{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261011-advance-20261015-r04-composition-0065",
  "pilot_number": "R0060-30665",
  "number": 30665,
  "family_id": "qf-density15-composition-noncommuting-affine",
  "title": {
    "zh-Hant": "第 30665 題：兩序仿射合成的差",
    "en": "Question 30665: Both Affine Composition Orders with Slope-First Coefficients"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261011-advance-20261015-r04-composition-0065",
    "batch_id": "batch-20261011-advance-20261015-r04-composition-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-composition"
    ],
    "title_zh": "兩序仿射合成的差",
    "stem_zh": "先f後g，明示中間域。Q總函數f(x)=(13)x+(1)，g(x)=(-2)x+(1)，有序輸入=[(0)/(1), (1)/(2), (-1)/(1)]。求 (g∘f的[一次係數,常數]順序, f∘g的[一次係數,常數]順序, 兩序各三個值, g∘f常數減f∘g常數)。",
    "answer": {
      "kind": "expression",
      "value": "([-26, -1], [-26, 14], [-1/1, -14/1, 25/1], [14/1, 1/1, 40/1], -15)",
      "explanation_zh": "兩序仿射合成與常數差=([-26, -1], [-26, 14], [-1/1, -14/1, 25/1], [14/1, 1/1, 40/1], -15)。各序都作兩次代入，並用輸入0與1的值差取得一次係數；不互換求值順序。合成不當乘法或任意交換，有限路徑非全部函數等式形式證明。"
    },
    "difficulty": {
      "band": "upper_secondary",
      "level": 2,
      "skills": [
        "兩序仿射合成的差"
      ]
    },
    "variant": {
      "family_id": "qf-density15-composition-noncommuting-affine",
      "seed": 202610154065,
      "parameters": {
        "a": 13,
        "b": 1,
        "c": -2,
        "d": 1,
        "inputs": [
          [
            0,
            1
          ],
          [
            1,
            2
          ],
          [
            -1,
            1
          ]
        ],
        "domain": "finite_explicit_composition_protocol",
        "operation": "density15_composition_noncommuting_affine",
        "semantic_template": "qf-density15-composition-noncommuting-affine/[[\"-26\",\"-1\"],[\"-26\",\"14\"],[\"-1/1\",\"-14/1\",\"25/1\"],[\"14/1\",\"1/1\",\"40/1\"]]"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_integer_rational_arithmetic",
        "producer_finite_enumeration",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-11",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261011-advance-20261015-r04-composition-0065",
    "batch_id": "batch-20261011-advance-20261015-r04-composition-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "ea81ec2029a853aef9b6d3e651168fe7e0e0923adc27d407632c934d77d9e2ee",
      "family_id": "qf-density15-composition-noncommuting-affine",
      "seed": 202610154065,
      "semantic_template": "qf-density15-composition-noncommuting-affine/[[\"-26\",\"-1\"],[\"-26\",\"14\"],[\"-1/1\",\"-14/1\",\"25/1\"],[\"14/1\",\"1/1\",\"40/1\"]]"
    },
    "fields": {
      "title": "Both Affine Composition Orders with Slope-First Coefficients",
      "stem": "Total Q functions f(x)=(13)x+(1), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).",
      "answer": {
        "kind": "expression",
        "value": "([-26, -1], [-26, 14], [-1/1, -14/1, 25/1], [14/1, 1/1, 40/1], -15)",
        "explanation": "Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-26, -1], [-26, 14], [-1/1, -14/1, 25/1], [14/1, 1/1, 40/1], -15)."
