{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20260927-linear-equation-form-validity-0013",
  "pilot_number": "P1000-0263",
  "number": 263,
  "family_id": "qf-linear-equation-form-validity",
  "title": {
    "zh-Hant": "第 263 題：一次方程形式有效性",
    "en": "Question 263: Validity of a Linear-Equation Form"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20260927-linear-equation-form-validity-0013",
    "batch_id": "batch-20260927-linear-equation-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-linear-equation"
    ],
    "title_zh": "一次方程形式有效性",
    "stem_zh": "在有理數論域，考慮 -1x+(0)=0。依本 MKO，這是否是一元一次方程？",
    "answer": {
      "kind": "choice",
      "value": "是",
      "choices": [
        "是",
        "否"
      ],
      "explanation_zh": "a=-1≠0，所以是本 MKO 的一元一次方程。"
    },
    "difficulty": {
      "band": "lower_secondary",
      "level": 2,
      "skills": [
        "一次係數",
        "方程形式"
      ]
    },
    "variant": {
      "family_id": "qf-linear-equation-form-validity",
      "seed": 20260940,
      "parameters": {
        "a": -1,
        "b": 0,
        "domain": "rationals",
        "operation": "form_validity",
        "semantic_template": "qf-linear-equation-form-validity/{\"a\":-1,\"b\":0,\"domain\":\"rationals\",\"operation\":\"form_validity\"}"
      }
    },
    "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-form-validity-0013",
    "batch_id": "batch-20260927-linear-equation-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
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      "family_id": "qf-linear-equation-form-validity",
      "seed": 20260940,
      "semantic_template": "qf-linear-equation-form-validity/{\"a\":-1,\"b\":0,\"domain\":\"rationals\",\"operation\":\"form_validity\"}"
    },
    "fields": {
      "title": "Validity of a Linear-Equation Form",
      "stem": "Over the rational numbers, consider -1x+(0)=0. Under this MKO, is it a linear equation in one variable?",
      "answer": {
        "kind": "choice",
        "value": "Yes",
        "choices": [
          "Yes",
          "No"
        ],
        "explanation": "a=-1≠0, so this is a linear equation in one variable under the MKO definition."
      },
      "skills": [
        "linear coefficient",
        "equation form"
      ],
      "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，考慮 -1x+(0)=0。依本 MKO，這是否是一元一次方程？",
        "en": "Over the rational numbers, consider -1x+(0)=0. Under this MKO, is it a linear equation in one variable?"
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "a=-1, b=0, x∈Q",
      "after": "equation: -1x+(0)=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": "a=-1",
      "after": "a≠0: 是",
      "relation": "implies",
      "rule_id": "nondegenerate_definition",
      "conditions": [],
      "explanation": {
        "zh-Hant": "一元一次方程式的定義要求一次係數非零。",
        "en": "The definition of a nondegenerate linear equation requires a nonzero leading coefficient."
      },
      "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": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "-1x+(0)=0",
      "after": "linear form qualified: 是",
      "relation": "implies",
      "rule_id": "definition_application",
      "conditions": [],
      "explanation": {
        "zh-Hant": "檢查的是式子是否符合一次方程定義，不是它是否剛好有解。",
        "en": "This checks the definition, not merely whether the equation has a solution."
      },
      "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": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "if a=0 then ax+b=b",
      "after": "constant equality is a separate boundary",
      "relation": "implies",
      "rule_id": "degenerate_boundary",
      "conditions": [],
      "explanation": {
        "zh-Hant": "a=0 時 b=0 可有全部解，b≠0 可無解，兩者仍不是非退化一次方程。",
        "en": "When a=0, b=0 gives every rational solution and b≠0 gives none; neither is nondegenerate linear form."
      },
      "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": true
  },
  "conclusion": {
    "canonical_value": "是",
    "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；一次方程定義要求 a≠0，a=0 是另行處理的退化常數等式。",
    "en": "Use the rational domain. A linear equation requires a≠0; a=0 is a separate degenerate constant equation."
  },
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    },
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    "original_candidate_line_sha256": "52c2d337e5922a400ddc6f52d0e371a14439e4cd086d538e827b1632852658b6"
  },
  "historical_source_state": {
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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"
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    "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": {
    "id": "review-20261026-003-01",
    "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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    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 4,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
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    "formalization": "not_formalized",
    "Lean": "NOT_RUN",
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    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
  }
}