{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261010-advance-20261014-r03-equation-0038",
  "pilot_number": "R0012-06688",
  "number": 6688,
  "family_id": "qf-density14-equation-zero-scale-solution-loss",
  "title": {
    "zh-Hant": "第 6688 題：乘數前後的解集分類",
    "en": "Question 6688: Solution Classes Before and After Scaling"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261010-advance-20261014-r03-equation-0038",
    "batch_id": "batch-20261010-advance-20261014-r03-equation-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-equation"
    ],
    "title_zh": "乘數前後的解集分類",
    "stem_zh": "共同有理論域；依原式定義資格；原式2x+(-9)=0，兩側乘k=2。依序答 (原解集class, 乘後class, 兩class是否相同)，class用全Q／無解／單解 n/d。",
    "answer": {
      "kind": "expression",
      "value": "(單解 9/2, 單解 9/2, 是)",
      "explanation_zh": "乘數前後解集與等價性=(單解 9/2, 單解 9/2, 是)。分別按一次或退化常數式分類；乘零不能一般保持解集。有限雙側核對非一般方程求解證明。"
    },
    "difficulty": {
      "band": "lower_secondary",
      "level": 2,
      "skills": [
        "乘數前後的解集分類"
      ]
    },
    "variant": {
      "family_id": "qf-density14-equation-zero-scale-solution-loss",
      "seed": 202610143038,
      "parameters": {
        "a": 2,
        "b": -9,
        "k": 2,
        "domain": "finite_Q_original_equation_protocol",
        "operation": "density14_equation_zero_scale_solution_loss",
        "semantic_template": "qf-density14-equation-zero-scale-solution-loss/[\"單解 9/2\",\"單解 9/2\"]"
      }
    },
    "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-0038",
    "batch_id": "batch-20261010-advance-20261014-r03-equation-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "0aaaac89b6cac01d250db95be0add39b323b10453718bead84d572517a8514ee",
      "family_id": "qf-density14-equation-zero-scale-solution-loss",
      "seed": 202610143038,
      "semantic_template": "qf-density14-equation-zero-scale-solution-loss/[\"單解 9/2\",\"單解 9/2\"]"
    },
    "fields": {
      "title": "Solution Classes Before and After Scaling",
      "stem": "Over Q, the original equation is (2)x+(-9)=0; multiply both sides by k=2. Give (original solution class, scaled solution class, are the classes equal?). Use all Q, no-solution, or single-solution n/d.",
      "answer": {
        "kind": "expression",
        "value": "(single-solution 9/2, single-solution 9/2, Yes)",
        "explanation": "Classify the original and scaled equations separately, including constant degeneracies; multiplication by zero does not generally preserve the solution set. The exact result, retaining the requested order, is (single-solution 9/2, single-solution 9/2, Yes)."
      },
      "skills": [
        "Solution Classes Before and After Scaling"
      ],
      "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；原式2x+(-9)=0，兩側乘k=2。依序答 (原解集class, 乘後class, 兩class是否相同)，class用全Q／無解／單解 n/d。",
        "en": "Over Q, the original equation is (2)x+(-9)=0; multiply both sides by k=2. Give (original solution class, scaled solution class, are the classes equal?). Use all Q, no-solution, or single-solution n/d."
      }
    }
  ],
  "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": "original negative constant := sub(0, -9)",
      "after": "original negative constant = 9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "由Ax+B=0移項得到Ax=−B。",
        "en": "Rearrange Ax+B=0 into Ax=−B."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 original negative constant，不以小數近似替代。",
          "en": "Recompute original negative constant with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "0",
          "-9"
        ],
        "output": "9",
        "label": "original negative constant"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "original unique root := div(9, 2)",
      "after": "original unique root = 9/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "A非零才可相除，得到唯一有理根。",
        "en": "Divide only because A is nonzero,to obtain the unique rational root."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 original unique root，不以小數近似替代。",
          "en": "Recompute original unique root with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "9",
          "2"
        ],
        "output": "9/2",
        "label": "original unique root"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "original reverse substitution linear product := mul(2, 9/2)",
      "after": "original reverse substitution linear product = 9",
      "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 original reverse substitution linear product，不以小數近似替代。",
          "en": "Recompute original reverse substitution linear product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "9/2"
        ],
        "output": "9",
        "label": "original reverse substitution linear product"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "original reverse substitution affine value := sum(9, -9)",
      "after": "original reverse substitution affine value = 0",
      "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 original reverse substitution affine value，不以小數近似替代。",
          "en": "Recompute original reverse substitution affine value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "9",
          "-9"
        ],
        "output": "0",
        "label": "original reverse substitution affine value"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "original:A=2≠0;root=9/2",
      "after": "Ax+B=0",
      "relation": "implies",
      "rule_id": "nonzero_affine_equation_unique_root",
      "conditions": [],
      "explanation": {
        "zh-Hant": "移項與除非零數皆可逆，故唯一根不是只驗一個候選後猜測其唯一性。",