      },
      "skills": [
        "Both Affine Composition Orders with Slope-First Coefficients"
      ],
      "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": "先f後g，明示中間域。Q總函數f(x)=(13)x+(1)，g(x)=(-2)x+(1)，有序輸入=[(0)/(1), (1)/(2), (-1)/(1)]。求 (g∘f的[一次係數,常數]順序, f∘g的[一次係數,常數]順序, 兩序各三個值, g∘f常數減f∘g常數)。",
        "en": "Total Q functions f(x)=(13)x+(1), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [qf-density15-composition-noncommuting-affine, density15_composition_noncommuting_affine, finite_explicit_composition_protocol]",
      "after": "derived = [[GF_SLOPE_constant, FG_SLOPE_constant, GF_values, FG_values, SIGNED_constant_difference]]",
      "relation": "implies",
      "rule_id": "R60_EXACT_perbank_operation_domain_FRAME_and_output_roles",
      "conditions": [],
      "explanation": {
        "zh-Hant": "density14/15用full-prefix underscores，density17用local hyphens。coordinate frame與circle frame不同，輸出欄序／型別依本family，不改原中英／JSON／status。",
        "en": "density14/15 use FULL-prefix underscores;density17 uses LOCAL hyphens.The coordinate and circle frames differ.Retain THIS family ordered answer roles/types and all original bilingual JSON/status."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "用原始資料、型別、索引與本步前提反向代回；一致性審查不等於可由結論唯一還原輸入。有限候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "source roles = [Composition original rational input0, [0, 1]]",
      "after": "derived = [0/1]",
      "relation": "implies",
      "rule_id": "R60_RAW_SAFE_rational_pair_NONZERO_den_SIGN_normalized",
      "conditions": [],
      "explanation": {
        "zh-Hant": "raw兩component SAFE、den非0，可負den，先正規化sign/gcd；衍生fraction不限SAFE或1e6，整數保/1。",
        "en": "Both RAW pair components are SAFE and the denominator is nonzero.Negative denominators normalize sign/gcd;derived fractions are not SAFE/1e6-capped.Integer fractions retain/1."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "用原始資料、型別、索引與本步前提反向代回；一致性審查不等於可由結論唯一還原輸入。有限候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "source roles = [Composition original rational input1, [1, 2]]",
      "after": "derived = [1/2]",
      "relation": "implies",
      "rule_id": "R60_RAW_SAFE_rational_pair_NONZERO_den_SIGN_normalized",
      "conditions": [],
      "explanation": {
        "zh-Hant": "raw兩component SAFE、den非0，可負den，先正規化sign/gcd；衍生fraction不限SAFE或1e6，整數保/1。",
        "en": "Both RAW pair components are SAFE and the denominator is nonzero.Negative denominators normalize sign/gcd;derived fractions are not SAFE/1e6-capped.Integer fractions retain/1."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "用原始資料、型別、索引與本步前提反向代回；一致性審查不等於可由結論唯一還原輸入。有限候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "source roles = [Composition original rational input2, [-1, 1]]",
      "after": "derived = [-1/1]",
      "relation": "implies",
      "rule_id": "R60_RAW_SAFE_rational_pair_NONZERO_den_SIGN_normalized",
      "conditions": [],
      "explanation": {
        "zh-Hant": "raw兩component SAFE、den非0，可負den，先正規化sign/gcd；衍生fraction不限SAFE或1e6，整數保/1。",
        "en": "Both RAW pair components are SAFE and the denominator is nonzero.Negative denominators normalize sign/gcd;derived fractions are not SAFE/1e6-capped.Integer fractions retain/1."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "用原始資料、型別、索引與本步前提反向代回；一致性審查不等於可由結論唯一還原輸入。有限候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "Affine GF ORIGINAL c times a slope := mul(-2, 13)",
      "after": "Affine GF ORIGINAL c times a slope = -26",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。",
        "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Affine GF ORIGINAL c times a slope，不以小數近似替代。",
          "en": "Recompute Affine GF ORIGINAL c times a slope with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-2",
          "13"
        ],
        "output": "-26",
        "label": "Affine GF ORIGINAL c times a slope"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "Affine GF ORIGINAL c times b := mul(-2, 1)",
      "after": "Affine GF ORIGINAL c times b = -2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。",
        "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Affine GF ORIGINAL c times b，不以小數近似替代。",
          "en": "Recompute Affine GF ORIGINAL c times b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-2",
          "1"
        ],
        "output": "-2",
        "label": "Affine GF ORIGINAL c times b"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "Affine GF ORIGINAL constant cb plus d := sum(-2, 1)",
      "after": "Affine GF ORIGINAL constant cb plus d = -1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。",
        "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Affine GF ORIGINAL constant cb plus d，不以小數近似替代。",
