{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0098",
  "pilot_number": "R0034-17598",
  "number": 17598,
  "family_id": "qf-mod34-pointwise-finite-prefix-boundary",
  "title": {
    "zh-Hant": "第 17598 題：函數列逐點收斂",
    "en": "Question 17598: A Long Prefix Still Does Not Replace the All-n Tail"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0098",
    "batch_id": "batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-pointwise-convergence-of-functions"
    ],
    "title_zh": "函數列逐點收斂",
    "stem_zh": "同一有限domain上的symbolic rational function tail。操作=pointwise-finite-prefix-boundary；資料={\"limit\":[22,23],\"coefficients\":[23,-24],\"shift\":2,\"prefix_length\":23}。請依 [prefix_length,value_rows,sufficient,boundary] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(23, [[0, [67/2, 89/3, 111/4, 133/5, 155/6, 177/7, 199/8, 221/9, 243/10, 265/11, 287/12, 309/13, 331/14, 353/15, 375/16, 397/17, 419/18, 441/19, 463/20, 485/21, 507/22, 23, 551/24]], [1, [11, 15, 17, 91/5, 19, 137/7, 20, 61/3, 103/5, 229/11, 21, 275/13, 149/7, 107/5, 43/2, 367/17, 65/3, 413/19, 109/5, 153/7, 241/11, 505/23, 22]]], 否, finite_prefix_not_pointwise_convergence_certificate)",
      "explanation_zh": "Wave5第三層核對=(23, [[0, [67/2, 89/3, 111/4, 133/5, 155/6, 177/7, 199/8, 221/9, 243/10, 265/11, 287/12, 309/13, 331/14, 353/15, 375/16, 397/17, 419/18, 441/19, 463/20, 485/21, 507/22, 23, 551/24]], [1, [11, 15, 17, 91/5, 19, 137/7, 20, 61/3, 103/5, 229/11, 21, 275/13, 149/7, 107/5, 43/2, 367/17, 65/3, 413/19, 109/5, 153/7, 241/11, 505/23, 22]]], 否, finite_prefix_not_pointwise_convergence_certificate)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 3,
      "skills": [
        "函數列逐點收斂"
      ]
    },
    "variant": {
      "family_id": "qf-mod34-pointwise-finite-prefix-boundary",
      "seed": 202611032098,
      "parameters": {
        "limit": [
          22,
          23
        ],
        "coefficients": [
          23,
          -24
        ],
        "shift": 2,
        "prefix_length": 23,
        "domain": "Finite_domain_symbolic_pointwise_tail",
        "operation": "pointwise-finite-prefix-boundary",
        "semantic_template": "qf-mod34-pointwise-finite-prefix-boundary/{\"coefficients\":[23,-24],\"limit\":[22,23],\"prefix_length\":23,\"shift\":2}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_or_symbolic_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-14",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0098",
    "batch_id": "batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "86e48b0fe10fcc38349cfe16999186513f3bf328010666780100c5f19e443601",
      "family_id": "qf-mod34-pointwise-finite-prefix-boundary",
      "seed": 202611032098,
      "semantic_template": "qf-mod34-pointwise-finite-prefix-boundary/{\"coefficients\":[23,-24],\"limit\":[22,23],\"prefix_length\":23,\"shift\":2}"
    },
    "fields": {
      "title": "A Long Prefix Still Does Not Replace the All-n Tail",
      "stem": "Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[22, 23], coefficients=[23, -24], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. prefix_length=23, beginning at n=0. Give (prefix_length,value_rows,sufficient?,finite_prefix_not_pointwise_convergence_certificate), with each row [point,values_in_n_order].",
      "answer": {
        "kind": "expression",
        "value": "(23, [[0, [67/2, 89/3, 111/4, 133/5, 155/6, 177/7, 199/8, 221/9, 243/10, 265/11, 287/12, 309/13, 331/14, 353/15, 375/16, 397/17, 419/18, 441/19, 463/20, 485/21, 507/22, 23, 551/24]], [1, [11, 15, 17, 91/5, 19, 137/7, 20, 61/3, 103/5, 229/11, 21, 275/13, 149/7, 107/5, 43/2, 367/17, 65/3, 413/19, 109/5, 153/7, 241/11, 505/23, 22]]], No, finite_prefix_not_pointwise_convergence_certificate)",
        "explanation": "Display only the requested finite values. The No sufficiency flag concerns inference from this prefix alone, not a claim that the separately declared symbolic tail fails to converge. The exact result, retaining the requested order, is (23, [[0, [67/2, 89/3, 111/4, 133/5, 155/6, 177/7, 199/8, 221/9, 243/10, 265/11, 287/12, 309/13, 331/14, 353/15, 375/16, 397/17, 419/18, 441/19, 463/20, 485/21, 507/22, 23, 551/24]], [1, [11, 15, 17, 91/5, 19, 137/7, 20, 61/3, 103/5, 229/11, 21, 275/13, 149/7, 107/5, 43/2, 367/17, 65/3, 413/19, 109/5, 153/7, 241/11, 505/23, 22]]], No, finite_prefix_not_pointwise_convergence_certificate)."
      },
      "skills": [
        "A Long Prefix Still Does Not Replace the All-n Tail"
      ],
      "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": "同一有限domain上的symbolic rational function tail。操作=pointwise-finite-prefix-boundary；資料={\"limit\":[22,23],\"coefficients\":[23,-24],\"shift\":2,\"prefix_length\":23}。請依 [prefix_length,value_rows,sufficient,boundary] 欄序回答。",
        "en": "Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[22, 23], coefficients=[23, -24], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. prefix_length=23, beginning at n=0. Give (prefix_length,value_rows,sufficient?,finite_prefix_not_pointwise_convergence_certificate), with each row [point,values_in_n_order]."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "same COMPLETE finite domain Fin(2);original aligned limits=[22, 23];signed coefficients=[23, -24];positive shift=2",
      "after": "declared for EVERY n>=0:f_n(x)=limit[x]+coefficient[x]/(n+shift);signed error=f_n(x)-limit[x]=coefficient[x]/(n+shift)",
      "relation": "implies",
      "rule_id": "pointwise_explicit_common_domain_all_n_tail_not_probability",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整共同有限domain與全尾公式是題目明示前提，不從樣本猜出。函數值與誤差不是機率，可帶負號或超過1；不截斷、不平移索引。",
        "en": "The complete common finite domain and all-n formula are declared premises,not extrapolated from samples.Function values and errors are not probabilities;they may be signed or exceed1.Do not clip or shift indices."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "finite prefix point0 index0 denominator := sum(0, 2)",
      "after": "finite prefix point0 index0 denominator = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index0 denominator，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index0 denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "2"
        ],
        "output": "2",
        "label": "finite prefix point0 index0 denominator"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "finite prefix point0 index0 signed correction := div(23, 2)",
      "after": "finite prefix point0 index0 signed correction = 23/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index0 signed correction，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index0 signed correction with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "23",
          "2"
        ],
        "output": "23/2",
        "label": "finite prefix point0 index0 signed correction"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "finite prefix point0 index0 function value := sum(22, 23/2)",
      "after": "finite prefix point0 index0 function value = 67/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index0 function value，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index0 function value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "22",
          "23/2"
        ],
        "output": "67/2",
        "label": "finite prefix point0 index0 function value"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "finite prefix point0 index0 reverse signed error := sub(67/2, 22)",
      "after": "finite prefix point0 index0 reverse signed error = 23/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index0 reverse signed error，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index0 reverse signed error with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "67/2",
          "22"
        ],
        "output": "23/2",
        "label": "finite prefix point0 index0 reverse signed error"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "finite prefix point0 index0:original point=0;n=0;limit=22;signed coefficient=23",
      "after": "f_n(x)=67/2;f_n(x)-limit=23/2;this error field remains SIGNED",
      "relation": "implies",
      "rule_id": "pointwise_selected_value_reverse_error_not_absolute_value",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先加修正項得到函數值，再同減limit反向核誤差；原evaluate欄位保留正負，只有epsilon界才取絕對值。",
        "en": "Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "finite prefix point0 index1 denominator := sum(1, 2)",
      "after": "finite prefix point0 index1 denominator = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index1 denominator，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index1 denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "2"
        ],
        "output": "3",
        "label": "finite prefix point0 index1 denominator"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "finite prefix point0 index1 signed correction := div(23, 3)",
      "after": "finite prefix point0 index1 signed correction = 23/3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index1 signed correction，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index1 signed correction with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "23",
          "3"
        ],
        "output": "23/3",
        "label": "finite prefix point0 index1 signed correction"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "finite prefix point0 index1 function value := sum(22, 23/3)",
      "after": "finite prefix point0 index1 function value = 89/3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index1 function value，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index1 function value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "22",
          "23/3"
        ],
        "output": "89/3",
        "label": "finite prefix point0 index1 function value"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "finite prefix point0 index1 reverse signed error := sub(89/3, 22)",
      "after": "finite prefix point0 index1 reverse signed error = 23/3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index1 reverse signed error，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index1 reverse signed error with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "89/3",
          "22"
        ],
        "output": "23/3",
        "label": "finite prefix point0 index1 reverse signed error"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "finite prefix point0 index1:original point=0;n=1;limit=22;signed coefficient=23",
      "after": "f_n(x)=89/3;f_n(x)-limit=23/3;this error field remains SIGNED",
      "relation": "implies",
      "rule_id": "pointwise_selected_value_reverse_error_not_absolute_value",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先加修正項得到函數值，再同減limit反向核誤差；原evaluate欄位保留正負，只有epsilon界才取絕對值。",
        "en": "Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "finite prefix point0 index2 denominator := sum(2, 2)",
      "after": "finite prefix point0 index2 denominator = 4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index2 denominator，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index2 denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "2",
          "2"
        ],
        "output": "4",
        "label": "finite prefix point0 index2 denominator"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "finite prefix point0 index2 signed correction := div(23, 4)",
      "after": "finite prefix point0 index2 signed correction = 23/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index2 signed correction，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index2 signed correction with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "23",
          "4"
        ],
        "output": "23/4",
        "label": "finite prefix point0 index2 signed correction"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "finite prefix point0 index2 function value := sum(22, 23/4)",
      "after": "finite prefix point0 index2 function value = 111/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index2 function value，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index2 function value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "22",
          "23/4"
        ],
        "output": "111/4",
        "label": "finite prefix point0 index2 function value"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "finite prefix point0 index2 reverse signed error := sub(111/4, 22)",
      "after": "finite prefix point0 index2 reverse signed error = 23/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index2 reverse signed error，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index2 reverse signed error with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "111/4",
          "22"
        ],
        "output": "23/4",
        "label": "finite prefix point0 index2 reverse signed error"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "finite prefix point0 index2:original point=0;n=2;limit=22;signed coefficient=23",
      "after": "f_n(x)=111/4;f_n(x)-limit=23/4;this error field remains SIGNED",
      "relation": "implies",
      "rule_id": "pointwise_selected_value_reverse_error_not_absolute_value",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先加修正項得到函數值，再同減limit反向核誤差；原evaluate欄位保留正負，只有epsilon界才取絕對值。",
        "en": "Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "finite prefix point0 index3 denominator := sum(3, 2)",
      "after": "finite prefix point0 index3 denominator = 5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index3 denominator，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index3 denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "3",
          "2"
        ],
        "output": "5",
        "label": "finite prefix point0 index3 denominator"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "finite prefix point0 index3 signed correction := div(23, 5)",
      "after": "finite prefix point0 index3 signed correction = 23/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index3 signed correction，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index3 signed correction with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "23",
          "5"
        ],
        "output": "23/5",
        "label": "finite prefix point0 index3 signed correction"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "finite prefix point0 index3 function value := sum(22, 23/5)",
      "after": "finite prefix point0 index3 function value = 133/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index3 function value，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index3 function value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "22",
          "23/5"
        ],
        "output": "133/5",
        "label": "finite prefix point0 index3 function value"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "finite prefix point0 index3 reverse signed error := sub(133/5, 22)",
      "after": "finite prefix point0 index3 reverse signed error = 23/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index3 reverse signed error，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index3 reverse signed error with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "133/5",
          "22"
        ],
        "output": "23/5",
        "label": "finite prefix point0 index3 reverse signed error"
      }
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "finite prefix point0 index3:original point=0;n=3;limit=22;signed coefficient=23",
      "after": "f_n(x)=133/5;f_n(x)-limit=23/5;this error field remains SIGNED",
      "relation": "implies",
      "rule_id": "pointwise_selected_value_reverse_error_not_absolute_value",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先加修正項得到函數值，再同減limit反向核誤差；原evaluate欄位保留正負，只有epsilon界才取絕對值。",
        "en": "Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "finite prefix point0 index4 denominator := sum(4, 2)",
      "after": "finite prefix point0 index4 denominator = 6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index4 denominator，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index4 denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "4",
          "2"
        ],
        "output": "6",
        "label": "finite prefix point0 index4 denominator"
      }
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "finite prefix point0 index4 signed correction := div(23, 6)",
      "after": "finite prefix point0 index4 signed correction = 23/6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index4 signed correction，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index4 signed correction with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "23",
          "6"
        ],
        "output": "23/6",
        "label": "finite prefix point0 index4 signed correction"
      }
    },
    {
      "id": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "finite prefix point0 index4 function value := sum(22, 23/6)",
      "after": "finite prefix point0 index4 function value = 155/6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index4 function value，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index4 function value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "22",
          "23/6"
        ],
        "output": "155/6",
        "label": "finite prefix point0 index4 function value"
      }
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "finite prefix point0 index4 reverse signed error := sub(155/6, 22)",
