{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261010-advance-20261014-r03-equation-0062",
  "pilot_number": "R0012-06637",
  "number": 6637,
  "family_id": "qf-density14-equation-denominator-exclusion-check",
  "title": {
    "zh-Hant": "第 6637 題：原式pole資格與交乘殘差",
    "en": "Question 6637: Original Denominator Guard versus Cross-Multiplication"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261010-advance-20261014-r03-equation-0062",
    "batch_id": "batch-20261010-advance-20261014-r03-equation-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-equation"
    ],
    "title_zh": "原式pole資格與交乘殘差",
    "stem_zh": "共同有理論域；依原式定義資格；(-1x+(12))/(x−(-1))=1；原式排除x=h，h=-1，queries=[-1/1, -1/2, 1/1]。依序列 (定義資格, 原式殘差或未定義, 交乘殘差, 原式解資格)，四個列表保留原序。",
    "answer": {
      "kind": "expression",
      "value": "([否, 是, 是], [未定義, 24/1, 9/2], [13/1, 12/1, 9/1], [否, 否, 否])",
      "explanation_zh": "原式定義資格與交乘殘差=([否, 是, 是], [未定義, 24/1, 9/2], [13/1, 12/1, 9/1], [否, 否, 否])。先檢原分母，再代原式；交乘值另列，不替代定義資格；交叉相乘不恢復排除點。有限雙側核對非一般方程求解證明。"
    },
    "difficulty": {
      "band": "lower_secondary",
      "level": 2,
      "skills": [
        "原式pole資格與交乘殘差"
      ]
    },
    "variant": {
      "family_id": "qf-density14-equation-denominator-exclusion-check",
      "seed": 202610143062,
      "parameters": {
        "a": -1,
        "b": 12,
        "c": 1,
        "h": -1,
        "queries": [
          [
            -1,
            1
          ],
          [
            -1,
            2
          ],
          [
            1,
            1
          ]
        ],
        "domain": "finite_Q_original_equation_protocol",
        "operation": "density14_equation_denominator_exclusion_check",
        "semantic_template": "qf-density14-equation-denominator-exclusion-check/[[false,\"undefined\",\"13/1\"],[true,\"24/1\",\"12/1\"],[true,\"9/2\",\"9/1\"]]"
      }
    },
    "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": "2026-10-10",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261010-advance-20261014-r03-equation-0062",
    "batch_id": "batch-20261010-advance-20261014-r03-equation-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "a458510c98c8d65dd57a506ac5af93f4d8ff3850843fe8b829b68cb57a1b5f3e",
      "family_id": "qf-density14-equation-denominator-exclusion-check",
      "seed": 202610143062,
      "semantic_template": "qf-density14-equation-denominator-exclusion-check/[[false,\"undefined\",\"13/1\"],[true,\"24/1\",\"12/1\"],[true,\"9/2\",\"9/1\"]]"
    },
    "fields": {
      "title": "Original Denominator Guard versus Cross-Multiplication",
      "stem": "Over Q, the original equation is ((-1)x+(12))/(x−(-1))=1. Exclude x=h=-1; ordered queries=[-1/1, -1/2, 1/1]. Give (domain-eligibility mask, original residuals or undefined, cross-multiplied residuals, original-solution mask), keeping each list’s order.",
      "answer": {
        "kind": "expression",
        "value": "([No, Yes, Yes], [undefined, 24/1, 9/2], [13/1, 12/1, 9/1], [No, No, No])",
        "explanation": "Check the original denominator before evaluating; a zero cross-multiplication residual does not restore an excluded undefined point or replace original-equation eligibility. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 24/1, 9/2], [13/1, 12/1, 9/1], [No, No, No])."