          "en": "Recompute Affine GF ORIGINAL constant cb plus d with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "1"
        ],
        "output": "-1",
        "label": "Affine GF ORIGINAL constant cb plus d"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "Affine FG ORIGINAL a times c slope := mul(13, -2)",
      "after": "Affine FG ORIGINAL a times c slope = -26",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。",
        "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Affine FG ORIGINAL a times c slope，不以小數近似替代。",
          "en": "Recompute Affine FG ORIGINAL a times c slope with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "13",
          "-2"
        ],
        "output": "-26",
        "label": "Affine FG ORIGINAL a times c slope"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "Affine FG ORIGINAL a times d := mul(13, 1)",
      "after": "Affine FG ORIGINAL a times d = 13",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。",
        "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Affine FG ORIGINAL a times d，不以小數近似替代。",
          "en": "Recompute Affine FG ORIGINAL a times d with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "13",
          "1"
        ],
        "output": "13",
        "label": "Affine FG ORIGINAL a times d"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "Affine FG ORIGINAL constant ad plus b := sum(13, 1)",
      "after": "Affine FG ORIGINAL constant ad plus b = 14",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。",
        "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Affine FG ORIGINAL constant ad plus b，不以小數近似替代。",
          "en": "Recompute Affine FG ORIGINAL constant ad plus b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "13",
          "1"
        ],
        "output": "14",
        "label": "Affine FG ORIGINAL constant ad plus b"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "Affine ORIGINAL f input0 multiply := mul(13, 0)",
      "after": "Affine ORIGINAL f input0 multiply = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。",
        "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Affine ORIGINAL f input0 multiply，不以小數近似替代。",
          "en": "Recompute Affine ORIGINAL f input0 multiply with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "13",
          "0"
        ],
        "output": "0",
        "label": "Affine ORIGINAL f input0 multiply"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "Affine ORIGINAL f input0 plus b := sum(0, 1)",
      "after": "Affine ORIGINAL f input0 plus b = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。",
        "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Affine ORIGINAL f input0 plus b，不以小數近似替代。",
          "en": "Recompute Affine ORIGINAL f input0 plus b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "1"
        ],
        "output": "1",
        "label": "Affine ORIGINAL f input0 plus b"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "Affine ORIGINAL g AFTER f input0 multiply := mul(-2, 1)",
      "after": "Affine ORIGINAL g AFTER f input0 multiply = -2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。",
        "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Affine ORIGINAL g AFTER f input0 multiply，不以小數近似替代。",
          "en": "Recompute Affine ORIGINAL g AFTER f input0 multiply with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-2",
          "1"
        ],
        "output": "-2",
        "label": "Affine ORIGINAL g AFTER f input0 multiply"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "Affine ORIGINAL g AFTER f input0 plus d := sum(-2, 1)",
      "after": "Affine ORIGINAL g AFTER f input0 plus d = -1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。",
        "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Affine ORIGINAL g AFTER f input0 plus d，不以小數近似替代。",
          "en": "Recompute Affine ORIGINAL g AFTER f input0 plus d with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "1"
        ],
        "output": "-1",
        "label": "Affine ORIGINAL g AFTER f input0 plus d"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "Affine ORIGINAL g input0 multiply := mul(-2, 0)",
      "after": "Affine ORIGINAL g input0 multiply = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。",
        "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Affine ORIGINAL g input0 multiply，不以小數近似替代。",
          "en": "Recompute Affine ORIGINAL g input0 multiply with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-2",
          "0"
        ],
        "output": "0",
        "label": "Affine ORIGINAL g input0 multiply"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "Affine ORIGINAL g input0 plus d := sum(0, 1)",