      "after": "finite prefix point0 index4 reverse signed error = 23/6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index4 reverse signed error，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index4 reverse signed error with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "155/6",
          "22"
        ],
        "output": "23/6",
        "label": "finite prefix point0 index4 reverse signed error"
      }
    },
    {
      "id": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "finite prefix point0 index4:original point=0;n=4;limit=22;signed coefficient=23",
      "after": "f_n(x)=155/6;f_n(x)-limit=23/6;this error field remains SIGNED",
      "relation": "implies",
      "rule_id": "pointwise_selected_value_reverse_error_not_absolute_value",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先加修正項得到函數值，再同減limit反向核誤差；原evaluate欄位保留正負，只有epsilon界才取絕對值。",
        "en": "Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "finite prefix point0 index5 denominator := sum(5, 2)",
      "after": "finite prefix point0 index5 denominator = 7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index5 denominator，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index5 denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "5",
          "2"
        ],
        "output": "7",
        "label": "finite prefix point0 index5 denominator"
      }
    },
    {
      "id": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "finite prefix point0 index5 signed correction := div(23, 7)",
      "after": "finite prefix point0 index5 signed correction = 23/7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index5 signed correction，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index5 signed correction with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "23",
          "7"
        ],
        "output": "23/7",
        "label": "finite prefix point0 index5 signed correction"
      }
    },
    {
      "id": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "finite prefix point0 index5 function value := sum(22, 23/7)",
      "after": "finite prefix point0 index5 function value = 177/7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index5 function value，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index5 function value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "22",
          "23/7"
        ],
        "output": "177/7",
        "label": "finite prefix point0 index5 function value"
      }
    },
    {
      "id": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "finite prefix point0 index5 reverse signed error := sub(177/7, 22)",
      "after": "finite prefix point0 index5 reverse signed error = 23/7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index5 reverse signed error，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index5 reverse signed error with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "177/7",
          "22"
        ],
        "output": "23/7",
        "label": "finite prefix point0 index5 reverse signed error"
      }
    },
    {
      "id": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "finite prefix point0 index5:original point=0;n=5;limit=22;signed coefficient=23",
      "after": "f_n(x)=177/7;f_n(x)-limit=23/7;this error field remains SIGNED",
      "relation": "implies",
      "rule_id": "pointwise_selected_value_reverse_error_not_absolute_value",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先加修正項得到函數值，再同減limit反向核誤差；原evaluate欄位保留正負，只有epsilon界才取絕對值。",
        "en": "Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S032",
      "uses": [
        "G001",
        "S031"
      ],
      "before": "finite prefix point0 index6 denominator := sum(6, 2)",
      "after": "finite prefix point0 index6 denominator = 8",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index6 denominator，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index6 denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "6",
          "2"
        ],
        "output": "8",
        "label": "finite prefix point0 index6 denominator"
      }
    },
    {
      "id": "S033",
      "uses": [
        "G001",
        "S032"
      ],
      "before": "finite prefix point0 index6 signed correction := div(23, 8)",
      "after": "finite prefix point0 index6 signed correction = 23/8",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index6 signed correction，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index6 signed correction with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "23",
          "8"
        ],
        "output": "23/8",
        "label": "finite prefix point0 index6 signed correction"
      }
    },
    {
      "id": "S034",
      "uses": [
        "G001",
        "S033"
      ],
      "before": "finite prefix point0 index6 function value := sum(22, 23/8)",
      "after": "finite prefix point0 index6 function value = 199/8",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index6 function value，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index6 function value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "22",
          "23/8"
        ],
        "output": "199/8",
        "label": "finite prefix point0 index6 function value"
      }
    },
    {
      "id": "S035",
      "uses": [
        "G001",
        "S034"
      ],
      "before": "finite prefix point0 index6 reverse signed error := sub(199/8, 22)",
      "after": "finite prefix point0 index6 reverse signed error = 23/8",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index6 reverse signed error，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index6 reverse signed error with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "199/8",
          "22"
        ],
        "output": "23/8",
        "label": "finite prefix point0 index6 reverse signed error"
      }
    },
    {
      "id": "S036",
      "uses": [
        "G001",
        "S035"
      ],
      "before": "finite prefix point0 index6:original point=0;n=6;limit=22;signed coefficient=23",
      "after": "f_n(x)=199/8;f_n(x)-limit=23/8;this error field remains SIGNED",
      "relation": "implies",
      "rule_id": "pointwise_selected_value_reverse_error_not_absolute_value",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先加修正項得到函數值，再同減limit反向核誤差；原evaluate欄位保留正負，只有epsilon界才取絕對值。",
        "en": "Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S037",
      "uses": [
        "G001",
        "S036"
      ],
      "before": "finite prefix point0 index7 denominator := sum(7, 2)",
      "after": "finite prefix point0 index7 denominator = 9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index7 denominator，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index7 denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "7",
          "2"
        ],
        "output": "9",
        "label": "finite prefix point0 index7 denominator"
      }
    },
    {
      "id": "S038",
      "uses": [
        "G001",
        "S037"
      ],
      "before": "finite prefix point0 index7 signed correction := div(23, 9)",
      "after": "finite prefix point0 index7 signed correction = 23/9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index7 signed correction，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index7 signed correction with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "23",
          "9"
        ],
        "output": "23/9",
        "label": "finite prefix point0 index7 signed correction"
      }
    },
    {
      "id": "S039",
      "uses": [
        "G001",
        "S038"
      ],
      "before": "finite prefix point0 index7 function value := sum(22, 23/9)",
      "after": "finite prefix point0 index7 function value = 221/9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index7 function value，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index7 function value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "22",
          "23/9"
        ],
        "output": "221/9",
        "label": "finite prefix point0 index7 function value"
      }
    },
    {
      "id": "S040",
      "uses": [
        "G001",
        "S039"
      ],
      "before": "finite prefix point0 index7 reverse signed error := sub(221/9, 22)",
      "after": "finite prefix point0 index7 reverse signed error = 23/9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index7 reverse signed error，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index7 reverse signed error with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "221/9",
          "22"
        ],
        "output": "23/9",
        "label": "finite prefix point0 index7 reverse signed error"
      }
    },
    {
      "id": "S041",
      "uses": [
        "G001",
        "S040"
      ],
      "before": "finite prefix point0 index7:original point=0;n=7;limit=22;signed coefficient=23",
      "after": "f_n(x)=221/9;f_n(x)-limit=23/9;this error field remains SIGNED",
      "relation": "implies",
      "rule_id": "pointwise_selected_value_reverse_error_not_absolute_value",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先加修正項得到函數值，再同減limit反向核誤差；原evaluate欄位保留正負，只有epsilon界才取絕對值。",
        "en": "Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S042",
      "uses": [
        "G001",
        "S041"
      ],
      "before": "finite prefix point0 index8 denominator := sum(8, 2)",
      "after": "finite prefix point0 index8 denominator = 10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index8 denominator，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index8 denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "8",
          "2"
        ],
        "output": "10",
        "label": "finite prefix point0 index8 denominator"
      }
    },
    {
      "id": "S043",
      "uses": [
        "G001",
        "S042"
      ],
      "before": "finite prefix point0 index8 signed correction := div(23, 10)",
      "after": "finite prefix point0 index8 signed correction = 23/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index8 signed correction，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index8 signed correction with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "23",
          "10"
        ],
        "output": "23/10",
        "label": "finite prefix point0 index8 signed correction"
      }
    },
    {
      "id": "S044",
      "uses": [
        "G001",
        "S043"
      ],
      "before": "finite prefix point0 index8 function value := sum(22, 23/10)",
      "after": "finite prefix point0 index8 function value = 243/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index8 function value，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index8 function value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "22",
          "23/10"
        ],
        "output": "243/10",
        "label": "finite prefix point0 index8 function value"
      }
    },
    {
      "id": "S045",
      "uses": [
        "G001",
        "S044"
      ],
      "before": "finite prefix point0 index8 reverse signed error := sub(243/10, 22)",
      "after": "finite prefix point0 index8 reverse signed error = 23/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index8 reverse signed error，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index8 reverse signed error with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "243/10",
          "22"
        ],
        "output": "23/10",
        "label": "finite prefix point0 index8 reverse signed error"
      }
    },
    {
      "id": "S046",
      "uses": [
        "G001",
        "S045"
      ],
      "before": "finite prefix point0 index8:original point=0;n=8;limit=22;signed coefficient=23",
      "after": "f_n(x)=243/10;f_n(x)-limit=23/10;this error field remains SIGNED",
      "relation": "implies",
      "rule_id": "pointwise_selected_value_reverse_error_not_absolute_value",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先加修正項得到函數值，再同減limit反向核誤差；原evaluate欄位保留正負，只有epsilon界才取絕對值。",
        "en": "Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S047",
      "uses": [
        "G001",
        "S046"
      ],
      "before": "finite prefix point0 index9 denominator := sum(9, 2)",
      "after": "finite prefix point0 index9 denominator = 11",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index9 denominator，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index9 denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "9",
          "2"
        ],
        "output": "11",
        "label": "finite prefix point0 index9 denominator"
      }
    },
    {
      "id": "S048",
      "uses": [
        "G001",
        "S047"
      ],
      "before": "finite prefix point0 index9 signed correction := div(23, 11)",
      "after": "finite prefix point0 index9 signed correction = 23/11",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index9 signed correction，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index9 signed correction with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "23",
          "11"
        ],
        "output": "23/11",
        "label": "finite prefix point0 index9 signed correction"
      }
    },
    {
      "id": "S049",
      "uses": [
        "G001",
        "S048"
      ],
      "before": "finite prefix point0 index9 function value := sum(22, 23/11)",
      "after": "finite prefix point0 index9 function value = 265/11",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index9 function value，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index9 function value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "22",
          "23/11"
        ],
        "output": "265/11",
        "label": "finite prefix point0 index9 function value"
      }
    },
    {
      "id": "S050",
      "uses": [
        "G001",
        "S049"
      ],
      "before": "finite prefix point0 index9 reverse signed error := sub(265/11, 22)",
      "after": "finite prefix point0 index9 reverse signed error = 23/11",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index9 reverse signed error，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index9 reverse signed error with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "265/11",
          "22"
        ],
        "output": "23/11",
        "label": "finite prefix point0 index9 reverse signed error"
      }
    },
    {
      "id": "S051",
      "uses": [
        "G001",
        "S050"
      ],
      "before": "finite prefix point0 index9:original point=0;n=9;limit=22;signed coefficient=23",
      "after": "f_n(x)=265/11;f_n(x)-limit=23/11;this error field remains SIGNED",
      "relation": "implies",
      "rule_id": "pointwise_selected_value_reverse_error_not_absolute_value",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先加修正項得到函數值，再同減limit反向核誤差；原evaluate欄位保留正負，只有epsilon界才取絕對值。",
        "en": "Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S052",
      "uses": [
        "G001",
        "S051"
      ],
      "before": "finite prefix point0 index10 denominator := sum(10, 2)",
      "after": "finite prefix point0 index10 denominator = 12",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index10 denominator，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index10 denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "10",
          "2"
        ],
        "output": "12",
        "label": "finite prefix point0 index10 denominator"
      }
    },
    {
      "id": "S053",
      "uses": [
        "G001",
        "S052"
      ],
      "before": "finite prefix point0 index10 signed correction := div(23, 12)",
      "after": "finite prefix point0 index10 signed correction = 23/12",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index10 signed correction，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index10 signed correction with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "23",
          "12"
        ],
        "output": "23/12",
        "label": "finite prefix point0 index10 signed correction"
      }
    },
    {
      "id": "S054",
      "uses": [
        "G001",
        "S053"
      ],
      "before": "finite prefix point0 index10 function value := sum(22, 23/12)",
      "after": "finite prefix point0 index10 function value = 287/12",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index10 function value，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index10 function value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "22",
          "23/12"
        ],
        "output": "287/12",
        "label": "finite prefix point0 index10 function value"
      }
    },
    {
      "id": "S055",
      "uses": [
        "G001",
        "S054"
      ],
      "before": "finite prefix point0 index10 reverse signed error := sub(287/12, 22)",
      "after": "finite prefix point0 index10 reverse signed error = 23/12",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index10 reverse signed error，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index10 reverse signed error with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "287/12",
          "22"
        ],
        "output": "23/12",
        "label": "finite prefix point0 index10 reverse signed error"
      }
    },
    {
      "id": "S056",
      "uses": [
        "G001",
        "S055"
      ],
      "before": "finite prefix point0 index10:original point=0;n=10;limit=22;signed coefficient=23",
