{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20260927-linear-equation-solution-verification-0018",
  "pilot_number": "P1000-0318",
  "number": 318,
  "family_id": "qf-linear-equation-solution-verification",
  "title": {
    "zh-Hant": "第 318 題：有理解的交叉乘積驗證",
    "en": "Question 318: Cross-Multiplication Check of a Rational Solution"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20260927-linear-equation-solution-verification-0018",
    "batch_id": "batch-20260927-linear-equation-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-linear-equation"
    ],
    "title_zh": "有理解的交叉乘積驗證",
    "stem_zh": "在有理數論域，方程 4x+(13)=0；候選 x=-3。候選值是否為解？",
    "answer": {
      "kind": "choice",
      "value": "否",
      "choices": [
        "是",
        "否"
      ],
      "explanation_zh": "q=1≠0；4×-3+(13)×1=1，所以不是解。"
    },
    "difficulty": {
      "band": "lower_secondary",
      "level": 1,
      "skills": [
        "有理數代入",
        "交叉乘積"
      ]
    },
    "variant": {
      "family_id": "qf-linear-equation-solution-verification",
      "seed": 20261145,
      "parameters": {
        "a": 4,
        "b": 13,
        "p": -3,
        "q": 1,
        "domain": "rationals",
        "operation": "cross_multiplication_substitution",
        "semantic_template": "qf-linear-equation-solution-verification/{\"a\":4,\"b\":13,\"domain\":\"rationals\",\"operation\":\"cross_multiplication_substitution\",\"p\":-3,\"q\":1}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "schema_pending",
        "independent_recomputation_pending"
      ],
      "warning_zh": "本題為 candidate；整數與有理數的有限精確重算不是形式證明或人工審查。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "GPT-5.6 Terra",
      "generated_at": "2026-09-27",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20260927-linear-equation-solution-verification-0018",
    "batch_id": "batch-20260927-linear-equation-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "7fa7274290ead16a261ffa182c658e4ec738d0465b1181153280187ffe711057",
      "family_id": "qf-linear-equation-solution-verification",
      "seed": 20261145,
      "semantic_template": "qf-linear-equation-solution-verification/{\"a\":4,\"b\":13,\"domain\":\"rationals\",\"operation\":\"cross_multiplication_substitution\",\"p\":-3,\"q\":1}"
    },
    "fields": {
      "title": "Cross-Multiplication Check of a Rational Solution",
      "stem": "Over the rational numbers, consider 4x+(13)=0 and the candidate x=-3/1. Is the candidate a solution?",
      "answer": {
        "kind": "choice",
        "value": "No",
        "choices": [
          "Yes",
          "No"
        ],
        "explanation": "q=1≠0 and 4×-3+(13)×1=1, so the candidate is not a solution."
      },
      "skills": [
        "rational substitution",
        "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": "在有理數論域，方程 4x+(13)=0；候選 x=-3。候選值是否為解？",
        "en": "Over the rational numbers, consider 4x+(13)=0 and the candidate x=-3/1. Is the candidate a solution?"
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "a=4, b=13, x∈Q",
      "after": "equation: 4x+(13)=0",
      "relation": "implies",
      "rule_id": "read_coefficients",
      "conditions": [],
      "explanation": {
        "zh-Hant": "明確讀取一次項、常數項與有理數域，不從式子表面名稱推斷非退化。",
        "en": "Read the linear coefficient, constant and rational domain explicitly; a name alone does not establish nondegeneracy."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "代回原式核對，除法前的非零條件不可省略。",
          "en": "Substitute into the original equation; retain every nonzero condition before division."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "candidate x := div(-3, 1)",
      "after": "candidate x = -3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先把候選有理數化為精確分數；分母不能為零。",
        "en": "Read the candidate as an exact rational; its denominator must be nonzero."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 candidate x，不以小數近似替代。",
          "en": "Recompute candidate x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "-3",
          "1"
        ],
        "output": "-3",
        "label": "candidate x"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "a·x := mul(4, -3)",
      "after": "a·x = -12",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原係數計算一次項。",
        "en": "Use the original coefficient in the linear term."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 a·x，不以小數近似替代。",
          "en": "Recompute a·x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "4",
          "-3"
        ],
        "output": "-12",
        "label": "a·x"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "a·x+b := sum(-12, 13)",
      "after": "a·x+b = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "殘差等於0才是原方程的解。",
        "en": "The candidate solves the original equation exactly when the residual is0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 a·x+b，不以小數近似替代。",
          "en": "Recompute a·x+b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-12",
          "13"
        ],
        "output": "1",
        "label": "a·x+b"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "a·p := mul(4, -3)",
      "after": "a·p = -12",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "另外交叉相乘核對整數側。",
        "en": "Independently form the first cross-multiplied integer term."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 a·p，不以小數近似替代。",
          "en": "Recompute a·p with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "4",
          "-3"
        ],
        "output": "-12",
        "label": "a·p"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "b·q := mul(13, 1)",
      "after": "b·q = 13",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "將常數也乘同一非零分母。",
        "en": "Multiply the constant by the same nonzero denominator."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 b·q，不以小數近似替代。",
          "en": "Recompute b·q with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "13",
          "1"
        ],
        "output": "13",
        "label": "b·q"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "a·p+b·q := sum(-12, 13)",
      "after": "a·p+b·q = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "非零 q 使這個等式與原殘差為零等價。",
        "en": "Nonzero q makes this equality equivalent to a zero original residual."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 a·p+b·q，不以小數近似替代。",
          "en": "Recompute a·p+b·q with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-12",
          "13"
        ],
        "output": "1",
        "label": "a·p+b·q"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "residual=1; cross=1",
      "after": "candidate solves equation: 否",
      "relation": "implies",
      "rule_id": "substitution_verdict",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩種精確核對一致，才採用候選解的真假結論。",
        "en": "The two exact checks agree on the candidate's validity."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "代回原式核對，除法前的非零條件不可省略。",
          "en": "Substitute into the original equation; retain every nonzero condition before division."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": false
  },
  "conclusion": {
    "canonical_value": "否",
    "explanation": {
      "zh-Hant": "上述完整代值與條件核對得到原題答案；欄位次序及真假標記保持原始合約。",
      "en": "The full substitutions and premise audits recover the original answer, retaining its field order and verdict meanings."
    }
  },
  "definition_note": {
    "zh-Hant": "有理數論域；候選 p/q 須 q≠0，殘差為零與交叉相乘等式等價。",
    "en": "Use the rational domain. A candidate p/q needs q≠0; a zero residual is equivalent to the cross-multiplied identity."
  },
  "source": {
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      "english_manifest_path": "public/data/i18n/editions/batch-20260927-linear-equation-foundations.en.json",
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    },
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    "original_candidate_line_sha256": "ac45b5923e69ed8a5b1bc939660feab5e546bf876cd384b289bffbbde08802f6"
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      "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": "來源綁定的教學推導候選；精確算術及有限列舉不是獨立AI審查或一般形式化證明。原題、答案、MKO、公式、AST及歷史保持不變。",
      "en": "Source-bound pedagogical derivation candidate: exact arithmetic and finite enumeration are not independent AI 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": {
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    "review_date": "2026-10-26",
    "owner_authorized_reader_publication": true,
    "scope": "Thirty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct sentence shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full answer tuples, model results 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.",
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    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 8,
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    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
  }
}