      "after": "Affine ORIGINAL g input0 plus d = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。",
        "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Affine ORIGINAL g input0 plus d，不以小數近似替代。",
          "en": "Recompute Affine ORIGINAL g input0 plus d with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "1"
        ],
        "output": "1",
        "label": "Affine ORIGINAL g input0 plus d"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "Affine ORIGINAL f AFTER g input0 multiply := mul(13, 1)",
      "after": "Affine ORIGINAL f AFTER g input0 multiply = 13",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。",
        "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Affine ORIGINAL f AFTER g input0 multiply，不以小數近似替代。",
          "en": "Recompute Affine ORIGINAL f AFTER g input0 multiply with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "13",
          "1"
        ],
        "output": "13",
        "label": "Affine ORIGINAL f AFTER g input0 multiply"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "Affine ORIGINAL f AFTER g input0 plus b := sum(13, 1)",
      "after": "Affine ORIGINAL f AFTER g input0 plus b = 14",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。",
        "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Affine ORIGINAL f AFTER g input0 plus b，不以小數近似替代。",
          "en": "Recompute Affine ORIGINAL f AFTER g input0 plus b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "13",
          "1"
        ],
        "output": "14",
        "label": "Affine ORIGINAL f AFTER g input0 plus b"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "Affine reverse GF coefficient input0 product := mul(-26, 0)",
      "after": "Affine reverse GF coefficient input0 product = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。",
        "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Affine reverse GF coefficient input0 product，不以小數近似替代。",
          "en": "Recompute Affine reverse GF coefficient input0 product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-26",
          "0"
        ],
        "output": "0",
        "label": "Affine reverse GF coefficient input0 product"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "Affine reverse GF coefficient input0 sum := sum(0, -1)",
      "after": "Affine reverse GF coefficient input0 sum = -1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。",
        "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Affine reverse GF coefficient input0 sum，不以小數近似替代。",
          "en": "Recompute Affine reverse GF coefficient input0 sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "-1"
        ],
        "output": "-1",
        "label": "Affine reverse GF coefficient input0 sum"
      }
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "Affine reverse FG coefficient input0 product := mul(-26, 0)",
      "after": "Affine reverse FG coefficient input0 product = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。",
        "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Affine reverse FG coefficient input0 product，不以小數近似替代。",
          "en": "Recompute Affine reverse FG coefficient input0 product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-26",
          "0"
        ],
        "output": "0",
        "label": "Affine reverse FG coefficient input0 product"
      }
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "Affine reverse FG coefficient input0 sum := sum(0, 14)",
      "after": "Affine reverse FG coefficient input0 sum = 14",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。",
        "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Affine reverse FG coefficient input0 sum，不以小數近似替代。",
          "en": "Recompute Affine reverse FG coefficient input0 sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "14"
        ],
        "output": "14",
        "label": "Affine reverse FG coefficient input0 sum"
      }
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "Affine ORIGINAL f input1 multiply := mul(13, 1/2)",
      "after": "Affine ORIGINAL f input1 multiply = 13/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。",
        "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Affine ORIGINAL f input1 multiply，不以小數近似替代。",
          "en": "Recompute Affine ORIGINAL f input1 multiply with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "13",
          "1/2"
        ],
        "output": "13/2",
        "label": "Affine ORIGINAL f input1 multiply"
      }
    },
    {
      "id": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "Affine ORIGINAL f input1 plus b := sum(13/2, 1)",