      },
      "skills": [
        "Original Denominator Guard versus Cross-Multiplication"
      ],
      "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": "共同有理論域；依原式定義資格；(-1x+(12))/(x−(-1))=1；原式排除x=h，h=-1，queries=[-1/1, -1/2, 1/1]。依序列 (定義資格, 原式殘差或未定義, 交乘殘差, 原式解資格)，四個列表保留原序。",
        "en": "Over Q, the original equation is ((-1)x+(12))/(x−(-1))=1. Exclude x=h=-1; ordered queries=[-1/1, -1/2, 1/1]. Give (domain-eligibility mask, original residuals or undefined, cross-multiplied residuals, original-solution mask), keeping each list’s order."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "common rational domain with original equation definitions",
      "after": "evaluate/classify the given model without enlarging its original domain",
      "relation": "implies",
      "rule_id": "original_rational_equation_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原式定義資格先於代入與變形；只核對明示模型，不把交乘或某點數值當成任意方程的一般求解證明。",
        "en": "Original definedness precedes substitution and transformations.Check the declared model,not arbitrary equation solving inferred from cross multiplication or one-point values."
      },
      "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": "query0 original denominator := sub(-1, -1)",
      "after": "query0 original denominator = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先求原分母x−h，0時原式排除這個點。",
        "en": "Evaluate the original denominator x−h first;zero excludes this point from the original equation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 query0 original denominator，不以小數近似替代。",
          "en": "Recompute query0 original denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-1",
          "-1"
        ],
        "output": "0",
        "label": "query0 original denominator"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "query0 numerator linear product := mul(-1, -1)",
      "after": "query0 numerator linear product = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先把指定x代入此側係數乘積。",
        "en": "Substitute the specified x into this side's coefficient product."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 query0 numerator linear product，不以小數近似替代。",
          "en": "Recompute query0 numerator linear product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "-1"
        ],
        "output": "1",
        "label": "query0 numerator linear product"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "query0 numerator affine value := sum(1, 12)",
      "after": "query0 numerator affine value = 13",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加此側自己的常數，不混用另一側。",
        "en": "Then add this side's own constant,not the other side's."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 query0 numerator affine value，不以小數近似替代。",
          "en": "Recompute query0 numerator affine value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "12"
        ],
        "output": "13",
        "label": "query0 numerator affine value"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "query0 cross right side := mul(1, 0)",
      "after": "query0 cross right side = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "交乘右側是c(x−h)，即使排除點也可作多項式計算。",
        "en": "The cross-multiplied right side is c(x−h),a polynomial value computable even at an excluded point."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 query0 cross right side，不以小數近似替代。",
          "en": "Recompute query0 cross right side with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "0"
        ],
        "output": "0",
        "label": "query0 cross right side"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "query0 cross residual := sub(13, 0)",
      "after": "query0 cross residual = 13",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "交乘殘差是分子減c(x−h)，與原式定義資格分列。",
        "en": "The cross residual is numerator minus c(x−h),reported separately from original definedness."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 query0 cross residual，不以小數近似替代。",
          "en": "Recompute query0 cross residual with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "13",
          "0"
        ],
        "output": "13",
        "label": "query0 cross residual"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "x=-1/1;x−h=0;cross residual=13",
      "after": "original defined=否;original residual=未定義;original solution=否",
      "relation": "implies",
      "rule_id": "original_denominator_exclusion_persists",
      "conditions": [],
      "explanation": {
        "zh-Hant": "交乘等價只在x≠h下成立；排除點即使分子0且交乘殘差0，也不能恢復為原式的解。",
        "en": "Cross multiplication is equivalent only when x≠h;even numerator0 and cross residual0 at an excluded point cannot restore an original solution."