      "after": "f_n(x)=287/12;f_n(x)-limit=23/12;this error field remains SIGNED",
      "relation": "implies",
      "rule_id": "pointwise_selected_value_reverse_error_not_absolute_value",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先加修正項得到函數值，再同減limit反向核誤差；原evaluate欄位保留正負，只有epsilon界才取絕對值。",
        "en": "Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S057",
      "uses": [
        "G001",
        "S056"
      ],
      "before": "finite prefix point0 index11 denominator := sum(11, 2)",
      "after": "finite prefix point0 index11 denominator = 13",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index11 denominator，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index11 denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "11",
          "2"
        ],
        "output": "13",
        "label": "finite prefix point0 index11 denominator"
      }
    },
    {
      "id": "S058",
      "uses": [
        "G001",
        "S057"
      ],
      "before": "finite prefix point0 index11 signed correction := div(23, 13)",
      "after": "finite prefix point0 index11 signed correction = 23/13",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index11 signed correction，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index11 signed correction with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "23",
          "13"
        ],
        "output": "23/13",
        "label": "finite prefix point0 index11 signed correction"
      }
    },
    {
      "id": "S059",
      "uses": [
        "G001",
        "S058"
      ],
      "before": "finite prefix point0 index11 function value := sum(22, 23/13)",
      "after": "finite prefix point0 index11 function value = 309/13",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index11 function value，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index11 function value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "22",
          "23/13"
        ],
        "output": "309/13",
        "label": "finite prefix point0 index11 function value"
      }
    },
    {
      "id": "S060",
      "uses": [
        "G001",
        "S059"
      ],
      "before": "finite prefix point0 index11 reverse signed error := sub(309/13, 22)",
      "after": "finite prefix point0 index11 reverse signed error = 23/13",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index11 reverse signed error，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index11 reverse signed error with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "309/13",
          "22"
        ],
        "output": "23/13",
        "label": "finite prefix point0 index11 reverse signed error"
      }
    },
    {
      "id": "S061",
      "uses": [
        "G001",
        "S060"
      ],
      "before": "finite prefix point0 index11:original point=0;n=11;limit=22;signed coefficient=23",
      "after": "f_n(x)=309/13;f_n(x)-limit=23/13;this error field remains SIGNED",
      "relation": "implies",
      "rule_id": "pointwise_selected_value_reverse_error_not_absolute_value",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先加修正項得到函數值，再同減limit反向核誤差；原evaluate欄位保留正負，只有epsilon界才取絕對值。",
        "en": "Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S062",
      "uses": [
        "G001",
        "S061"
      ],
      "before": "finite prefix point0 index12 denominator := sum(12, 2)",
      "after": "finite prefix point0 index12 denominator = 14",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index12 denominator，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index12 denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "12",
          "2"
        ],
        "output": "14",
        "label": "finite prefix point0 index12 denominator"
      }
    },
    {
      "id": "S063",
      "uses": [
        "G001",
        "S062"
      ],
      "before": "finite prefix point0 index12 signed correction := div(23, 14)",
      "after": "finite prefix point0 index12 signed correction = 23/14",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index12 signed correction，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index12 signed correction with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "23",
          "14"
        ],
        "output": "23/14",
        "label": "finite prefix point0 index12 signed correction"
      }
    },
    {
      "id": "S064",
      "uses": [
        "G001",
        "S063"
      ],
      "before": "finite prefix point0 index12 function value := sum(22, 23/14)",
      "after": "finite prefix point0 index12 function value = 331/14",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index12 function value，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index12 function value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "22",
          "23/14"
        ],
        "output": "331/14",
        "label": "finite prefix point0 index12 function value"
      }
    },
    {
      "id": "S065",
      "uses": [
        "G001",
        "S064"
      ],
      "before": "finite prefix point0 index12 reverse signed error := sub(331/14, 22)",
      "after": "finite prefix point0 index12 reverse signed error = 23/14",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index12 reverse signed error，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index12 reverse signed error with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "331/14",
          "22"
        ],
        "output": "23/14",
        "label": "finite prefix point0 index12 reverse signed error"
      }
    },
    {
      "id": "S066",
      "uses": [
        "G001",
        "S065"
      ],
      "before": "finite prefix point0 index12:original point=0;n=12;limit=22;signed coefficient=23",
      "after": "f_n(x)=331/14;f_n(x)-limit=23/14;this error field remains SIGNED",
      "relation": "implies",
      "rule_id": "pointwise_selected_value_reverse_error_not_absolute_value",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先加修正項得到函數值，再同減limit反向核誤差；原evaluate欄位保留正負，只有epsilon界才取絕對值。",
        "en": "Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S067",
      "uses": [
        "G001",
        "S066"
      ],
      "before": "finite prefix point0 index13 denominator := sum(13, 2)",
      "after": "finite prefix point0 index13 denominator = 15",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index13 denominator，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index13 denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "13",
          "2"
        ],
        "output": "15",
        "label": "finite prefix point0 index13 denominator"
      }
    },
    {
      "id": "S068",
      "uses": [
        "G001",
        "S067"
      ],
      "before": "finite prefix point0 index13 signed correction := div(23, 15)",
      "after": "finite prefix point0 index13 signed correction = 23/15",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index13 signed correction，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index13 signed correction with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "23",
          "15"
        ],
        "output": "23/15",
        "label": "finite prefix point0 index13 signed correction"
      }
    },
    {
      "id": "S069",
      "uses": [
        "G001",
        "S068"
      ],
      "before": "finite prefix point0 index13 function value := sum(22, 23/15)",
      "after": "finite prefix point0 index13 function value = 353/15",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index13 function value，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index13 function value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "22",
          "23/15"
        ],
        "output": "353/15",
        "label": "finite prefix point0 index13 function value"
      }
    },
    {
      "id": "S070",
      "uses": [
        "G001",
        "S069"
      ],
      "before": "finite prefix point0 index13 reverse signed error := sub(353/15, 22)",
      "after": "finite prefix point0 index13 reverse signed error = 23/15",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index13 reverse signed error，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index13 reverse signed error with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "353/15",
          "22"
        ],
        "output": "23/15",
        "label": "finite prefix point0 index13 reverse signed error"
      }
    },
    {
      "id": "S071",
      "uses": [
        "G001",
        "S070"
      ],
      "before": "finite prefix point0 index13:original point=0;n=13;limit=22;signed coefficient=23",
      "after": "f_n(x)=353/15;f_n(x)-limit=23/15;this error field remains SIGNED",
      "relation": "implies",
      "rule_id": "pointwise_selected_value_reverse_error_not_absolute_value",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先加修正項得到函數值，再同減limit反向核誤差；原evaluate欄位保留正負，只有epsilon界才取絕對值。",
        "en": "Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S072",
      "uses": [
        "G001",
        "S071"
      ],
      "before": "finite prefix point0 index14 denominator := sum(14, 2)",
      "after": "finite prefix point0 index14 denominator = 16",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index14 denominator，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index14 denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "14",
          "2"
        ],
        "output": "16",
        "label": "finite prefix point0 index14 denominator"
      }
    },
    {
      "id": "S073",
      "uses": [
        "G001",
        "S072"
      ],
      "before": "finite prefix point0 index14 signed correction := div(23, 16)",
      "after": "finite prefix point0 index14 signed correction = 23/16",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index14 signed correction，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index14 signed correction with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "23",
          "16"
        ],
        "output": "23/16",
        "label": "finite prefix point0 index14 signed correction"
      }
    },
    {
      "id": "S074",
      "uses": [
        "G001",
        "S073"
      ],
      "before": "finite prefix point0 index14 function value := sum(22, 23/16)",
      "after": "finite prefix point0 index14 function value = 375/16",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index14 function value，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index14 function value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "22",
          "23/16"
        ],
        "output": "375/16",
        "label": "finite prefix point0 index14 function value"
      }
    },
    {
      "id": "S075",
      "uses": [
        "G001",
        "S074"
      ],
      "before": "finite prefix point0 index14 reverse signed error := sub(375/16, 22)",
      "after": "finite prefix point0 index14 reverse signed error = 23/16",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index14 reverse signed error，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index14 reverse signed error with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "375/16",
          "22"
        ],
        "output": "23/16",
        "label": "finite prefix point0 index14 reverse signed error"
      }
    },
    {
      "id": "S076",
      "uses": [
        "G001",
        "S075"
      ],
      "before": "finite prefix point0 index14:original point=0;n=14;limit=22;signed coefficient=23",
      "after": "f_n(x)=375/16;f_n(x)-limit=23/16;this error field remains SIGNED",
      "relation": "implies",
      "rule_id": "pointwise_selected_value_reverse_error_not_absolute_value",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先加修正項得到函數值，再同減limit反向核誤差；原evaluate欄位保留正負，只有epsilon界才取絕對值。",
        "en": "Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S077",
      "uses": [
        "G001",
        "S076"
      ],
      "before": "finite prefix point0 index15 denominator := sum(15, 2)",
      "after": "finite prefix point0 index15 denominator = 17",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index15 denominator，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index15 denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "15",
          "2"
        ],
        "output": "17",
        "label": "finite prefix point0 index15 denominator"
      }
    },
    {
      "id": "S078",
      "uses": [
        "G001",
        "S077"
      ],
      "before": "finite prefix point0 index15 signed correction := div(23, 17)",
      "after": "finite prefix point0 index15 signed correction = 23/17",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index15 signed correction，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index15 signed correction with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "23",
          "17"
        ],
        "output": "23/17",
        "label": "finite prefix point0 index15 signed correction"
      }
    },
    {
      "id": "S079",
      "uses": [
        "G001",
        "S078"
      ],
      "before": "finite prefix point0 index15 function value := sum(22, 23/17)",
      "after": "finite prefix point0 index15 function value = 397/17",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index15 function value，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index15 function value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "22",
          "23/17"
        ],
        "output": "397/17",
        "label": "finite prefix point0 index15 function value"
      }
    },
    {
      "id": "S080",
      "uses": [
        "G001",
        "S079"
      ],
      "before": "finite prefix point0 index15 reverse signed error := sub(397/17, 22)",
      "after": "finite prefix point0 index15 reverse signed error = 23/17",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index15 reverse signed error，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index15 reverse signed error with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "397/17",
          "22"
        ],
        "output": "23/17",
        "label": "finite prefix point0 index15 reverse signed error"
      }
    },
    {
      "id": "S081",
      "uses": [
        "G001",
        "S080"
      ],
      "before": "finite prefix point0 index15:original point=0;n=15;limit=22;signed coefficient=23",
      "after": "f_n(x)=397/17;f_n(x)-limit=23/17;this error field remains SIGNED",
      "relation": "implies",
      "rule_id": "pointwise_selected_value_reverse_error_not_absolute_value",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先加修正項得到函數值，再同減limit反向核誤差；原evaluate欄位保留正負，只有epsilon界才取絕對值。",
        "en": "Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S082",
      "uses": [
        "G001",
        "S081"
      ],
      "before": "finite prefix point0 index16 denominator := sum(16, 2)",
      "after": "finite prefix point0 index16 denominator = 18",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index16 denominator，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index16 denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "16",
          "2"
        ],
        "output": "18",
        "label": "finite prefix point0 index16 denominator"
      }
    },
    {
      "id": "S083",
      "uses": [
        "G001",
        "S082"
      ],
      "before": "finite prefix point0 index16 signed correction := div(23, 18)",
      "after": "finite prefix point0 index16 signed correction = 23/18",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index16 signed correction，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index16 signed correction with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "23",
          "18"
        ],
        "output": "23/18",
        "label": "finite prefix point0 index16 signed correction"
      }
    },
    {
      "id": "S084",
      "uses": [
        "G001",
        "S083"
      ],
      "before": "finite prefix point0 index16 function value := sum(22, 23/18)",
      "after": "finite prefix point0 index16 function value = 419/18",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index16 function value，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index16 function value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "22",
          "23/18"
        ],
        "output": "419/18",
        "label": "finite prefix point0 index16 function value"
      }
    },
    {
      "id": "S085",
      "uses": [
        "G001",
        "S084"
      ],
      "before": "finite prefix point0 index16 reverse signed error := sub(419/18, 22)",
      "after": "finite prefix point0 index16 reverse signed error = 23/18",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index16 reverse signed error，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index16 reverse signed error with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "419/18",
          "22"
        ],
        "output": "23/18",
        "label": "finite prefix point0 index16 reverse signed error"
      }
    },
    {
      "id": "S086",
      "uses": [
        "G001",
        "S085"
      ],
      "before": "finite prefix point0 index16:original point=0;n=16;limit=22;signed coefficient=23",
      "after": "f_n(x)=419/18;f_n(x)-limit=23/18;this error field remains SIGNED",
      "relation": "implies",
      "rule_id": "pointwise_selected_value_reverse_error_not_absolute_value",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先加修正項得到函數值，再同減limit反向核誤差；原evaluate欄位保留正負，只有epsilon界才取絕對值。",