        "en": "Rearrangement and division by a nonzero number are reversible,so uniqueness does not come merely from testing one candidate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "scaled coefficient := mul(2, 2)",
      "after": "scaled coefficient = 4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩側乘k後，x係數也乘同一k。",
        "en": "Multiplying both sides by k also multiplies the x-coefficient by that same k."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 scaled coefficient，不以小數近似替代。",
          "en": "Recompute scaled coefficient with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "2"
        ],
        "output": "4",
        "label": "scaled coefficient"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "scaled constant := mul(2, -9)",
      "after": "scaled constant = -18",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "常數項同樣乘k，不只改一個係數。",
        "en": "Multiply the constant term by k as well,not just one coefficient."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 scaled constant，不以小數近似替代。",
          "en": "Recompute scaled constant with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "-9"
        ],
        "output": "-18",
        "label": "scaled constant"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "scaled negative constant := sub(0, -18)",
      "after": "scaled negative constant = 18",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "由Ax+B=0移項得到Ax=−B。",
        "en": "Rearrange Ax+B=0 into Ax=−B."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 scaled negative constant，不以小數近似替代。",
          "en": "Recompute scaled negative constant with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "0",
          "-18"
        ],
        "output": "18",
        "label": "scaled negative constant"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "scaled unique root := div(18, 4)",
      "after": "scaled unique root = 9/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "A非零才可相除，得到唯一有理根。",
        "en": "Divide only because A is nonzero,to obtain the unique rational root."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 scaled unique root，不以小數近似替代。",
          "en": "Recompute scaled unique root with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "18",
          "4"
        ],
        "output": "9/2",
        "label": "scaled unique root"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "scaled reverse substitution linear product := mul(4, 9/2)",
      "after": "scaled reverse substitution linear product = 18",
      "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 scaled reverse substitution linear product，不以小數近似替代。",
          "en": "Recompute scaled reverse substitution linear product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "4",
          "9/2"
        ],
        "output": "18",
        "label": "scaled reverse substitution linear product"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "scaled reverse substitution affine value := sum(18, -18)",
      "after": "scaled reverse substitution affine value = 0",
      "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 scaled reverse substitution affine value，不以小數近似替代。",
          "en": "Recompute scaled reverse substitution affine value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "18",
          "-18"
        ],
        "output": "0",
        "label": "scaled reverse substitution affine value"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "scaled:A=4≠0;root=9/2",
      "after": "Ax+B=0",
      "relation": "implies",
      "rule_id": "nonzero_affine_equation_unique_root",
      "conditions": [],
      "explanation": {
        "zh-Hant": "移項與除非零數皆可逆，故唯一根不是只驗一個候選後猜測其唯一性。",
        "en": "Rearrangement and division by a nonzero number are reversible,so uniqueness does not come merely from testing one candidate."
      },
      "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": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "k=2;before=單解 9/2;after=單解 9/2",
      "after": "same solution class=是",
      "relation": "implies",
      "rule_id": "zero_scaling_solution_equivalence_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "非零k可除回而保留解集；k=0將式子變0=0，可能從無解或單解變全Q。原式本已全Q時才不因此改類別。",
        "en": "A nonzero k can be divided back,retaining solutions;k=0 turns the equation into0=0 and can change no-solution or singleton into all Q.If the original was already all Q,this does not change its class."
      },
      "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": [
      "單解 9/2",
      "單解 9/2",
      true
    ]
  },
  "conclusion": {
    "canonical_value": "(單解 9/2, 單解 9/2, 是)",
    "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": 38,
      "original_line_sha256": "3df8d60280e707404ec456fc4600755ff849eff419af9d0bfa214e58a33dde4f",
      "question_compact_sha256": "0aaaac89b6cac01d250db95be0add39b323b10453718bead84d572517a8514ee",
      "english_row_compact_sha256": "4e5e384c8097c987043ce8727ff31312095b6e57c0cee51594f43e3b32456d4e",
      "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"
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    "mko_bindings": {
      "mko-equation": {
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        "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": "f2b6585dc949f80b058e319423f8be7e7400da8cf51e85642f1136322d5eb775",
    "original_candidate_line_sha256": "f2b6585dc949f80b058e319423f8be7e7400da8cf51e85642f1136322d5eb775"
  },
  "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": 14,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "e00210dc2a2180fc8d5e7ea127be50cfb2cf53c0a2f63447edaff3a4bb759d83",
      "arithmetic_steps": 10
    },
    "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."
  }
}