      },
      "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": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "query1 original denominator := sub(-1/2, -1)",
      "after": "query1 original denominator = 1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先求原分母x−h，0時原式排除這個點。",
        "en": "Evaluate the original denominator x−h first;zero excludes this point from the original equation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 query1 original denominator，不以小數近似替代。",
          "en": "Recompute query1 original denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-1/2",
          "-1"
        ],
        "output": "1/2",
        "label": "query1 original denominator"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "query1 numerator linear product := mul(-1, -1/2)",
      "after": "query1 numerator linear product = 1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先把指定x代入此側係數乘積。",
        "en": "Substitute the specified x into this side's coefficient product."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 query1 numerator linear product，不以小數近似替代。",
          "en": "Recompute query1 numerator linear product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "-1/2"
        ],
        "output": "1/2",
        "label": "query1 numerator linear product"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "query1 numerator affine value := sum(1/2, 12)",
      "after": "query1 numerator affine value = 25/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加此側自己的常數，不混用另一側。",
        "en": "Then add this side's own constant,not the other side's."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 query1 numerator affine value，不以小數近似替代。",
          "en": "Recompute query1 numerator affine value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1/2",
          "12"
        ],
        "output": "25/2",
        "label": "query1 numerator affine value"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "query1 cross right side := mul(1, 1/2)",
      "after": "query1 cross right side = 1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "交乘右側是c(x−h)，即使排除點也可作多項式計算。",
        "en": "The cross-multiplied right side is c(x−h),a polynomial value computable even at an excluded point."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 query1 cross right side，不以小數近似替代。",
          "en": "Recompute query1 cross right side with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1/2"
        ],
        "output": "1/2",
        "label": "query1 cross right side"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "query1 cross residual := sub(25/2, 1/2)",
      "after": "query1 cross residual = 12",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "交乘殘差是分子減c(x−h)，與原式定義資格分列。",
        "en": "The cross residual is numerator minus c(x−h),reported separately from original definedness."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 query1 cross residual，不以小數近似替代。",
          "en": "Recompute query1 cross residual with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "25/2",
          "1/2"
        ],
        "output": "12",
        "label": "query1 cross residual"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "query1 original quotient := div(25/2, 1/2)",
      "after": "query1 original quotient = 25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "僅在原分母非零時讀原商。",
        "en": "Evaluate the original quotient only with a nonzero original denominator."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 query1 original quotient，不以小數近似替代。",
          "en": "Recompute query1 original quotient with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "25/2",
          "1/2"
        ],
        "output": "25",
        "label": "query1 original quotient"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "query1 original residual := sub(25, 1)",
      "after": "query1 original residual = 24",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原商減c是原方程殘差。",
        "en": "Original quotient minus c is the original-equation residual."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 query1 original residual，不以小數近似替代。",
          "en": "Recompute query1 original residual with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "25",
          "1"
        ],
        "output": "24",
        "label": "query1 original residual"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "x=-1/2;x−h=1/2;cross residual=12",
      "after": "original defined=是;original residual=24/1;original solution=否",
      "relation": "implies",
      "rule_id": "original_denominator_exclusion_persists",
      "conditions": [],
      "explanation": {
        "zh-Hant": "交乘等價只在x≠h下成立；排除點即使分子0且交乘殘差0，也不能恢復為原式的解。",
        "en": "Cross multiplication is equivalent only when x≠h;even numerator0 and cross residual0 at an excluded point cannot restore an original solution."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "query2 original denominator := sub(1, -1)",
      "after": "query2 original denominator = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先求原分母x−h，0時原式排除這個點。",
        "en": "Evaluate the original denominator x−h first;zero excludes this point from the original equation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 query2 original denominator，不以小數近似替代。",
          "en": "Recompute query2 original denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "1",
          "-1"
        ],
        "output": "2",
        "label": "query2 original denominator"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "query2 numerator linear product := mul(-1, 1)",
      "after": "query2 numerator linear product = -1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先把指定x代入此側係數乘積。",
        "en": "Substitute the specified x into this side's coefficient product."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 query2 numerator linear product，不以小數近似替代。",
          "en": "Recompute query2 numerator linear product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "1"
        ],
        "output": "-1",
