{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261008-advance-20261011-r03-variable-expression-0096",
  "pilot_number": "R0008-04721",
  "number": 4721,
  "family_id": "qf-density11-expression-domain-candidate",
  "title": {
    "zh-Hant": "第 4721 題：原式定義域資格與有序求值",
    "en": "Question 4721: Original Rational-Expression Domain Guard"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261008-advance-20261011-r03-variable-expression-0096",
    "batch_id": "batch-20261008-advance-20261011-r03-variable-expression-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-variable-expression"
    ],
    "title_zh": "原式定義域資格與有序求值",
    "stem_zh": "有理數Q；表達式非方程式，原式E(x)=19/(x−(-2))。原式定義域排除x=-2；有序候選=[-2/1, 17/2, -24/1]。依序答 (三項合法資格列表, 三項E值列表)，排除點寫未定義。",
    "answer": {
      "kind": "expression",
      "value": "([否, 是, 是], [未定義, 38/21, -19/22])",
      "explanation_zh": "定義域資格與求值=([否, 是, 是], [未定義, 38/21, -19/22])。先檢查原式排除點再求值；零分子也不能消除原分母為零的空洞，不擴張原式定義域。"
    },
    "difficulty": {
      "band": "upper_secondary",
      "level": 3,
      "skills": [
        "原式定義域資格與有序求值"
      ]
    },
    "variant": {
      "family_id": "qf-density11-expression-domain-candidate",
      "seed": 202610113096,
      "parameters": {
        "a": 19,
        "b": -2,
        "candidates": [
          [
            -2,
            1
          ],
          [
            17,
            2
          ],
          [
            -24,
            1
          ]
        ],
        "operation": "density11_expression_domain_candidate",
        "domain": "Q_partial_rational_expression",
        "semantic_template": "qf-density11-expression-domain-candidate/[19,[\"0/1\",\"21/2\",\"-22/1\"]]"
      }
    },
    "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": "2026-10-08",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261008-advance-20261011-r03-variable-expression-0096",
    "batch_id": "batch-20261008-advance-20261011-r03-variable-expression-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "337d6cef2db71bf613255b571089e43564fa5752d2615184dbf45fb093fafce0",
      "family_id": "qf-density11-expression-domain-candidate",
      "seed": 202610113096,
      "semantic_template": "qf-density11-expression-domain-candidate/[19,[\"0/1\",\"21/2\",\"-22/1\"]]"
    },
    "fields": {
      "title": "Original Rational-Expression Domain Guard",
      "stem": "All operations and evaluations are exact in Q, not truncated natural-number arithmetic. The original expression is E(x)=(19)/(x−(-2)), excluding x=-2. Ordered candidates=[-2/1, 17/2, -24/1]. Give (three domain-eligibility values, three E values or undefined).",
      "answer": {
        "kind": "expression",
        "value": "([No, Yes, Yes], [undefined, 38/21, -19/22])",
        "explanation": "Check the original excluded point before evaluating; even a zero numerator cannot fill a denominator-zero hole or extend the original domain. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 38/21, -19/22])."
      },
      "skills": [
        "Original Rational-Expression Domain Guard"
      ],
      "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": "有理數Q；表達式非方程式，原式E(x)=19/(x−(-2))。原式定義域排除x=-2；有序候選=[-2/1, 17/2, -24/1]。依序答 (三項合法資格列表, 三項E值列表)，排除點寫未定義。",
        "en": "All operations and evaluations are exact in Q, not truncated natural-number arithmetic. The original expression is E(x)=(19)/(x−(-2)), excluding x=-2. Ordered candidates=[-2/1, 17/2, -24/1]. Give (three domain-eligibility values, three E values or undefined)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "typed rational expression and explicitly given environment",
      "after": "evaluation,not solving an equation",
      "relation": "implies",
      "rule_id": "expression_environment_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "表達式求值不是求解方程式；只把明示環境代入，不反推未知根或擅改綁定。",
        "en": "Evaluating an expression is not solving an equation;substitute only the supplied environment without inventing roots or changing bindings."
      },
      "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": "candidate0 original denominator := sub(-2, -2)",
      "after": "candidate0 original denominator = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先代入原式分母x−b，不先用零分子約掉空洞。",
        "en": "Evaluate the original denominator x−b first;do not cancel a hole just because the numerator is0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 candidate0 original denominator，不以小數近似替代。",
          "en": "Recompute candidate0 original denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-2",
          "-2"
        ],
        "output": "0",
        "label": "candidate0 original denominator"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "x=-2;x−b=0",
      "after": "legal=否;E(x)=未定義",
      "relation": "implies",
      "rule_id": "original_expression_domain_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "合法性取自原式定義域；a=0也不能讓排除點變成合法，排除點的值仍未定義。",
        "en": "Eligibility comes from the original expression's domain;even a=0 does not legalize the excluded point,and its value remains undefined."
