0d7c8860fd85df19cbe183b",
      "question_first_witness_sha256": "066bfeab8fc279706861faade71b9a18a61a25235fc48700092f0488d2a6da6a"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 354,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "0ebd6cace7174a9ef27b85cf1d0cc3ca8ebd54581f3cce0b15b99492426e7918",
      "arithmetic_steps": 302
    },
    "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."
  }
}