      "after": "Affine ORIGINAL f input1 plus b = 15/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。",
        "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Affine ORIGINAL f input1 plus b，不以小數近似替代。",
          "en": "Recompute Affine ORIGINAL f input1 plus b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "13/2",
          "1"
        ],
        "output": "15/2",
        "label": "Affine ORIGINAL f input1 plus b"
      }
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "Affine ORIGINAL g AFTER f input1 multiply := mul(-2, 15/2)",
      "after": "Affine ORIGINAL g AFTER f input1 multiply = -15",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。",
        "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Affine ORIGINAL g AFTER f input1 multiply，不以小數近似替代。",
          "en": "Recompute Affine ORIGINAL g AFTER f input1 multiply with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-2",
          "15/2"
        ],
        "output": "-15",
        "label": "Affine ORIGINAL g AFTER f input1 multiply"
      }
    },
    {
      "id": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "Affine ORIGINAL g AFTER f input1 plus d := sum(-15, 1)",
      "after": "Affine ORIGINAL g AFTER f input1 plus d = -14",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。",
        "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Affine ORIGINAL g AFTER f input1 plus d，不以小數近似替代。",
          "en": "Recompute Affine ORIGINAL g AFTER f input1 plus d with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-15",
          "1"
        ],
        "output": "-14",
        "label": "Affine ORIGINAL g AFTER f input1 plus d"
      }
    },
    {
      "id": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "Affine ORIGINAL g input1 multiply := mul(-2, 1/2)",
      "after": "Affine ORIGINAL g input1 multiply = -1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。",
        "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Affine ORIGINAL g input1 multiply，不以小數近似替代。",
          "en": "Recompute Affine ORIGINAL g input1 multiply with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-2",
          "1/2"
        ],
        "output": "-1",
        "label": "Affine ORIGINAL g input1 multiply"
      }
    },
    {
      "id": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "Affine ORIGINAL g input1 plus d := sum(-1, 1)",
      "after": "Affine ORIGINAL g input1 plus d = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。",
        "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Affine ORIGINAL g input1 plus d，不以小數近似替代。",
          "en": "Recompute Affine ORIGINAL g input1 plus d with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-1",
          "1"
        ],
        "output": "0",
        "label": "Affine ORIGINAL g input1 plus d"
      }
    },
    {
      "id": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "Affine ORIGINAL f AFTER g input1 multiply := mul(13, 0)",
      "after": "Affine ORIGINAL f AFTER g input1 multiply = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。",
        "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Affine ORIGINAL f AFTER g input1 multiply，不以小數近似替代。",
          "en": "Recompute Affine ORIGINAL f AFTER g input1 multiply with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "13",
          "0"
        ],
        "output": "0",
        "label": "Affine ORIGINAL f AFTER g input1 multiply"
      }
    },
    {
      "id": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "Affine ORIGINAL f AFTER g input1 plus b := sum(0, 1)",
      "after": "Affine ORIGINAL f AFTER g input1 plus b = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。",
        "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Affine ORIGINAL f AFTER g input1 plus b，不以小數近似替代。",
          "en": "Recompute Affine ORIGINAL f AFTER g input1 plus b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "1"
        ],
        "output": "1",
        "label": "Affine ORIGINAL f AFTER g input1 plus b"
      }
    },
    {
      "id": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "Affine reverse GF coefficient input1 product := mul(-26, 1/2)",
      "after": "Affine reverse GF coefficient input1 product = -13",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。",
        "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Affine reverse GF coefficient input1 product，不以小數近似替代。",
          "en": "Recompute Affine reverse GF coefficient input1 product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-26",
          "1/2"
        ],
        "output": "-13",
        "label": "Affine reverse GF coefficient input1 product"
      }
    },
    {
      "id": "S032",
      "uses": [
        "G001",
        "S031"
      ],
