{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261007-advance-20261009-r03-function-limit-0092",
  "pilot_number": "R0006-03992",
  "number": 3992,
  "family_id": "qf-density-limit-punctured-quotient-value",
  "title": {
    "zh-Hant": "第 3992 題：原商在合法去心點的值",
    "en": "Question 3992: Original Quotient Value at a Permitted Punctured Point"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261007-advance-20261009-r03-function-limit-0092",
    "batch_id": "batch-20261007-advance-20261009-r03-function-limit-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-function-limit"
    ],
    "title_zh": "原商在合法去心點的值",
    "stem_zh": "實數模型；點值與去心極限分別解讀。a=4，原商 f(x)=(x²−a²)/(x−a) 僅在 x≠a 定義。求 x=5 的值，採最簡正分母分數；這不是未定義中心點值或去心極限的替代。",
    "answer": {
      "kind": "expression",
      "value": "9/1",
      "explanation_zh": "原商合法點值=9/1。本次指定點分母非零，直接以整數平方差相除；未填原商在a的值。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "原商在合法去心點的值"
      ]
    },
    "variant": {
      "family_id": "qf-density-limit-punctured-quotient-value",
      "seed": 202610093092,
      "parameters": {
        "a": 4,
        "x": 5,
        "operation": "density_limit_punctured_quotient_value",
        "domain": "punctured_reals",
        "semantic_template": "qf-density-limit-punctured-quotient-value/[4,5]"
      }
    },
    "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": "2026-10-07",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261007-advance-20261009-r03-function-limit-0092",
    "batch_id": "batch-20261007-advance-20261009-r03-function-limit-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "9b4b0fc0d7da8ade6e54231c055d0762dd4d7508ac3a5deafe1e41196820b11e",
      "family_id": "qf-density-limit-punctured-quotient-value",
      "seed": 202610093092,
      "semantic_template": "qf-density-limit-punctured-quotient-value/[4,5]"
    },
    "fields": {
      "title": "Original Quotient Value at a Permitted Punctured Point",
      "stem": "The original quotient (x²−(4)²)/(x−(4)) has punctured real domain x≠a. Evaluate it at the permitted x=5. Use reduced positive-denominator fractions, retaining /1 for integers.",
      "answer": {
        "kind": "expression",
        "value": "9/1",
        "explanation": "The denominator is nonzero here, so the exact original quotient value is 9/1. This does not fill the excluded center or assign a division-by-zero default."
      },
      "skills": [
        "Original Quotient Value at a Permitted Punctured Point"
      ],
      "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": "實數模型；點值與去心極限分別解讀。a=4，原商 f(x)=(x²−a²)/(x−a) 僅在 x≠a 定義。求 x=5 的值，採最簡正分母分數；這不是未定義中心點值或去心極限的替代。",
        "en": "The original quotient (x²−(4)²)/(x−(4)) has punctured real domain x≠a. Evaluate it at the permitted x=5. Use reduced positive-denominator fractions, retaining /1 for integers."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "declared real model with an excluded center for limit data",
      "after": "point value,legal off-center evaluation and limit are distinct",
      "relation": "implies",
      "rule_id": "punctured_limit_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "去心極限不讀中心點值；指定合法點的求值也不冒充極限。這些明示仿射或可約多項式模型不代表任意函數。",
        "en": "A punctured limit does not read the center value,and evaluation at one legal point is not a limit.These declared affine or factorizable models do not represent arbitrary functions."
      },
      "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": "x squared := square(5)",
      "after": "x squared = 25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留指定合法點的完整平方。",
        "en": "Square the entire specified legal input."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 x squared，不以小數近似替代。",
          "en": "Recompute x squared with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "square",
        "inputs": [
          "5"
        ],
        "output": "25",
        "label": "x squared"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "center squared := square(4)",
      "after": "center squared = 16",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "負中心也先完整平方。",
        "en": "Square the entire center,including any negative sign."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 center squared，不以小數近似替代。",
          "en": "Recompute center squared with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "square",
        "inputs": [
          "4"
        ],
        "output": "16",
        "label": "center squared"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "difference of squares := sub(25, 16)",
      "after": "difference of squares = 9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "分子取x²−a²。",
        "en": "The numerator is x²−a²."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 difference of squares，不以小數近似替代。",
          "en": "Recompute difference of squares with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "25",
          "16"
        ],
        "output": "9",
        "label": "difference of squares"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "nonzero punctured denominator := sub(5, 4)",
      "after": "nonzero punctured denominator = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先核對x−a≠0，才准相除。",
        "en": "Check x−a≠0 before division."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 nonzero punctured denominator，不以小數近似替代。",
          "en": "Recompute nonzero punctured denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "5",
          "4"
        ],
        "output": "1",
        "label": "nonzero punctured denominator"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "legal quotient value := div(9, 1)",
      "after": "legal quotient value = 9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只在原商已定義的點精確相除，不補中心值。",
        "en": "Divide exactly only at the legal point of the original quotient;do not fill its center value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 legal quotient value，不以小數近似替代。",
          "en": "Recompute legal quotient value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "9",
          "1"
        ],
        "output": "9",
        "label": "legal quotient value"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "x=5≠a=4",
      "after": "(x²−a²)/(x−a)=x+a=9",
      "relation": "implies",
      "rule_id": "legal_quotient_factor_audit",
      "conditions": [],
      "explanation": {
        "zh-Hant": "平方差因式分解可反核結果；約分的等式仍保留x≠a，不等於原商在a有定義。",
        "en": "Factorization audits the result;the cancellation identity retains x≠a and does not define the original quotient at a."
      },
      "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": "9",
    "denominator": "1"
  },
  "conclusion": {
    "canonical_value": "9/1",
    "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": {
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    "question_binding": {
      "batch_id": "batch-20261007-advance-20261009-r03-function-limit-foundations",
      "batch_path": "public/data/questions/batches/batch-20261007-advance-20261009-r03-function-limit-foundations/questions.jsonl",
      "batch_sha256": "b4e71b191c5c0c73b5a782d764be0c2938b563b18f840324154a28dda4ced9fa",
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      "english_manifest_path": "public/data/i18n/editions/batch-20261007-advance-20261009-r03-function-limit-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261007-advance-20261009-r03-function-limit-foundations.en.jsonl"
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        "original_path": "public/data/mko/function-limit.json",
        "english_path": "public/data/i18n/editions/mko-function-limit.en.json"
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    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "6bac5f83f28cdd9c992a469306bfdf5c5415de17d11847cb9b17b4d13934c7e6",
    "original_candidate_sha256": "bc2c37325b5abe808fd3451acaf12a63ffb042e7bcc9d652591018e16fa61d9c",
    "original_candidate_line_sha256": "bc2c37325b5abe808fd3451acaf12a63ffb042e7bcc9d652591018e16fa61d9c"
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  "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"
    },
    "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-p7",
    "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": {
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    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 7,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
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    "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."
  }
}