        "label": "query2 numerator linear product"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "query2 numerator affine value := sum(-1, 12)",
      "after": "query2 numerator affine value = 11",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加此側自己的常數，不混用另一側。",
        "en": "Then add this side's own constant,not the other side's."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 query2 numerator affine value，不以小數近似替代。",
          "en": "Recompute query2 numerator affine value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-1",
          "12"
        ],
        "output": "11",
        "label": "query2 numerator affine value"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "query2 cross right side := mul(1, 2)",
      "after": "query2 cross right side = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "交乘右側是c(x−h)，即使排除點也可作多項式計算。",
        "en": "The cross-multiplied right side is c(x−h),a polynomial value computable even at an excluded point."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 query2 cross right side，不以小數近似替代。",
          "en": "Recompute query2 cross right side with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "2"
        ],
        "output": "2",
        "label": "query2 cross right side"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "query2 cross residual := sub(11, 2)",
      "after": "query2 cross residual = 9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "交乘殘差是分子減c(x−h)，與原式定義資格分列。",
        "en": "The cross residual is numerator minus c(x−h),reported separately from original definedness."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 query2 cross residual，不以小數近似替代。",
          "en": "Recompute query2 cross residual with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "2"
        ],
        "output": "9",
        "label": "query2 cross residual"
      }
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "query2 original quotient := div(11, 2)",
      "after": "query2 original quotient = 11/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "僅在原分母非零時讀原商。",
        "en": "Evaluate the original quotient only with a nonzero original denominator."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 query2 original quotient，不以小數近似替代。",
          "en": "Recompute query2 original quotient with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "11",
          "2"
        ],
        "output": "11/2",
        "label": "query2 original quotient"
      }
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "query2 original residual := sub(11/2, 1)",
      "after": "query2 original residual = 9/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原商減c是原方程殘差。",
        "en": "Original quotient minus c is the original-equation residual."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 query2 original residual，不以小數近似替代。",
          "en": "Recompute query2 original residual with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11/2",
          "1"
        ],
        "output": "9/2",
        "label": "query2 original residual"
      }
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "x=1/1;x−h=2;cross residual=9",
      "after": "original defined=是;original residual=9/2;original solution=否",
      "relation": "implies",
      "rule_id": "original_denominator_exclusion_persists",
      "conditions": [],
      "explanation": {
        "zh-Hant": "交乘等價只在x≠h下成立；排除點即使分子0且交乘殘差0，也不能恢復為原式的解。",
        "en": "Cross multiplication is equivalent only when x≠h;even numerator0 and cross residual0 at an excluded point cannot restore an original solution."
      },
      "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": [
      [
        false,
        true,
        true
      ],
      [
        "未定義",
        "24",
        "9/2"
      ],
      [
        "13",
        "12",
        "9"
      ],
      [
        false,
        false,
        false
      ]
    ]
  },
  "conclusion": {
    "canonical_value": "([否, 是, 是], [未定義, 24/1, 9/2], [13/1, 12/1, 9/1], [否, 否, 否])",
    "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-20261010-advance-20261014-r03-equation-foundations",
      "batch_path": "public/data/questions/batches/batch-20261010-advance-20261014-r03-equation-foundations/questions.jsonl",
      "batch_sha256": "58ec4ad9a6f12d07886bb2094cca830c943fd9ca604ef7406ae3c2babeb79c02",
      "original_line": 62,
      "original_line_sha256": "80277cd1383acee21881d87790de95b8eb528284c2bce6177851408f617950c4",
      "question_compact_sha256": "a458510c98c8d65dd57a506ac5af93f4d8ff3850843fe8b829b68cb57a1b5f3e",
      "english_row_compact_sha256": "60f3607e2b146a95ab67e49e834da531509c505274dd43aa69ec97daef206801",
      "english_manifest_path": "public/data/i18n/editions/batch-20261010-advance-20261014-r03-equation-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261010-advance-20261014-r03-equation-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-equation": {
        "sha256": "6f8d6ca379da35727c14f320d99d0bc8c53c47ad478d0f462d30a4127aea0a78",
        "english_sha256": "de50265f38fde7fafcc45aa22db603721664273dbafb01a18c77a7d51b393bdd",
        "original_path": "public/data/mko/equation.json",
        "english_path": "public/data/i18n/editions/mko-equation.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "c5c612ca91797c328f4489939ec6f7d885385e63ea21b334c09c421f17933d99",
    "original_candidate_sha256": "f6295b0df5cf00883840f6cda09af41ba2578268831c70fc6ebe27c025347825",
    "original_candidate_line_sha256": "f6295b0df5cf00883840f6cda09af41ba2578268831c70fc6ebe27c025347825"
  },
  "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-20261026-005-p13",
    "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": "90dbc494b2d34ef51f412dcff63cfcc1a2215d038bff71f38f5dded4c64faf00",
      "question_first_witness_sha256": "b72a3b8bcf10f584396e0dd5198c657b0934a71864e5a3fa6321d7a613aef9f9"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 23,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "7fd8336471fbf1394d359e0e6183000f8a7020782bfc557c52e5633a0d5caf53",
      "arithmetic_steps": 19
    },
    "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."
  }
}