      "before": "Affine reverse GF coefficient input1 sum := sum(-13, -1)",
      "after": "Affine reverse GF coefficient input1 sum = -14",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。",
        "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Affine reverse GF coefficient input1 sum，不以小數近似替代。",
          "en": "Recompute Affine reverse GF coefficient input1 sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-13",
          "-1"
        ],
        "output": "-14",
        "label": "Affine reverse GF coefficient input1 sum"
      }
    },
    {
      "id": "S033",
      "uses": [
        "G001",
        "S032"
      ],
      "before": "Affine reverse FG coefficient input1 product := mul(-26, 1/2)",
      "after": "Affine reverse FG coefficient input1 product = -13",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。",
        "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Affine reverse FG coefficient input1 product，不以小數近似替代。",
          "en": "Recompute Affine reverse FG coefficient input1 product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-26",
          "1/2"
        ],
        "output": "-13",
        "label": "Affine reverse FG coefficient input1 product"
      }
    },
    {
      "id": "S034",
      "uses": [
        "G001",
        "S033"
      ],
      "before": "Affine reverse FG coefficient input1 sum := sum(-13, 14)",
      "after": "Affine reverse FG coefficient input1 sum = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。",
        "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Affine reverse FG coefficient input1 sum，不以小數近似替代。",
          "en": "Recompute Affine reverse FG coefficient input1 sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-13",
          "14"
        ],
        "output": "1",
        "label": "Affine reverse FG coefficient input1 sum"
      }
    },
    {
      "id": "S035",
      "uses": [
        "G001",
        "S034"
      ],
      "before": "Affine ORIGINAL f input2 multiply := mul(13, -1)",
      "after": "Affine ORIGINAL f input2 multiply = -13",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。",
        "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Affine ORIGINAL f input2 multiply，不以小數近似替代。",
          "en": "Recompute Affine ORIGINAL f input2 multiply with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "13",
          "-1"
        ],
        "output": "-13",
        "label": "Affine ORIGINAL f input2 multiply"
      }
    },
    {
      "id": "S036",
      "uses": [
        "G001",
        "S035"
      ],
      "before": "Affine ORIGINAL f input2 plus b := sum(-13, 1)",
      "after": "Affine ORIGINAL f input2 plus b = -12",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。",
        "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Affine ORIGINAL f input2 plus b，不以小數近似替代。",
          "en": "Recompute Affine ORIGINAL f input2 plus b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-13",
          "1"
        ],
        "output": "-12",
        "label": "Affine ORIGINAL f input2 plus b"
      }
    },
    {
      "id": "S037",
      "uses": [
        "G001",
        "S036"
      ],
      "before": "Affine ORIGINAL g AFTER f input2 multiply := mul(-2, -12)",
      "after": "Affine ORIGINAL g AFTER f input2 multiply = 24",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。",
        "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Affine ORIGINAL g AFTER f input2 multiply，不以小數近似替代。",
          "en": "Recompute Affine ORIGINAL g AFTER f input2 multiply with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-2",
          "-12"
        ],
        "output": "24",
        "label": "Affine ORIGINAL g AFTER f input2 multiply"
      }
    },
    {
      "id": "S038",
      "uses": [
        "G001",
        "S037"
      ],
      "before": "Affine ORIGINAL g AFTER f input2 plus d := sum(24, 1)",
      "after": "Affine ORIGINAL g AFTER f input2 plus d = 25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。",
        "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Affine ORIGINAL g AFTER f input2 plus d，不以小數近似替代。",
          "en": "Recompute Affine ORIGINAL g AFTER f input2 plus d with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "24",
          "1"
        ],
        "output": "25",
        "label": "Affine ORIGINAL g AFTER f input2 plus d"
      }
    },
    {
      "id": "S039",
      "uses": [
        "G001",
        "S038"
      ],
      "before": "Affine ORIGINAL g input2 multiply := mul(-2, -1)",
      "after": "Affine ORIGINAL g input2 multiply = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。",
        "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Affine ORIGINAL g input2 multiply，不以小數近似替代。",