        "en": "Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S087",
      "uses": [
        "G001",
        "S086"
      ],
      "before": "finite prefix point0 index17 denominator := sum(17, 2)",
      "after": "finite prefix point0 index17 denominator = 19",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index17 denominator，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index17 denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "17",
          "2"
        ],
        "output": "19",
        "label": "finite prefix point0 index17 denominator"
      }
    },
    {
      "id": "S088",
      "uses": [
        "G001",
        "S087"
      ],
      "before": "finite prefix point0 index17 signed correction := div(23, 19)",
      "after": "finite prefix point0 index17 signed correction = 23/19",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index17 signed correction，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index17 signed correction with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "23",
          "19"
        ],
        "output": "23/19",
        "label": "finite prefix point0 index17 signed correction"
      }
    },
    {
      "id": "S089",
      "uses": [
        "G001",
        "S088"
      ],
      "before": "finite prefix point0 index17 function value := sum(22, 23/19)",
      "after": "finite prefix point0 index17 function value = 441/19",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index17 function value，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index17 function value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "22",
          "23/19"
        ],
        "output": "441/19",
        "label": "finite prefix point0 index17 function value"
      }
    },
    {
      "id": "S090",
      "uses": [
        "G001",
        "S089"
      ],
      "before": "finite prefix point0 index17 reverse signed error := sub(441/19, 22)",
      "after": "finite prefix point0 index17 reverse signed error = 23/19",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index17 reverse signed error，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index17 reverse signed error with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "441/19",
          "22"
        ],
        "output": "23/19",
        "label": "finite prefix point0 index17 reverse signed error"
      }
    },
    {
      "id": "S091",
      "uses": [
        "G001",
        "S090"
      ],
      "before": "finite prefix point0 index17:original point=0;n=17;limit=22;signed coefficient=23",
      "after": "f_n(x)=441/19;f_n(x)-limit=23/19;this error field remains SIGNED",
      "relation": "implies",
      "rule_id": "pointwise_selected_value_reverse_error_not_absolute_value",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先加修正項得到函數值，再同減limit反向核誤差；原evaluate欄位保留正負，只有epsilon界才取絕對值。",
        "en": "Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S092",
      "uses": [
        "G001",
        "S091"
      ],
      "before": "finite prefix point0 index18 denominator := sum(18, 2)",
      "after": "finite prefix point0 index18 denominator = 20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index18 denominator，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index18 denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "18",
          "2"
        ],
        "output": "20",
        "label": "finite prefix point0 index18 denominator"
      }
    },
    {
      "id": "S093",
      "uses": [
        "G001",
        "S092"
      ],
      "before": "finite prefix point0 index18 signed correction := div(23, 20)",
      "after": "finite prefix point0 index18 signed correction = 23/20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index18 signed correction，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index18 signed correction with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "23",
          "20"
        ],
        "output": "23/20",
        "label": "finite prefix point0 index18 signed correction"
      }
    },
    {
      "id": "S094",
      "uses": [
        "G001",
        "S093"
      ],
      "before": "finite prefix point0 index18 function value := sum(22, 23/20)",
      "after": "finite prefix point0 index18 function value = 463/20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index18 function value，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index18 function value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "22",
          "23/20"
        ],
        "output": "463/20",
        "label": "finite prefix point0 index18 function value"
      }
    },
    {
      "id": "S095",
      "uses": [
        "G001",
        "S094"
      ],
      "before": "finite prefix point0 index18 reverse signed error := sub(463/20, 22)",
      "after": "finite prefix point0 index18 reverse signed error = 23/20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index18 reverse signed error，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index18 reverse signed error with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "463/20",
          "22"
        ],
        "output": "23/20",
        "label": "finite prefix point0 index18 reverse signed error"
      }
    },
    {
      "id": "S096",
      "uses": [
        "G001",
        "S095"
      ],
      "before": "finite prefix point0 index18:original point=0;n=18;limit=22;signed coefficient=23",
      "after": "f_n(x)=463/20;f_n(x)-limit=23/20;this error field remains SIGNED",
      "relation": "implies",
      "rule_id": "pointwise_selected_value_reverse_error_not_absolute_value",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先加修正項得到函數值，再同減limit反向核誤差；原evaluate欄位保留正負，只有epsilon界才取絕對值。",
        "en": "Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S097",
      "uses": [
        "G001",
        "S096"
      ],
      "before": "finite prefix point0 index19 denominator := sum(19, 2)",
      "after": "finite prefix point0 index19 denominator = 21",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index19 denominator，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index19 denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "19",
          "2"
        ],
        "output": "21",
        "label": "finite prefix point0 index19 denominator"
      }
    },
    {
      "id": "S098",
      "uses": [
        "G001",
        "S097"
      ],
      "before": "finite prefix point0 index19 signed correction := div(23, 21)",
      "after": "finite prefix point0 index19 signed correction = 23/21",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index19 signed correction，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index19 signed correction with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "23",
          "21"
        ],
        "output": "23/21",
        "label": "finite prefix point0 index19 signed correction"
      }
    },
    {
      "id": "S099",
      "uses": [
        "G001",
        "S098"
      ],
      "before": "finite prefix point0 index19 function value := sum(22, 23/21)",
      "after": "finite prefix point0 index19 function value = 485/21",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index19 function value，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index19 function value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "22",
          "23/21"
        ],
        "output": "485/21",
        "label": "finite prefix point0 index19 function value"
      }
    },
    {
      "id": "S100",
      "uses": [
        "G001",
        "S099"
      ],
      "before": "finite prefix point0 index19 reverse signed error := sub(485/21, 22)",
      "after": "finite prefix point0 index19 reverse signed error = 23/21",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index19 reverse signed error，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index19 reverse signed error with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "485/21",
          "22"
        ],
        "output": "23/21",
        "label": "finite prefix point0 index19 reverse signed error"
      }
    },
    {
      "id": "S101",
      "uses": [
        "G001",
        "S100"
      ],
      "before": "finite prefix point0 index19:original point=0;n=19;limit=22;signed coefficient=23",
      "after": "f_n(x)=485/21;f_n(x)-limit=23/21;this error field remains SIGNED",
      "relation": "implies",
      "rule_id": "pointwise_selected_value_reverse_error_not_absolute_value",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先加修正項得到函數值，再同減limit反向核誤差；原evaluate欄位保留正負，只有epsilon界才取絕對值。",
        "en": "Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S102",
      "uses": [
        "G001",
        "S101"
      ],
      "before": "finite prefix point0 index20 denominator := sum(20, 2)",
      "after": "finite prefix point0 index20 denominator = 22",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index20 denominator，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index20 denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "20",
          "2"
        ],
        "output": "22",
        "label": "finite prefix point0 index20 denominator"
      }
    },
    {
      "id": "S103",
      "uses": [
        "G001",
        "S102"
      ],
      "before": "finite prefix point0 index20 signed correction := div(23, 22)",
      "after": "finite prefix point0 index20 signed correction = 23/22",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index20 signed correction，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index20 signed correction with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "23",
          "22"
        ],
        "output": "23/22",
        "label": "finite prefix point0 index20 signed correction"
      }
    },
    {
      "id": "S104",
      "uses": [
        "G001",
        "S103"
      ],
      "before": "finite prefix point0 index20 function value := sum(22, 23/22)",
      "after": "finite prefix point0 index20 function value = 507/22",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index20 function value，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index20 function value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "22",
          "23/22"
        ],
        "output": "507/22",
        "label": "finite prefix point0 index20 function value"
      }
    },
    {
      "id": "S105",
      "uses": [
        "G001",
        "S104"
      ],
      "before": "finite prefix point0 index20 reverse signed error := sub(507/22, 22)",
      "after": "finite prefix point0 index20 reverse signed error = 23/22",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index20 reverse signed error，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index20 reverse signed error with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "507/22",
          "22"
        ],
        "output": "23/22",
        "label": "finite prefix point0 index20 reverse signed error"
      }
    },
    {
      "id": "S106",
      "uses": [
        "G001",
        "S105"
      ],
      "before": "finite prefix point0 index20:original point=0;n=20;limit=22;signed coefficient=23",
      "after": "f_n(x)=507/22;f_n(x)-limit=23/22;this error field remains SIGNED",
      "relation": "implies",
      "rule_id": "pointwise_selected_value_reverse_error_not_absolute_value",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先加修正項得到函數值，再同減limit反向核誤差；原evaluate欄位保留正負，只有epsilon界才取絕對值。",
        "en": "Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S107",
      "uses": [
        "G001",
        "S106"
      ],
      "before": "finite prefix point0 index21 denominator := sum(21, 2)",
      "after": "finite prefix point0 index21 denominator = 23",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index21 denominator，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index21 denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "21",
          "2"
        ],
        "output": "23",
        "label": "finite prefix point0 index21 denominator"
      }
    },
    {
      "id": "S108",
      "uses": [
        "G001",
        "S107"
      ],
      "before": "finite prefix point0 index21 signed correction := div(23, 23)",
      "after": "finite prefix point0 index21 signed correction = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index21 signed correction，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index21 signed correction with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "23",
          "23"
        ],
        "output": "1",
        "label": "finite prefix point0 index21 signed correction"
      }
    },
    {
      "id": "S109",
      "uses": [
        "G001",
        "S108"
      ],
      "before": "finite prefix point0 index21 function value := sum(22, 1)",
      "after": "finite prefix point0 index21 function value = 23",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index21 function value，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index21 function value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "22",
          "1"
        ],
        "output": "23",
        "label": "finite prefix point0 index21 function value"
      }
    },
    {
      "id": "S110",
      "uses": [
        "G001",
        "S109"
      ],
      "before": "finite prefix point0 index21 reverse signed error := sub(23, 22)",
      "after": "finite prefix point0 index21 reverse signed error = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index21 reverse signed error，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index21 reverse signed error with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "23",
          "22"
        ],
        "output": "1",
        "label": "finite prefix point0 index21 reverse signed error"
      }
    },
    {
      "id": "S111",
      "uses": [
        "G001",
        "S110"
      ],
      "before": "finite prefix point0 index21:original point=0;n=21;limit=22;signed coefficient=23",
      "after": "f_n(x)=23;f_n(x)-limit=1;this error field remains SIGNED",
      "relation": "implies",
      "rule_id": "pointwise_selected_value_reverse_error_not_absolute_value",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先加修正項得到函數值，再同減limit反向核誤差；原evaluate欄位保留正負，只有epsilon界才取絕對值。",
        "en": "Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S112",
      "uses": [
        "G001",
        "S111"
      ],
      "before": "finite prefix point0 index22 denominator := sum(22, 2)",
      "after": "finite prefix point0 index22 denominator = 24",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index22 denominator，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index22 denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "22",
          "2"
        ],
        "output": "24",
        "label": "finite prefix point0 index22 denominator"
      }
    },
    {
      "id": "S113",
      "uses": [
        "G001",
        "S112"
      ],
      "before": "finite prefix point0 index22 signed correction := div(23, 24)",
      "after": "finite prefix point0 index22 signed correction = 23/24",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index22 signed correction，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index22 signed correction with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "23",
          "24"
        ],
        "output": "23/24",
        "label": "finite prefix point0 index22 signed correction"
      }
    },
    {
      "id": "S114",
      "uses": [
        "G001",
        "S113"
      ],
      "before": "finite prefix point0 index22 function value := sum(22, 23/24)",
      "after": "finite prefix point0 index22 function value = 551/24",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index22 function value，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index22 function value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "22",
          "23/24"
        ],
        "output": "551/24",
        "label": "finite prefix point0 index22 function value"
      }
    },
    {
      "id": "S115",
      "uses": [
        "G001",
        "S114"
      ],
      "before": "finite prefix point0 index22 reverse signed error := sub(551/24, 22)",
      "after": "finite prefix point0 index22 reverse signed error = 23/24",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point0 index22 reverse signed error，不以小數近似替代。",
          "en": "Recompute finite prefix point0 index22 reverse signed error with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "551/24",
          "22"
        ],
        "output": "23/24",
        "label": "finite prefix point0 index22 reverse signed error"
      }
    },
    {
      "id": "S116",
      "uses": [
        "G001",
        "S115"
      ],
      "before": "finite prefix point0 index22:original point=0;n=22;limit=22;signed coefficient=23",
      "after": "f_n(x)=551/24;f_n(x)-limit=23/24;this error field remains SIGNED",
      "relation": "implies",
      "rule_id": "pointwise_selected_value_reverse_error_not_absolute_value",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先加修正項得到函數值，再同減limit反向核誤差；原evaluate欄位保留正負，只有epsilon界才取絕對值。",
        "en": "Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S117",
      "uses": [
        "G001",
        "S116"
      ],
      "before": "finite prefix point1 index0 denominator := sum(0, 2)",