      },
      "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": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "candidate1 original denominator := sub(17/2, -2)",
      "after": "candidate1 original denominator = 21/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先代入原式分母x−b，不先用零分子約掉空洞。",
        "en": "Evaluate the original denominator x−b first;do not cancel a hole just because the numerator is0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 candidate1 original denominator，不以小數近似替代。",
          "en": "Recompute candidate1 original denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "17/2",
          "-2"
        ],
        "output": "21/2",
        "label": "candidate1 original denominator"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "candidate1 E(x) := div(19, 21/2)",
      "after": "candidate1 E(x) = 38/21",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "分母非零才求原式值。",
        "en": "Evaluate the original expression only with a nonzero denominator."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 candidate1 E(x)，不以小數近似替代。",
          "en": "Recompute candidate1 E(x) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "19",
          "21/2"
        ],
        "output": "38/21",
        "label": "candidate1 E(x)"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "x=17/2;x−b=21/2",
      "after": "legal=是;E(x)=38/21",
      "relation": "implies",
      "rule_id": "original_expression_domain_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "合法性取自原式定義域；a=0也不能讓排除點變成合法，排除點的值仍未定義。",
        "en": "Eligibility comes from the original expression's domain;even a=0 does not legalize the excluded point,and its value remains undefined."
      },
      "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": "candidate2 original denominator := sub(-24, -2)",
      "after": "candidate2 original denominator = -22",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先代入原式分母x−b，不先用零分子約掉空洞。",
        "en": "Evaluate the original denominator x−b first;do not cancel a hole just because the numerator is0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 candidate2 original denominator，不以小數近似替代。",
          "en": "Recompute candidate2 original denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-24",
          "-2"
        ],
        "output": "-22",
        "label": "candidate2 original denominator"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "candidate2 E(x) := div(19, -22)",
      "after": "candidate2 E(x) = -19/22",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "分母非零才求原式值。",
        "en": "Evaluate the original expression only with a nonzero denominator."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 candidate2 E(x)，不以小數近似替代。",
          "en": "Recompute candidate2 E(x) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "19",
          "-22"
        ],
        "output": "-19/22",
        "label": "candidate2 E(x)"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "x=-24;x−b=-22",
      "after": "legal=是;E(x)=-19/22",
      "relation": "implies",
      "rule_id": "original_expression_domain_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "合法性取自原式定義域；a=0也不能讓排除點變成合法，排除點的值仍未定義。",
        "en": "Eligibility comes from the original expression's domain;even a=0 does not legalize the excluded point,and its value remains undefined."
      },
      "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",
        "38/21",
        "-19/22"
      ]
    ],
    "original_denominators": [
      "0",
      "21/2",
      "-22"
    ]
  },
  "conclusion": {
    "canonical_value": "([否, 是, 是], [未定義, 38/21, -19/22])",
    "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-20261008-advance-20261011-r03-variable-expression-foundations",
      "batch_path": "public/data/questions/batches/batch-20261008-advance-20261011-r03-variable-expression-foundations/questions.jsonl",
      "batch_sha256": "62cfe30aaffc9d75ac212c2a5a4338aa58a8acb9aa7a328ac228475f96d02fb6",
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      "original_line_sha256": "0e552e269016c393f75218f16addf713910ec964e61bab3b76f9b59d153ad670",
      "question_compact_sha256": "337d6cef2db71bf613255b571089e43564fa5752d2615184dbf45fb093fafce0",
      "english_row_compact_sha256": "9ae8ae75721e88e2def5a219d562a8ae8ac1f173317a11288a2376637f3f56ac",
      "english_manifest_path": "public/data/i18n/editions/batch-20261008-advance-20261011-r03-variable-expression-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261008-advance-20261011-r03-variable-expression-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-variable-expression": {
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        "english_sha256": "4a8cfbcc1fa5d73c360d18a95f35f69578250d0104baf76036f4efc8de9d7e81",
        "original_path": "public/data/mko/variable-expression.json",
        "english_path": "public/data/i18n/editions/mko-variable-expression.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "6dc34e24b0f43c50e49365ca1a1153175f8661d5290c9236df5527bb047157eb",
    "original_candidate_sha256": "cc40d12095ee5346f65c925c9ff31a39f0c5df7cd834a6b1f7d112674f186dc7",
    "original_candidate_line_sha256": "cc40d12095ee5346f65c925c9ff31a39f0c5df7cd834a6b1f7d112674f186dc7"
  },
  "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-20261026-004-p9",
    "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": "b6e05a3e8193a93db505dbf337528fca72e2e41528b1e92b698a3f0683d74289",
      "question_first_witness_sha256": "caa8693baf0b91acd09a20215d1aa6954ae513fc0b0814f26ac1c8c834b60d67"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 9,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "98baed689c0c5641f6f623a759bcd76624f6d3a63eaeac467a1c4c6c333e600d",
      "arithmetic_steps": 5
    },
    "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."
  }
}