          "en": "Recompute Affine ORIGINAL g input2 multiply with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-2",
          "-1"
        ],
        "output": "2",
        "label": "Affine ORIGINAL g input2 multiply"
      }
    },
    {
      "id": "S040",
      "uses": [
        "G001",
        "S039"
      ],
      "before": "Affine ORIGINAL g input2 plus d := sum(2, 1)",
      "after": "Affine ORIGINAL g input2 plus d = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。",
        "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Affine ORIGINAL g input2 plus d，不以小數近似替代。",
          "en": "Recompute Affine ORIGINAL g input2 plus d with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "2",
          "1"
        ],
        "output": "3",
        "label": "Affine ORIGINAL g input2 plus d"
      }
    },
    {
      "id": "S041",
      "uses": [
        "G001",
        "S040"
      ],
      "before": "Affine ORIGINAL f AFTER g input2 multiply := mul(13, 3)",
      "after": "Affine ORIGINAL f AFTER g input2 multiply = 39",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。",
        "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Affine ORIGINAL f AFTER g input2 multiply，不以小數近似替代。",
          "en": "Recompute Affine ORIGINAL f AFTER g input2 multiply with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "13",
          "3"
        ],
        "output": "39",
        "label": "Affine ORIGINAL f AFTER g input2 multiply"
      }
    },
    {
      "id": "S042",
      "uses": [
        "G001",
        "S041"
      ],
      "before": "Affine ORIGINAL f AFTER g input2 plus b := sum(39, 1)",
      "after": "Affine ORIGINAL f AFTER g input2 plus b = 40",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。",
        "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Affine ORIGINAL f AFTER g input2 plus b，不以小數近似替代。",
          "en": "Recompute Affine ORIGINAL f AFTER g input2 plus b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "39",
          "1"
        ],
        "output": "40",
        "label": "Affine ORIGINAL f AFTER g input2 plus b"
      }
    },
    {
      "id": "S043",
      "uses": [
        "G001",
        "S042"
      ],
      "before": "Affine reverse GF coefficient input2 product := mul(-26, -1)",
      "after": "Affine reverse GF coefficient input2 product = 26",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。",
        "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Affine reverse GF coefficient input2 product，不以小數近似替代。",
          "en": "Recompute Affine reverse GF coefficient input2 product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-26",
          "-1"
        ],
        "output": "26",
        "label": "Affine reverse GF coefficient input2 product"
      }
    },
    {
      "id": "S044",
      "uses": [
        "G001",
        "S043"
      ],
      "before": "Affine reverse GF coefficient input2 sum := sum(26, -1)",
      "after": "Affine reverse GF coefficient input2 sum = 25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。",
        "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Affine reverse GF coefficient input2 sum，不以小數近似替代。",
          "en": "Recompute Affine reverse GF coefficient input2 sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "26",
          "-1"
        ],
        "output": "25",
        "label": "Affine reverse GF coefficient input2 sum"
      }
    },
    {
      "id": "S045",
      "uses": [
        "G001",
        "S044"
      ],
      "before": "Affine reverse FG coefficient input2 product := mul(-26, -1)",
      "after": "Affine reverse FG coefficient input2 product = 26",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。",
        "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Affine reverse FG coefficient input2 product，不以小數近似替代。",
          "en": "Recompute Affine reverse FG coefficient input2 product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-26",
          "-1"
        ],
        "output": "26",
        "label": "Affine reverse FG coefficient input2 product"
      }
    },
    {
      "id": "S046",
      "uses": [
        "G001",
        "S045"
      ],
      "before": "Affine reverse FG coefficient input2 sum := sum(26, 14)",
      "after": "Affine reverse FG coefficient input2 sum = 40",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。",
        "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Affine reverse FG coefficient input2 sum，不以小數近似替代。",
          "en": "Recompute Affine reverse FG coefficient input2 sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "26",
          "14"
        ],
        "output": "40",