      "after": "finite prefix point1 index0 denominator = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index0 denominator，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index0 denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "2"
        ],
        "output": "2",
        "label": "finite prefix point1 index0 denominator"
      }
    },
    {
      "id": "S118",
      "uses": [
        "G001",
        "S117"
      ],
      "before": "finite prefix point1 index0 signed correction := div(-24, 2)",
      "after": "finite prefix point1 index0 signed correction = -12",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index0 signed correction，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index0 signed correction with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "-24",
          "2"
        ],
        "output": "-12",
        "label": "finite prefix point1 index0 signed correction"
      }
    },
    {
      "id": "S119",
      "uses": [
        "G001",
        "S118"
      ],
      "before": "finite prefix point1 index0 function value := sum(23, -12)",
      "after": "finite prefix point1 index0 function value = 11",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index0 function value，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index0 function value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "23",
          "-12"
        ],
        "output": "11",
        "label": "finite prefix point1 index0 function value"
      }
    },
    {
      "id": "S120",
      "uses": [
        "G001",
        "S119"
      ],
      "before": "finite prefix point1 index0 reverse signed error := sub(11, 23)",
      "after": "finite prefix point1 index0 reverse signed error = -12",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index0 reverse signed error，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index0 reverse signed error with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "23"
        ],
        "output": "-12",
        "label": "finite prefix point1 index0 reverse signed error"
      }
    },
    {
      "id": "S121",
      "uses": [
        "G001",
        "S120"
      ],
      "before": "finite prefix point1 index0:original point=1;n=0;limit=23;signed coefficient=-24",
      "after": "f_n(x)=11;f_n(x)-limit=-12;this error field remains SIGNED",
      "relation": "implies",
      "rule_id": "pointwise_selected_value_reverse_error_not_absolute_value",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先加修正項得到函數值，再同減limit反向核誤差；原evaluate欄位保留正負，只有epsilon界才取絕對值。",
        "en": "Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S122",
      "uses": [
        "G001",
        "S121"
      ],
      "before": "finite prefix point1 index1 denominator := sum(1, 2)",
      "after": "finite prefix point1 index1 denominator = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index1 denominator，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index1 denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "2"
        ],
        "output": "3",
        "label": "finite prefix point1 index1 denominator"
      }
    },
    {
      "id": "S123",
      "uses": [
        "G001",
        "S122"
      ],
      "before": "finite prefix point1 index1 signed correction := div(-24, 3)",
      "after": "finite prefix point1 index1 signed correction = -8",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index1 signed correction，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index1 signed correction with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "-24",
          "3"
        ],
        "output": "-8",
        "label": "finite prefix point1 index1 signed correction"
      }
    },
    {
      "id": "S124",
      "uses": [
        "G001",
        "S123"
      ],
      "before": "finite prefix point1 index1 function value := sum(23, -8)",
      "after": "finite prefix point1 index1 function value = 15",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index1 function value，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index1 function value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "23",
          "-8"
        ],
        "output": "15",
        "label": "finite prefix point1 index1 function value"
      }
    },
    {
      "id": "S125",
      "uses": [
        "G001",
        "S124"
      ],
      "before": "finite prefix point1 index1 reverse signed error := sub(15, 23)",
      "after": "finite prefix point1 index1 reverse signed error = -8",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index1 reverse signed error，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index1 reverse signed error with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "15",
          "23"
        ],
        "output": "-8",
        "label": "finite prefix point1 index1 reverse signed error"
      }
    },
    {
      "id": "S126",
      "uses": [
        "G001",
        "S125"
      ],
      "before": "finite prefix point1 index1:original point=1;n=1;limit=23;signed coefficient=-24",
      "after": "f_n(x)=15;f_n(x)-limit=-8;this error field remains SIGNED",
      "relation": "implies",
      "rule_id": "pointwise_selected_value_reverse_error_not_absolute_value",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先加修正項得到函數值，再同減limit反向核誤差；原evaluate欄位保留正負，只有epsilon界才取絕對值。",
        "en": "Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S127",
      "uses": [
        "G001",
        "S126"
      ],
      "before": "finite prefix point1 index2 denominator := sum(2, 2)",
      "after": "finite prefix point1 index2 denominator = 4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index2 denominator，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index2 denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "2",
          "2"
        ],
        "output": "4",
        "label": "finite prefix point1 index2 denominator"
      }
    },
    {
      "id": "S128",
      "uses": [
        "G001",
        "S127"
      ],
      "before": "finite prefix point1 index2 signed correction := div(-24, 4)",
      "after": "finite prefix point1 index2 signed correction = -6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index2 signed correction，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index2 signed correction with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "-24",
          "4"
        ],
        "output": "-6",
        "label": "finite prefix point1 index2 signed correction"
      }
    },
    {
      "id": "S129",
      "uses": [
        "G001",
        "S128"
      ],
      "before": "finite prefix point1 index2 function value := sum(23, -6)",
      "after": "finite prefix point1 index2 function value = 17",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index2 function value，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index2 function value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "23",
          "-6"
        ],
        "output": "17",
        "label": "finite prefix point1 index2 function value"
      }
    },
    {
      "id": "S130",
      "uses": [
        "G001",
        "S129"
      ],
      "before": "finite prefix point1 index2 reverse signed error := sub(17, 23)",
      "after": "finite prefix point1 index2 reverse signed error = -6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index2 reverse signed error，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index2 reverse signed error with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "17",
          "23"
        ],
        "output": "-6",
        "label": "finite prefix point1 index2 reverse signed error"
      }
    },
    {
      "id": "S131",
      "uses": [
        "G001",
        "S130"
      ],
      "before": "finite prefix point1 index2:original point=1;n=2;limit=23;signed coefficient=-24",
      "after": "f_n(x)=17;f_n(x)-limit=-6;this error field remains SIGNED",
      "relation": "implies",
      "rule_id": "pointwise_selected_value_reverse_error_not_absolute_value",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先加修正項得到函數值，再同減limit反向核誤差；原evaluate欄位保留正負，只有epsilon界才取絕對值。",
        "en": "Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S132",
      "uses": [
        "G001",
        "S131"
      ],
      "before": "finite prefix point1 index3 denominator := sum(3, 2)",
      "after": "finite prefix point1 index3 denominator = 5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index3 denominator，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index3 denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "3",
          "2"
        ],
        "output": "5",
        "label": "finite prefix point1 index3 denominator"
      }
    },
    {
      "id": "S133",
      "uses": [
        "G001",
        "S132"
      ],
      "before": "finite prefix point1 index3 signed correction := div(-24, 5)",
      "after": "finite prefix point1 index3 signed correction = -24/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index3 signed correction，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index3 signed correction with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "-24",
          "5"
        ],
        "output": "-24/5",
        "label": "finite prefix point1 index3 signed correction"
      }
    },
    {
      "id": "S134",
      "uses": [
        "G001",
        "S133"
      ],
      "before": "finite prefix point1 index3 function value := sum(23, -24/5)",
      "after": "finite prefix point1 index3 function value = 91/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index3 function value，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index3 function value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "23",
          "-24/5"
        ],
        "output": "91/5",
        "label": "finite prefix point1 index3 function value"
      }
    },
    {
      "id": "S135",
      "uses": [
        "G001",
        "S134"
      ],
      "before": "finite prefix point1 index3 reverse signed error := sub(91/5, 23)",
      "after": "finite prefix point1 index3 reverse signed error = -24/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index3 reverse signed error，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index3 reverse signed error with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "91/5",
          "23"
        ],
        "output": "-24/5",
        "label": "finite prefix point1 index3 reverse signed error"
      }
    },
    {
      "id": "S136",
      "uses": [
        "G001",
        "S135"
      ],
      "before": "finite prefix point1 index3:original point=1;n=3;limit=23;signed coefficient=-24",
      "after": "f_n(x)=91/5;f_n(x)-limit=-24/5;this error field remains SIGNED",
      "relation": "implies",
      "rule_id": "pointwise_selected_value_reverse_error_not_absolute_value",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先加修正項得到函數值，再同減limit反向核誤差；原evaluate欄位保留正負，只有epsilon界才取絕對值。",
        "en": "Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S137",
      "uses": [
        "G001",
        "S136"
      ],
      "before": "finite prefix point1 index4 denominator := sum(4, 2)",
      "after": "finite prefix point1 index4 denominator = 6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index4 denominator，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index4 denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "4",
          "2"
        ],
        "output": "6",
        "label": "finite prefix point1 index4 denominator"
      }
    },
    {
      "id": "S138",
      "uses": [
        "G001",
        "S137"
      ],
      "before": "finite prefix point1 index4 signed correction := div(-24, 6)",
      "after": "finite prefix point1 index4 signed correction = -4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index4 signed correction，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index4 signed correction with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "-24",
          "6"
        ],
        "output": "-4",
        "label": "finite prefix point1 index4 signed correction"
      }
    },
    {
      "id": "S139",
      "uses": [
        "G001",
        "S138"
      ],
      "before": "finite prefix point1 index4 function value := sum(23, -4)",
      "after": "finite prefix point1 index4 function value = 19",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index4 function value，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index4 function value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "23",
          "-4"
        ],
        "output": "19",
        "label": "finite prefix point1 index4 function value"
      }
    },
    {
      "id": "S140",
      "uses": [
        "G001",
        "S139"
      ],
      "before": "finite prefix point1 index4 reverse signed error := sub(19, 23)",
      "after": "finite prefix point1 index4 reverse signed error = -4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index4 reverse signed error，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index4 reverse signed error with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "19",
          "23"
        ],
        "output": "-4",
        "label": "finite prefix point1 index4 reverse signed error"
      }
    },
    {
      "id": "S141",
      "uses": [
        "G001",
        "S140"
      ],
      "before": "finite prefix point1 index4:original point=1;n=4;limit=23;signed coefficient=-24",
      "after": "f_n(x)=19;f_n(x)-limit=-4;this error field remains SIGNED",
      "relation": "implies",
      "rule_id": "pointwise_selected_value_reverse_error_not_absolute_value",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先加修正項得到函數值，再同減limit反向核誤差；原evaluate欄位保留正負，只有epsilon界才取絕對值。",
        "en": "Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S142",
      "uses": [
        "G001",
        "S141"
      ],
      "before": "finite prefix point1 index5 denominator := sum(5, 2)",
      "after": "finite prefix point1 index5 denominator = 7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index5 denominator，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index5 denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "5",
          "2"
        ],
        "output": "7",
        "label": "finite prefix point1 index5 denominator"
      }
    },
    {
      "id": "S143",
      "uses": [
        "G001",
        "S142"
      ],
      "before": "finite prefix point1 index5 signed correction := div(-24, 7)",
      "after": "finite prefix point1 index5 signed correction = -24/7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index5 signed correction，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index5 signed correction with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "-24",
          "7"
        ],
        "output": "-24/7",
        "label": "finite prefix point1 index5 signed correction"
      }
    },
    {
      "id": "S144",
      "uses": [
        "G001",
        "S143"
      ],
      "before": "finite prefix point1 index5 function value := sum(23, -24/7)",
      "after": "finite prefix point1 index5 function value = 137/7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index5 function value，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index5 function value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "23",
          "-24/7"
        ],
        "output": "137/7",
        "label": "finite prefix point1 index5 function value"
      }
    },
    {
      "id": "S145",
      "uses": [
        "G001",
        "S144"
      ],
      "before": "finite prefix point1 index5 reverse signed error := sub(137/7, 23)",
      "after": "finite prefix point1 index5 reverse signed error = -24/7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index5 reverse signed error，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index5 reverse signed error with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "137/7",
          "23"
        ],
        "output": "-24/7",
        "label": "finite prefix point1 index5 reverse signed error"
      }
    },
    {
      "id": "S146",
      "uses": [
        "G001",
        "S145"
      ],
      "before": "finite prefix point1 index5:original point=1;n=5;limit=23;signed coefficient=-24",
      "after": "f_n(x)=137/7;f_n(x)-limit=-24/7;this error field remains SIGNED",
      "relation": "implies",
      "rule_id": "pointwise_selected_value_reverse_error_not_absolute_value",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先加修正項得到函數值，再同減limit反向核誤差；原evaluate欄位保留正負，只有epsilon界才取絕對值。",
        "en": "Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S147",
      "uses": [
        "G001",
        "S146"
      ],
      "before": "finite prefix point1 index6 denominator := sum(6, 2)",
      "after": "finite prefix point1 index6 denominator = 8",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index6 denominator，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index6 denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "6",
          "2"
        ],
        "output": "8",
        "label": "finite prefix point1 index6 denominator"
      }
    },
    {
      "id": "S148",
      "uses": [
        "G001",
        "S147"
      ],
      "before": "finite prefix point1 index6 signed correction := div(-24, 8)",
      "after": "finite prefix point1 index6 signed correction = -3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index6 signed correction，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index6 signed correction with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "-24",
          "8"
        ],
        "output": "-3",
        "label": "finite prefix point1 index6 signed correction"
      }
    },
    {
      "id": "S149",
      "uses": [
        "G001",
        "S148"
      ],
      "before": "finite prefix point1 index6 function value := sum(23, -3)",
      "after": "finite prefix point1 index6 function value = 20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index6 function value，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index6 function value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "23",
          "-3"
        ],
        "output": "20",
        "label": "finite prefix point1 index6 function value"
      }
    },
    {
      "id": "S150",
      "uses": [
        "G001",
        "S149"
      ],
      "before": "finite prefix point1 index6 reverse signed error := sub(20, 23)",
      "after": "finite prefix point1 index6 reverse signed error = -3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index6 reverse signed error，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index6 reverse signed error with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "20",
          "23"
        ],
        "output": "-3",
        "label": "finite prefix point1 index6 reverse signed error"
      }
    },
    {
      "id": "S151",
      "uses": [
        "G001",
        "S150"
      ],
      "before": "finite prefix point1 index6:original point=1;n=6;limit=23;signed coefficient=-24",
      "after": "f_n(x)=20;f_n(x)-limit=-3;this error field remains SIGNED",
      "relation": "implies",
      "rule_id": "pointwise_selected_value_reverse_error_not_absolute_value",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先加修正項得到函數值，再同減limit反向核誤差；原evaluate欄位保留正負，只有epsilon界才取絕對值。",
        "en": "Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S152",
      "uses": [
        "G001",
        "S151"
      ],
      "before": "finite prefix point1 index7 denominator := sum(7, 2)",
      "after": "finite prefix point1 index7 denominator = 9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index7 denominator，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index7 denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "7",
          "2"
        ],
        "output": "9",
        "label": "finite prefix point1 index7 denominator"
      }
    },
    {
      "id": "S153",
      "uses": [
        "G001",
        "S152"
      ],
      "before": "finite prefix point1 index7 signed correction := div(-24, 9)",
      "after": "finite prefix point1 index7 signed correction = -8/3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index7 signed correction，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index7 signed correction with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "-24",
          "9"
        ],
        "output": "-8/3",
        "label": "finite prefix point1 index7 signed correction"
      }
    },
    {
      "id": "S154",
      "uses": [
        "G001",
        "S153"
      ],
      "before": "finite prefix point1 index7 function value := sum(23, -8/3)",
      "after": "finite prefix point1 index7 function value = 61/3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index7 function value，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index7 function value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "23",
          "-8/3"
        ],
        "output": "61/3",
        "label": "finite prefix point1 index7 function value"
      }
    },
    {
      "id": "S155",
      "uses": [
        "G001",
        "S154"
      ],
      "before": "finite prefix point1 index7 reverse signed error := sub(61/3, 23)",
      "after": "finite prefix point1 index7 reverse signed error = -8/3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index7 reverse signed error，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index7 reverse signed error with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "61/3",
          "23"
        ],
        "output": "-8/3",
        "label": "finite prefix point1 index7 reverse signed error"
      }
    },
    {
      "id": "S156",
      "uses": [
        "G001",
        "S155"
      ],
      "before": "finite prefix point1 index7:original point=1;n=7;limit=23;signed coefficient=-24",
      "after": "f_n(x)=61/3;f_n(x)-limit=-8/3;this error field remains SIGNED",
      "relation": "implies",
      "rule_id": "pointwise_selected_value_reverse_error_not_absolute_value",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先加修正項得到函數值，再同減limit反向核誤差；原evaluate欄位保留正負，只有epsilon界才取絕對值。",
        "en": "Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S157",
      "uses": [
        "G001",
        "S156"
      ],
      "before": "finite prefix point1 index8 denominator := sum(8, 2)",
      "after": "finite prefix point1 index8 denominator = 10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index8 denominator，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index8 denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "8",
          "2"
        ],
        "output": "10",
        "label": "finite prefix point1 index8 denominator"
      }
    },
    {
      "id": "S158",
      "uses": [
        "G001",
        "S157"
      ],
      "before": "finite prefix point1 index8 signed correction := div(-24, 10)",
      "after": "finite prefix point1 index8 signed correction = -12/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index8 signed correction，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index8 signed correction with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "-24",
          "10"
        ],
        "output": "-12/5",
        "label": "finite prefix point1 index8 signed correction"
      }
    },
    {
      "id": "S159",
      "uses": [
        "G001",
        "S158"
      ],
      "before": "finite prefix point1 index8 function value := sum(23, -12/5)",
      "after": "finite prefix point1 index8 function value = 103/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index8 function value，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index8 function value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "23",
          "-12/5"
        ],
        "output": "103/5",
        "label": "finite prefix point1 index8 function value"
      }
    },
    {
      "id": "S160",
      "uses": [
        "G001",
        "S159"
      ],
      "before": "finite prefix point1 index8 reverse signed error := sub(103/5, 23)",
      "after": "finite prefix point1 index8 reverse signed error = -12/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index8 reverse signed error，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index8 reverse signed error with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "103/5",
          "23"
        ],
        "output": "-12/5",
        "label": "finite prefix point1 index8 reverse signed error"
      }
    },
    {
      "id": "S161",
      "uses": [
        "G001",
        "S160"
      ],
      "before": "finite prefix point1 index8:original point=1;n=8;limit=23;signed coefficient=-24",
      "after": "f_n(x)=103/5;f_n(x)-limit=-12/5;this error field remains SIGNED",
      "relation": "implies",
      "rule_id": "pointwise_selected_value_reverse_error_not_absolute_value",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先加修正項得到函數值，再同減limit反向核誤差；原evaluate欄位保留正負，只有epsilon界才取絕對值。",
        "en": "Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S162",
      "uses": [
        "G001",
        "S161"
      ],
      "before": "finite prefix point1 index9 denominator := sum(9, 2)",
      "after": "finite prefix point1 index9 denominator = 11",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index9 denominator，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index9 denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "9",
          "2"
        ],
        "output": "11",
        "label": "finite prefix point1 index9 denominator"
      }
    },
    {
      "id": "S163",
      "uses": [
        "G001",
        "S162"
      ],
      "before": "finite prefix point1 index9 signed correction := div(-24, 11)",
      "after": "finite prefix point1 index9 signed correction = -24/11",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index9 signed correction，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index9 signed correction with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "-24",
          "11"
        ],
        "output": "-24/11",
        "label": "finite prefix point1 index9 signed correction"
      }
    },
    {
      "id": "S164",
      "uses": [
        "G001",
        "S163"
      ],
      "before": "finite prefix point1 index9 function value := sum(23, -24/11)",
      "after": "finite prefix point1 index9 function value = 229/11",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index9 function value，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index9 function value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "23",
          "-24/11"
        ],
        "output": "229/11",
        "label": "finite prefix point1 index9 function value"
      }
    },
    {
      "id": "S165",
      "uses": [
        "G001",
        "S164"
      ],
      "before": "finite prefix point1 index9 reverse signed error := sub(229/11, 23)",
      "after": "finite prefix point1 index9 reverse signed error = -24/11",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index9 reverse signed error，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index9 reverse signed error with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "229/11",
          "23"
        ],
        "output": "-24/11",
        "label": "finite prefix point1 index9 reverse signed error"
      }
    },
    {
      "id": "S166",
      "uses": [
        "G001",
        "S165"
      ],
      "before": "finite prefix point1 index9:original point=1;n=9;limit=23;signed coefficient=-24",
      "after": "f_n(x)=229/11;f_n(x)-limit=-24/11;this error field remains SIGNED",
      "relation": "implies",
      "rule_id": "pointwise_selected_value_reverse_error_not_absolute_value",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先加修正項得到函數值，再同減limit反向核誤差；原evaluate欄位保留正負，只有epsilon界才取絕對值。",
        "en": "Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S167",
      "uses": [
        "G001",
        "S166"
      ],
      "before": "finite prefix point1 index10 denominator := sum(10, 2)",
      "after": "finite prefix point1 index10 denominator = 12",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index10 denominator，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index10 denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "10",
          "2"
        ],
        "output": "12",
        "label": "finite prefix point1 index10 denominator"
      }
    },
    {
      "id": "S168",
      "uses": [
        "G001",
        "S167"
      ],
      "before": "finite prefix point1 index10 signed correction := div(-24, 12)",
      "after": "finite prefix point1 index10 signed correction = -2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index10 signed correction，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index10 signed correction with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "-24",
          "12"
        ],
        "output": "-2",
        "label": "finite prefix point1 index10 signed correction"
      }
    },
    {
      "id": "S169",
      "uses": [
        "G001",
        "S168"
      ],
      "before": "finite prefix point1 index10 function value := sum(23, -2)",
      "after": "finite prefix point1 index10 function value = 21",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index10 function value，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index10 function value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "23",
          "-2"
        ],
        "output": "21",
        "label": "finite prefix point1 index10 function value"
      }
    },
    {
      "id": "S170",
      "uses": [
        "G001",
        "S169"
      ],
      "before": "finite prefix point1 index10 reverse signed error := sub(21, 23)",
      "after": "finite prefix point1 index10 reverse signed error = -2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index10 reverse signed error，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index10 reverse signed error with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "21",
          "23"
        ],
        "output": "-2",
        "label": "finite prefix point1 index10 reverse signed error"
      }
    },
    {
      "id": "S171",
      "uses": [
        "G001",
        "S170"
      ],
      "before": "finite prefix point1 index10:original point=1;n=10;limit=23;signed coefficient=-24",
      "after": "f_n(x)=21;f_n(x)-limit=-2;this error field remains SIGNED",
      "relation": "implies",
      "rule_id": "pointwise_selected_value_reverse_error_not_absolute_value",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先加修正項得到函數值，再同減limit反向核誤差；原evaluate欄位保留正負，只有epsilon界才取絕對值。",
        "en": "Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S172",
      "uses": [
        "G001",
        "S171"
      ],
      "before": "finite prefix point1 index11 denominator := sum(11, 2)",
      "after": "finite prefix point1 index11 denominator = 13",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index11 denominator，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index11 denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "11",
          "2"
        ],
        "output": "13",
        "label": "finite prefix point1 index11 denominator"
      }
    },
    {
      "id": "S173",
      "uses": [
        "G001",
        "S172"
      ],
      "before": "finite prefix point1 index11 signed correction := div(-24, 13)",
      "after": "finite prefix point1 index11 signed correction = -24/13",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index11 signed correction，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index11 signed correction with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "-24",
          "13"
        ],
        "output": "-24/13",
        "label": "finite prefix point1 index11 signed correction"
      }
    },
    {
      "id": "S174",
      "uses": [
        "G001",
        "S173"
      ],
      "before": "finite prefix point1 index11 function value := sum(23, -24/13)",
      "after": "finite prefix point1 index11 function value = 275/13",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index11 function value，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index11 function value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "23",
          "-24/13"
        ],
        "output": "275/13",
        "label": "finite prefix point1 index11 function value"
      }
    },
    {
      "id": "S175",
      "uses": [
        "G001",
        "S174"
      ],
      "before": "finite prefix point1 index11 reverse signed error := sub(275/13, 23)",
      "after": "finite prefix point1 index11 reverse signed error = -24/13",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index11 reverse signed error，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index11 reverse signed error with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "275/13",
          "23"
        ],
        "output": "-24/13",
        "label": "finite prefix point1 index11 reverse signed error"
      }
    },
    {
      "id": "S176",
      "uses": [
        "G001",
        "S175"
      ],
      "before": "finite prefix point1 index11:original point=1;n=11;limit=23;signed coefficient=-24",
      "after": "f_n(x)=275/13;f_n(x)-limit=-24/13;this error field remains SIGNED",
      "relation": "implies",
      "rule_id": "pointwise_selected_value_reverse_error_not_absolute_value",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先加修正項得到函數值，再同減limit反向核誤差；原evaluate欄位保留正負，只有epsilon界才取絕對值。",
        "en": "Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S177",
      "uses": [
        "G001",
        "S176"
      ],
      "before": "finite prefix point1 index12 denominator := sum(12, 2)",
      "after": "finite prefix point1 index12 denominator = 14",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index12 denominator，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index12 denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "12",
          "2"
        ],
        "output": "14",
        "label": "finite prefix point1 index12 denominator"
      }
    },
    {
      "id": "S178",
      "uses": [
        "G001",
        "S177"
      ],
      "before": "finite prefix point1 index12 signed correction := div(-24, 14)",
      "after": "finite prefix point1 index12 signed correction = -12/7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index12 signed correction，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index12 signed correction with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "-24",
          "14"
        ],
        "output": "-12/7",
        "label": "finite prefix point1 index12 signed correction"
      }
    },
    {
      "id": "S179",
      "uses": [
        "G001",
        "S178"
      ],
      "before": "finite prefix point1 index12 function value := sum(23, -12/7)",