        "label": "Affine reverse FG coefficient input2 sum"
      }
    },
    {
      "id": "S047",
      "uses": [
        "G001",
        "S046"
      ],
      "before": "Affine ORIGINAL GF constant minus FG constant := sub(-1, 14)",
      "after": "Affine ORIGINAL GF constant minus FG constant = -15",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。",
        "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Affine ORIGINAL GF constant minus FG constant，不以小數近似替代。",
          "en": "Recompute Affine ORIGINAL GF constant minus FG constant with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-1",
          "14"
        ],
        "output": "-15",
        "label": "Affine ORIGINAL GF constant minus FG constant"
      }
    },
    {
      "id": "S048",
      "uses": [
        "G001",
        "S047"
      ],
      "before": "source roles = [13, 1, -2, 1, [0/1, 1/2, -1/1]]",
      "after": "derived = [[-26, -1], [-26, 14], [-1/1, -14/1, 25/1], [14/1, 1/1, 40/1], -15]",
      "relation": "implies",
      "rule_id": "Composition_TOTAL_Q_affine_SUBSTITUTE_each_order_SLOPE_FIRST",
      "conditions": [],
      "explanation": {
        "zh-Hant": "由完整Q仿射law代兩次：g(f(x))=(ca)x+(cb+d)，f(g(x))=(ac)x+(ad+b)。pair是[slope,constant]不是升冪；可有commuting或constant實例，三個sample本身不證任意函數可交換。",
        "en": "Substitute twice in the WHOLE Q-affine laws:g(f(x))=(ca)x+(cb+d),f(g(x))=(ac)x+(ad+b).Pairs are[SLOPE,constant],NOT ascending powers.Commuting/constant instances are legal;three samples do not prove arbitrary functions commute."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "用原始資料、型別、索引與本步前提反向代回；一致性審查不等於可由結論唯一還原輸入。有限候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      [
        "-26",
        "-1"
      ],
      [
        "-26",
        "14"
      ],
      [
        "-1/1",
        "-14/1",
        "25/1"
      ],
      [
        "14/1",
        "1/1",
        "40/1"
      ],
      "-15"
    ]
  },
  "conclusion": {
    "canonical_value": "([-26, -1], [-26, 14], [-1/1, -14/1, 25/1], [14/1, 1/1, 40/1], -15)",
    "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-20261011-advance-20261015-r04-composition-foundations",
      "batch_path": "public/data/questions/batches/batch-20261011-advance-20261015-r04-composition-foundations/questions.jsonl",
      "batch_sha256": "ae1bde6f45fb411b7205bf61b2d3de85ef2f3d38c4b040606cda5c16a8f9e12e",
      "original_line": 65,
      "original_line_sha256": "945a3c9bc0d2f159a8bd3646b95c4d24fdc72093ffa8b8a2e169464b12858e3f",
      "question_compact_sha256": "ea81ec2029a853aef9b6d3e651168fe7e0e0923adc27d407632c934d77d9e2ee",
      "english_row_compact_sha256": "51f25fb8e161ddcabc85613ba3e2fd742b85f9cd5a5ae69a6694026bbe3002ed",
      "english_manifest_path": "public/data/i18n/editions/batch-20261011-advance-20261015-r04-composition-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261011-advance-20261015-r04-composition-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-composition": {
        "sha256": "a915b980c52ed5cfa4fd9ef59e2a23ef237a0eebbe262cd6f1a67b15f052bfa9",
        "english_sha256": "702841fe3d6cacc8f4b48338fbd4fc96731540292253b3030c43dce8e1cf1979",
        "original_path": "public/data/mko/composition.json",
        "english_path": "public/data/i18n/editions/mko-composition.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "322533190c1eb10144c4c59b6c70cc2f8105108b5f4f3a39dd577712e7185c02",
    "original_candidate_sha256": "472ac1e3f18541c5e7b1bb2474cec061c52157886d2d9de4115976b45001229f",
    "original_candidate_line_sha256": "472ac1e3f18541c5e7b1bb2474cec061c52157886d2d9de4115976b45001229f"
  },
  "historical_source_state": {
    "review_state": {
      "derivation": "candidate_authored_and_locally_checked",
      "source_question_state": "candidate",
      "independent_AI_or_human_review": "NOT_PERFORMED",
      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
    },
    "scope_note": {
      "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。",
      "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged."
    },
    "label": "Frozen producer metadata; current per-question review is separate below."
  },
  "revision": {
    "id": "review-20261026-011-p61",
    "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": "d6b5a409bf933281a46617cd715d8016fd5ca3d6fb76f77d179ac2ecd9d608bf",
      "question_first_witness_sha256": "f80ecf598c6893a580af018e1a82849b33ee6887efb48a6a1d69569aa17f12e4"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 48,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "672871ad4a3c0ab0e9ebf81b1ff6ba77fd2f288f2ad231cacbab2fe78be08fc8",
      "arithmetic_steps": 43
    },
    "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."
  }
}