      "after": "finite prefix point1 index12 function value = 149/7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index12 function value，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index12 function value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "23",
          "-12/7"
        ],
        "output": "149/7",
        "label": "finite prefix point1 index12 function value"
      }
    },
    {
      "id": "S180",
      "uses": [
        "G001",
        "S179"
      ],
      "before": "finite prefix point1 index12 reverse signed error := sub(149/7, 23)",
      "after": "finite prefix point1 index12 reverse signed error = -12/7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index12 reverse signed error，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index12 reverse signed error with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "149/7",
          "23"
        ],
        "output": "-12/7",
        "label": "finite prefix point1 index12 reverse signed error"
      }
    },
    {
      "id": "S181",
      "uses": [
        "G001",
        "S180"
      ],
      "before": "finite prefix point1 index12:original point=1;n=12;limit=23;signed coefficient=-24",
      "after": "f_n(x)=149/7;f_n(x)-limit=-12/7;this error field remains SIGNED",
      "relation": "implies",
      "rule_id": "pointwise_selected_value_reverse_error_not_absolute_value",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先加修正項得到函數值，再同減limit反向核誤差；原evaluate欄位保留正負，只有epsilon界才取絕對值。",
        "en": "Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S182",
      "uses": [
        "G001",
        "S181"
      ],
      "before": "finite prefix point1 index13 denominator := sum(13, 2)",
      "after": "finite prefix point1 index13 denominator = 15",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index13 denominator，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index13 denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "13",
          "2"
        ],
        "output": "15",
        "label": "finite prefix point1 index13 denominator"
      }
    },
    {
      "id": "S183",
      "uses": [
        "G001",
        "S182"
      ],
      "before": "finite prefix point1 index13 signed correction := div(-24, 15)",
      "after": "finite prefix point1 index13 signed correction = -8/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index13 signed correction，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index13 signed correction with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "-24",
          "15"
        ],
        "output": "-8/5",
        "label": "finite prefix point1 index13 signed correction"
      }
    },
    {
      "id": "S184",
      "uses": [
        "G001",
        "S183"
      ],
      "before": "finite prefix point1 index13 function value := sum(23, -8/5)",
      "after": "finite prefix point1 index13 function value = 107/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index13 function value，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index13 function value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "23",
          "-8/5"
        ],
        "output": "107/5",
        "label": "finite prefix point1 index13 function value"
      }
    },
    {
      "id": "S185",
      "uses": [
        "G001",
        "S184"
      ],
      "before": "finite prefix point1 index13 reverse signed error := sub(107/5, 23)",
      "after": "finite prefix point1 index13 reverse signed error = -8/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index13 reverse signed error，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index13 reverse signed error with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "107/5",
          "23"
        ],
        "output": "-8/5",
        "label": "finite prefix point1 index13 reverse signed error"
      }
    },
    {
      "id": "S186",
      "uses": [
        "G001",
        "S185"
      ],
      "before": "finite prefix point1 index13:original point=1;n=13;limit=23;signed coefficient=-24",
      "after": "f_n(x)=107/5;f_n(x)-limit=-8/5;this error field remains SIGNED",
      "relation": "implies",
      "rule_id": "pointwise_selected_value_reverse_error_not_absolute_value",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先加修正項得到函數值，再同減limit反向核誤差；原evaluate欄位保留正負，只有epsilon界才取絕對值。",
        "en": "Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S187",
      "uses": [
        "G001",
        "S186"
      ],
      "before": "finite prefix point1 index14 denominator := sum(14, 2)",
      "after": "finite prefix point1 index14 denominator = 16",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index14 denominator，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index14 denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "14",
          "2"
        ],
        "output": "16",
        "label": "finite prefix point1 index14 denominator"
      }
    },
    {
      "id": "S188",
      "uses": [
        "G001",
        "S187"
      ],
      "before": "finite prefix point1 index14 signed correction := div(-24, 16)",
      "after": "finite prefix point1 index14 signed correction = -3/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index14 signed correction，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index14 signed correction with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "-24",
          "16"
        ],
        "output": "-3/2",
        "label": "finite prefix point1 index14 signed correction"
      }
    },
    {
      "id": "S189",
      "uses": [
        "G001",
        "S188"
      ],
      "before": "finite prefix point1 index14 function value := sum(23, -3/2)",
      "after": "finite prefix point1 index14 function value = 43/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index14 function value，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index14 function value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "23",
          "-3/2"
        ],
        "output": "43/2",
        "label": "finite prefix point1 index14 function value"
      }
    },
    {
      "id": "S190",
      "uses": [
        "G001",
        "S189"
      ],
      "before": "finite prefix point1 index14 reverse signed error := sub(43/2, 23)",
      "after": "finite prefix point1 index14 reverse signed error = -3/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index14 reverse signed error，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index14 reverse signed error with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "43/2",
          "23"
        ],
        "output": "-3/2",
        "label": "finite prefix point1 index14 reverse signed error"
      }
    },
    {
      "id": "S191",
      "uses": [
        "G001",
        "S190"
      ],
      "before": "finite prefix point1 index14:original point=1;n=14;limit=23;signed coefficient=-24",
      "after": "f_n(x)=43/2;f_n(x)-limit=-3/2;this error field remains SIGNED",
      "relation": "implies",
      "rule_id": "pointwise_selected_value_reverse_error_not_absolute_value",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先加修正項得到函數值，再同減limit反向核誤差；原evaluate欄位保留正負，只有epsilon界才取絕對值。",
        "en": "Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S192",
      "uses": [
        "G001",
        "S191"
      ],
      "before": "finite prefix point1 index15 denominator := sum(15, 2)",
      "after": "finite prefix point1 index15 denominator = 17",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index15 denominator，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index15 denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "15",
          "2"
        ],
        "output": "17",
        "label": "finite prefix point1 index15 denominator"
      }
    },
    {
      "id": "S193",
      "uses": [
        "G001",
        "S192"
      ],
      "before": "finite prefix point1 index15 signed correction := div(-24, 17)",
      "after": "finite prefix point1 index15 signed correction = -24/17",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index15 signed correction，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index15 signed correction with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "-24",
          "17"
        ],
        "output": "-24/17",
        "label": "finite prefix point1 index15 signed correction"
      }
    },
    {
      "id": "S194",
      "uses": [
        "G001",
        "S193"
      ],
      "before": "finite prefix point1 index15 function value := sum(23, -24/17)",
      "after": "finite prefix point1 index15 function value = 367/17",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index15 function value，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index15 function value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "23",
          "-24/17"
        ],
        "output": "367/17",
        "label": "finite prefix point1 index15 function value"
      }
    },
    {
      "id": "S195",
      "uses": [
        "G001",
        "S194"
      ],
      "before": "finite prefix point1 index15 reverse signed error := sub(367/17, 23)",
      "after": "finite prefix point1 index15 reverse signed error = -24/17",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index15 reverse signed error，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index15 reverse signed error with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "367/17",
          "23"
        ],
        "output": "-24/17",
        "label": "finite prefix point1 index15 reverse signed error"
      }
    },
    {
      "id": "S196",
      "uses": [
        "G001",
        "S195"
      ],
      "before": "finite prefix point1 index15:original point=1;n=15;limit=23;signed coefficient=-24",
      "after": "f_n(x)=367/17;f_n(x)-limit=-24/17;this error field remains SIGNED",
      "relation": "implies",
      "rule_id": "pointwise_selected_value_reverse_error_not_absolute_value",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先加修正項得到函數值，再同減limit反向核誤差；原evaluate欄位保留正負，只有epsilon界才取絕對值。",
        "en": "Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S197",
      "uses": [
        "G001",
        "S196"
      ],
      "before": "finite prefix point1 index16 denominator := sum(16, 2)",
      "after": "finite prefix point1 index16 denominator = 18",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index16 denominator，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index16 denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "16",
          "2"
        ],
        "output": "18",
        "label": "finite prefix point1 index16 denominator"
      }
    },
    {
      "id": "S198",
      "uses": [
        "G001",
        "S197"
      ],
      "before": "finite prefix point1 index16 signed correction := div(-24, 18)",
      "after": "finite prefix point1 index16 signed correction = -4/3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index16 signed correction，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index16 signed correction with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "-24",
          "18"
        ],
        "output": "-4/3",
        "label": "finite prefix point1 index16 signed correction"
      }
    },
    {
      "id": "S199",
      "uses": [
        "G001",
        "S198"
      ],
      "before": "finite prefix point1 index16 function value := sum(23, -4/3)",
      "after": "finite prefix point1 index16 function value = 65/3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index16 function value，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index16 function value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "23",
          "-4/3"
        ],
        "output": "65/3",
        "label": "finite prefix point1 index16 function value"
      }
    },
    {
      "id": "S200",
      "uses": [
        "G001",
        "S199"
      ],
      "before": "finite prefix point1 index16 reverse signed error := sub(65/3, 23)",
      "after": "finite prefix point1 index16 reverse signed error = -4/3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index16 reverse signed error，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index16 reverse signed error with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "65/3",
          "23"
        ],
        "output": "-4/3",
        "label": "finite prefix point1 index16 reverse signed error"
      }
    },
    {
      "id": "S201",
      "uses": [
        "G001",
        "S200"
      ],
      "before": "finite prefix point1 index16:original point=1;n=16;limit=23;signed coefficient=-24",
      "after": "f_n(x)=65/3;f_n(x)-limit=-4/3;this error field remains SIGNED",
      "relation": "implies",
      "rule_id": "pointwise_selected_value_reverse_error_not_absolute_value",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先加修正項得到函數值，再同減limit反向核誤差；原evaluate欄位保留正負，只有epsilon界才取絕對值。",
        "en": "Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S202",
      "uses": [
        "G001",
        "S201"
      ],
      "before": "finite prefix point1 index17 denominator := sum(17, 2)",
      "after": "finite prefix point1 index17 denominator = 19",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index17 denominator，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index17 denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "17",
          "2"
        ],
        "output": "19",
        "label": "finite prefix point1 index17 denominator"
      }
    },
    {
      "id": "S203",
      "uses": [
        "G001",
        "S202"
      ],
      "before": "finite prefix point1 index17 signed correction := div(-24, 19)",
      "after": "finite prefix point1 index17 signed correction = -24/19",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index17 signed correction，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index17 signed correction with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "-24",
          "19"
        ],
        "output": "-24/19",
        "label": "finite prefix point1 index17 signed correction"
      }
    },
    {
      "id": "S204",
      "uses": [
        "G001",
        "S203"
      ],
      "before": "finite prefix point1 index17 function value := sum(23, -24/19)",
      "after": "finite prefix point1 index17 function value = 413/19",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index17 function value，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index17 function value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "23",
          "-24/19"
        ],
        "output": "413/19",
        "label": "finite prefix point1 index17 function value"
      }
    },
    {
      "id": "S205",
      "uses": [
        "G001",
        "S204"
      ],
      "before": "finite prefix point1 index17 reverse signed error := sub(413/19, 23)",
      "after": "finite prefix point1 index17 reverse signed error = -24/19",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index17 reverse signed error，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index17 reverse signed error with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "413/19",
          "23"
        ],
        "output": "-24/19",
        "label": "finite prefix point1 index17 reverse signed error"
      }
    },
    {
      "id": "S206",
      "uses": [
        "G001",
        "S205"
      ],
      "before": "finite prefix point1 index17:original point=1;n=17;limit=23;signed coefficient=-24",
      "after": "f_n(x)=413/19;f_n(x)-limit=-24/19;this error field remains SIGNED",
      "relation": "implies",
      "rule_id": "pointwise_selected_value_reverse_error_not_absolute_value",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先加修正項得到函數值，再同減limit反向核誤差；原evaluate欄位保留正負，只有epsilon界才取絕對值。",
        "en": "Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S207",
      "uses": [
        "G001",
        "S206"
      ],
      "before": "finite prefix point1 index18 denominator := sum(18, 2)",
      "after": "finite prefix point1 index18 denominator = 20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index18 denominator，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index18 denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "18",
          "2"
        ],
        "output": "20",
        "label": "finite prefix point1 index18 denominator"
      }
    },
    {
      "id": "S208",
      "uses": [
        "G001",
        "S207"
      ],
      "before": "finite prefix point1 index18 signed correction := div(-24, 20)",
      "after": "finite prefix point1 index18 signed correction = -6/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index18 signed correction，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index18 signed correction with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "-24",
          "20"
        ],
        "output": "-6/5",
        "label": "finite prefix point1 index18 signed correction"
      }
    },
    {
      "id": "S209",
      "uses": [
        "G001",
        "S208"
      ],
      "before": "finite prefix point1 index18 function value := sum(23, -6/5)",
      "after": "finite prefix point1 index18 function value = 109/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index18 function value，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index18 function value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "23",
          "-6/5"
        ],
        "output": "109/5",
        "label": "finite prefix point1 index18 function value"
      }
    },
    {
      "id": "S210",
      "uses": [
        "G001",
        "S209"
      ],
      "before": "finite prefix point1 index18 reverse signed error := sub(109/5, 23)",
      "after": "finite prefix point1 index18 reverse signed error = -6/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index18 reverse signed error，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index18 reverse signed error with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "109/5",
          "23"
        ],
        "output": "-6/5",
        "label": "finite prefix point1 index18 reverse signed error"
      }
    },
    {
      "id": "S211",
      "uses": [
        "G001",
        "S210"
      ],
      "before": "finite prefix point1 index18:original point=1;n=18;limit=23;signed coefficient=-24",
      "after": "f_n(x)=109/5;f_n(x)-limit=-6/5;this error field remains SIGNED",
      "relation": "implies",
      "rule_id": "pointwise_selected_value_reverse_error_not_absolute_value",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先加修正項得到函數值，再同減limit反向核誤差；原evaluate欄位保留正負，只有epsilon界才取絕對值。",
        "en": "Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S212",
      "uses": [
        "G001",
        "S211"
      ],
      "before": "finite prefix point1 index19 denominator := sum(19, 2)",
      "after": "finite prefix point1 index19 denominator = 21",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index19 denominator，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index19 denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "19",
          "2"
        ],
        "output": "21",
        "label": "finite prefix point1 index19 denominator"
      }
    },
    {
      "id": "S213",
      "uses": [
        "G001",
        "S212"
      ],
      "before": "finite prefix point1 index19 signed correction := div(-24, 21)",
      "after": "finite prefix point1 index19 signed correction = -8/7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index19 signed correction，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index19 signed correction with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "-24",
          "21"
        ],
        "output": "-8/7",
        "label": "finite prefix point1 index19 signed correction"
      }
    },
    {
      "id": "S214",
      "uses": [
        "G001",
        "S213"
      ],
      "before": "finite prefix point1 index19 function value := sum(23, -8/7)",
      "after": "finite prefix point1 index19 function value = 153/7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index19 function value，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index19 function value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "23",
          "-8/7"
        ],
        "output": "153/7",
        "label": "finite prefix point1 index19 function value"
      }
    },
    {
      "id": "S215",
      "uses": [
        "G001",
        "S214"
      ],
      "before": "finite prefix point1 index19 reverse signed error := sub(153/7, 23)",
      "after": "finite prefix point1 index19 reverse signed error = -8/7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index19 reverse signed error，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index19 reverse signed error with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "153/7",
          "23"
        ],
        "output": "-8/7",
        "label": "finite prefix point1 index19 reverse signed error"
      }
    },
    {
      "id": "S216",
      "uses": [
        "G001",
        "S215"
      ],
      "before": "finite prefix point1 index19:original point=1;n=19;limit=23;signed coefficient=-24",
      "after": "f_n(x)=153/7;f_n(x)-limit=-8/7;this error field remains SIGNED",
      "relation": "implies",
      "rule_id": "pointwise_selected_value_reverse_error_not_absolute_value",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先加修正項得到函數值，再同減limit反向核誤差；原evaluate欄位保留正負，只有epsilon界才取絕對值。",
        "en": "Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S217",
      "uses": [
        "G001",
        "S216"
      ],
      "before": "finite prefix point1 index20 denominator := sum(20, 2)",
      "after": "finite prefix point1 index20 denominator = 22",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index20 denominator，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index20 denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "20",
          "2"
        ],
        "output": "22",
        "label": "finite prefix point1 index20 denominator"
      }
    },
    {
      "id": "S218",
      "uses": [
        "G001",
        "S217"
      ],
      "before": "finite prefix point1 index20 signed correction := div(-24, 22)",
      "after": "finite prefix point1 index20 signed correction = -12/11",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index20 signed correction，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index20 signed correction with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "-24",
          "22"
        ],
        "output": "-12/11",
        "label": "finite prefix point1 index20 signed correction"
      }
    },
    {
      "id": "S219",
      "uses": [
        "G001",
        "S218"
      ],
      "before": "finite prefix point1 index20 function value := sum(23, -12/11)",
      "after": "finite prefix point1 index20 function value = 241/11",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index20 function value，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index20 function value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "23",
          "-12/11"
        ],
        "output": "241/11",
        "label": "finite prefix point1 index20 function value"
      }
    },
    {
      "id": "S220",
      "uses": [
        "G001",
        "S219"
      ],
      "before": "finite prefix point1 index20 reverse signed error := sub(241/11, 23)",
      "after": "finite prefix point1 index20 reverse signed error = -12/11",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index20 reverse signed error，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index20 reverse signed error with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "241/11",
          "23"
        ],
        "output": "-12/11",
        "label": "finite prefix point1 index20 reverse signed error"
      }
    },
    {
      "id": "S221",
      "uses": [
        "G001",
        "S220"
      ],
      "before": "finite prefix point1 index20:original point=1;n=20;limit=23;signed coefficient=-24",
      "after": "f_n(x)=241/11;f_n(x)-limit=-12/11;this error field remains SIGNED",
      "relation": "implies",
      "rule_id": "pointwise_selected_value_reverse_error_not_absolute_value",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先加修正項得到函數值，再同減limit反向核誤差；原evaluate欄位保留正負，只有epsilon界才取絕對值。",
        "en": "Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S222",
      "uses": [
        "G001",
        "S221"
      ],
      "before": "finite prefix point1 index21 denominator := sum(21, 2)",
      "after": "finite prefix point1 index21 denominator = 23",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index21 denominator，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index21 denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "21",
          "2"
        ],
        "output": "23",
        "label": "finite prefix point1 index21 denominator"
      }
    },
    {
      "id": "S223",
      "uses": [
        "G001",
        "S222"
      ],
      "before": "finite prefix point1 index21 signed correction := div(-24, 23)",
      "after": "finite prefix point1 index21 signed correction = -24/23",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index21 signed correction，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index21 signed correction with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "-24",
          "23"
        ],
        "output": "-24/23",
        "label": "finite prefix point1 index21 signed correction"
      }
    },
    {
      "id": "S224",
      "uses": [
        "G001",
        "S223"
      ],
      "before": "finite prefix point1 index21 function value := sum(23, -24/23)",
      "after": "finite prefix point1 index21 function value = 505/23",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index21 function value，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index21 function value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "23",
          "-24/23"
        ],
        "output": "505/23",
        "label": "finite prefix point1 index21 function value"
      }
    },
    {
      "id": "S225",
      "uses": [
        "G001",
        "S224"
      ],
      "before": "finite prefix point1 index21 reverse signed error := sub(505/23, 23)",
      "after": "finite prefix point1 index21 reverse signed error = -24/23",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index21 reverse signed error，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index21 reverse signed error with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "505/23",
          "23"
        ],
        "output": "-24/23",
        "label": "finite prefix point1 index21 reverse signed error"
      }
    },
    {
      "id": "S226",
      "uses": [
        "G001",
        "S225"
      ],
      "before": "finite prefix point1 index21:original point=1;n=21;limit=23;signed coefficient=-24",
      "after": "f_n(x)=505/23;f_n(x)-limit=-24/23;this error field remains SIGNED",
      "relation": "implies",
      "rule_id": "pointwise_selected_value_reverse_error_not_absolute_value",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先加修正項得到函數值，再同減limit反向核誤差；原evaluate欄位保留正負，只有epsilon界才取絕對值。",
        "en": "Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S227",
      "uses": [
        "G001",
        "S226"
      ],
      "before": "finite prefix point1 index22 denominator := sum(22, 2)",
      "after": "finite prefix point1 index22 denominator = 24",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index22 denominator，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index22 denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "22",
          "2"
        ],
        "output": "24",
        "label": "finite prefix point1 index22 denominator"
      }
    },
    {
      "id": "S228",
      "uses": [
        "G001",
        "S227"
      ],
      "before": "finite prefix point1 index22 signed correction := div(-24, 24)",
      "after": "finite prefix point1 index22 signed correction = -1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index22 signed correction，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index22 signed correction with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "-24",
          "24"
        ],
        "output": "-1",
        "label": "finite prefix point1 index22 signed correction"
      }
    },
    {
      "id": "S229",
      "uses": [
        "G001",
        "S228"
      ],
      "before": "finite prefix point1 index22 function value := sum(23, -1)",
      "after": "finite prefix point1 index22 function value = 22",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index22 function value，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index22 function value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "23",
          "-1"
        ],
        "output": "22",
        "label": "finite prefix point1 index22 function value"
      }
    },
    {
      "id": "S230",
      "uses": [
        "G001",
        "S229"
      ],
      "before": "finite prefix point1 index22 reverse signed error := sub(22, 23)",
      "after": "finite prefix point1 index22 reverse signed error = -1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite prefix point1 index22 reverse signed error，不以小數近似替代。",
          "en": "Recompute finite prefix point1 index22 reverse signed error with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "22",
          "23"
        ],
        "output": "-1",
        "label": "finite prefix point1 index22 reverse signed error"
      }
    },
    {
      "id": "S231",
      "uses": [
        "G001",
        "S230"
      ],
      "before": "finite prefix point1 index22:original point=1;n=22;limit=23;signed coefficient=-24",
      "after": "f_n(x)=22;f_n(x)-limit=-1;this error field remains SIGNED",
      "relation": "implies",
      "rule_id": "pointwise_selected_value_reverse_error_not_absolute_value",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先加修正項得到函數值，再同減limit反向核誤差；原evaluate欄位保留正負，只有epsilon界才取絕對值。",
        "en": "Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S232",
      "uses": [
        "G001",
        "S231"
      ],
      "before": "finite observed length=23;complete domain-order rows=[[0, [67/2, 89/3, 111/4, 133/5, 155/6, 177/7, 199/8, 221/9, 243/10, 265/11, 287/12, 309/13, 331/14, 353/15, 375/16, 397/17, 419/18, 441/19, 463/20, 485/21, 507/22, 23, 551/24]], [1, [11, 15, 17, 91/5, 19, 137/7, 20, 61/3, 103/5, 229/11, 21, 275/13, 149/7, 107/5, 43/2, 367/17, 65/3, 413/19, 109/5, 153/7, 241/11, 505/23, 22]]]",
      "after": "prefix-alone sufficiency=否;finite_prefix_not_pointwise_convergence_certificate",
      "relation": "implies",
      "rule_id": "pointwise_finite_prefix_no_not_declared_model_failure",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這個No只限制從有限prefix本身的推論，不否定另給的全尾公式；逐點收斂的N可依x與epsilon，還須涵蓋所有晚n。",
        "en": "This No limits inference from the finite prefix alone,not the separate declared all-n formula.A pointwise witness may depend on x and epsilon and must cover every later n."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      23,
      [
        [
          0,
          [
            "67/2",
            "89/3",
            "111/4",
            "133/5",
            "155/6",
            "177/7",
            "199/8",
            "221/9",
            "243/10",
            "265/11",
            "287/12",
            "309/13",
            "331/14",
            "353/15",
            "375/16",
            "397/17",
            "419/18",
            "441/19",
            "463/20",
            "485/21",
            "507/22",
            "23",
            "551/24"
          ]
        ],
        [
          1,
          [
            "11",
            "15",
            "17",
            "91/5",
            "19",
            "137/7",
            "20",
            "61/3",
            "103/5",
            "229/11",
            "21",
            "275/13",
            "149/7",
            "107/5",
            "43/2",
            "367/17",
            "65/3",
            "413/19",
            "109/5",
            "153/7",
            "241/11",
            "505/23",
            "22"
          ]
        ]
      ],
      false,
      "finite_prefix_not_pointwise_convergence_certificate"
    ]
  },
  "conclusion": {
    "canonical_value": "(23, [[0, [67/2, 89/3, 111/4, 133/5, 155/6, 177/7, 199/8, 221/9, 243/10, 265/11, 287/12, 309/13, 331/14, 353/15, 375/16, 397/17, 419/18, 441/19, 463/20, 485/21, 507/22, 23, 551/24]], [1, [11, 15, 17, 91/5, 19, 137/7, 20, 61/3, 103/5, 229/11, 21, 275/13, 149/7, 107/5, 43/2, 367/17, 65/3, 413/19, 109/5, 153/7, 241/11, 505/23, 22]]], 否, finite_prefix_not_pointwise_convergence_certificate)",
    "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-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations",
      "batch_path": "public/data/questions/batches/batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations/questions.jsonl",
      "batch_sha256": "f429e64ae7ee8526b132b9c8fff40d92432ea2ab268c5e6a328d91542e0c6bfa",
      "original_line": 98,
      "original_line_sha256": "9ff5ffc814eff478594521cf9d07c9ee4f65be7a9fb94516272c2eae56f25c6c",
      "question_compact_sha256": "86e48b0fe10fcc38349cfe16999186513f3bf328010666780100c5f19e443601",
      "english_row_compact_sha256": "8abb9fbe680f9e7cb5b56ea33058b9b8f8881c03fecbe3d00f31da59f576b2ed",
      "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-pointwise-convergence-of-functions": {
        "sha256": "dde2c47154f28c5cd5dac55ff9f8a148e7761d085014f6ee6d2f72acbfb6012c",
        "english_sha256": "fd5f5de5b9ebd60cc4bb9c9abc7007107dd5d3642bc91e1ce97dc06e4f06b6f2",
        "original_path": "public/data/mko/pointwise-convergence-of-functions.json",
        "english_path": "public/data/i18n/editions/mko-pointwise-convergence-of-functions.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "d5b0037cbaeef26e3828d04bf29e0e0f6e466926098b6b09e3237d1fdbdbd0e2",
    "original_candidate_sha256": "d0a98936528e2333507955a0301791a018e30a90324d9309cf4fc73f611027ab",
    "original_candidate_line_sha256": "d0a98936528e2333507955a0301791a018e30a90324d9309cf4fc73f611027ab"
  },
  "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-20261026-008-p35",
    "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": "f73d099ebe3ff1c69a5d96f08c75b11b9e5b05120af40cf3ef388fce3b5c5336",
      "question_first_witness_sha256": "22675ffb6209b4fea076032b29b96e3dc19aad4244c54493f5f65f5e5d875699"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 232,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "4c08e8841c63bba53b6f46c20479cc6c712936883b231da693f481996430888d",
      "arithmetic_steps": 184
    },
    "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."
  }
}