{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261007-advance-20261008-r03-continuity-0058",
  "pilot_number": "R0005-03408",
  "number": 3408,
  "family_id": "qf-continuity-removable-extension",
  "title": {
    "zh-Hant": "第 3408 題：原商式的連續補定值",
    "en": "Question 3408: Center Value for a Continuous Extension"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261007-advance-20261008-r03-continuity-0058",
    "batch_id": "batch-20261007-advance-20261008-r03-continuity-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-continuity"
    ],
    "title_zh": "原商式的連續補定值",
    "stem_zh": "全實數論域，以下函數／點均以明示實數模型解讀。原商 (x²−(-5)²)/(x−(-5)) 在 x≠-5 定義。另建 g(a)=V 的全域延拓，哪個 V 使 g 在 a 連續？",
    "answer": {
      "kind": "short_text",
      "value": "-10",
      "explanation_zh": "補定點值=-10。原商未在 a 定義；答案指定的是新延拓 g 的補定值，不追改原商或把未定義值設零。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "原商式的連續補定值"
      ]
    },
    "variant": {
      "family_id": "qf-continuity-removable-extension",
      "seed": 202610083058,
      "parameters": {
        "a": -5,
        "operation": "removable_continuous_extension",
        "domain": "reals",
        "semantic_template": "qf-continuity-removable-extension/{\"inputs\":[\"-5\"],\"operation\":\"removable_continuous_extension\"}"
      }
    },
    "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-20261008-r03-continuity-0058",
    "batch_id": "batch-20261007-advance-20261008-r03-continuity-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "9f15f6f94b75a07cb9301aca97551106b11eeb5eee98d27b3c0db2501071a3a8",
      "family_id": "qf-continuity-removable-extension",
      "seed": 202610083058,
      "semantic_template": "qf-continuity-removable-extension/{\"inputs\":[\"-5\"],\"operation\":\"removable_continuous_extension\"}"
    },
    "fields": {
      "title": "Center Value for a Continuous Extension",
      "stem": "The original quotient (x²−(-5)²)/(x−(-5)) is defined only away from -5. Choose the center value of a new continuous extension to all R.",
      "answer": {
        "kind": "short_text",
        "value": "-10",
        "explanation": "The punctured limit is 2a=-10, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified."
      },
      "skills": [
        "Center Value for a Continuous Extension"
      ],
      "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": "全實數論域，以下函數／點均以明示實數模型解讀。原商 (x²−(-5)²)/(x−(-5)) 在 x≠-5 定義。另建 g(a)=V 的全域延拓，哪個 V 使 g 在 a 連續？",
        "en": "The original quotient (x²−(-5)²)/(x−(-5)) is defined only away from -5. Choose the center value of a new continuous extension to all R."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "declared real model;center a=-5",
      "after": "distinguish punctured limits,point values and a new extension",
      "relation": "implies",
      "rule_id": "continuity_model_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只使用明示實數公式；點值、去心極限及另建延拓的值不是同一欄位。",
        "en": "Use the declared real formulas;point values,punctured limits and values of a newly defined extension are different data."
      },
      "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": "(x²−(-5)²)/(x−(-5));x≠-5",
      "after": "(x−(-5))(x+(-5))/(x−(-5))=x+(-5)",
      "relation": "implies",
      "rule_id": "punctured_factor_cancellation",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先因式分解，僅因x≠a才可約去x−a；原商在a仍未定義。",
        "en": "Factor first and cancel x−a only because x≠a;the original quotient remains undefined 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
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "extension center value := sum(-5, -5)",
      "after": "extension center value = -10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "去心處等於x+a，故當x趨a時極限為a+a。",
        "en": "On the punctured domain the quotient equals x+a,whose limit as x approaches a is a+a."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 extension center value，不以小數近似替代。",
          "en": "Recompute extension center value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-5",
          "-5"
        ],
        "output": "-10",
        "label": "extension center value"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "|(x+a)−2a|=|x−a|;a=-5",
      "after": "δ=ε>0 proves the punctured limit -10",
      "relation": "implies",
      "rule_id": "extension_limit_epsilon_witness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任意ε>0取δ=ε，0<|x−a|<δ便使函數與2a相差小於ε，非僅有限取樣。",
        "en": "For any ε>0 choose δ=ε;0<|x−a|<δ makes the difference from2a smaller than ε.This is not finite sampling."
      },
      "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": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "new extension g equals the original quotient off a",
      "after": "continuity at a requires and is achieved by g(a)=-10",
      "relation": "implies",
      "rule_id": "extension_value_not_original_quotient",
      "conditions": [],
      "explanation": {
        "zh-Hant": "極限唯一，所以補值必須等於2a；如此新g就是全實數上的x+a。這不追改原商在a未定義的事實。",
        "en": "Uniqueness of the limit forces the new value2a;with that value,g equals x+a on all reals.This does not alter the original quotient's undefined value 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": "-10",
    "claim_scope": "new_continuous_extension"
  },
  "conclusion": {
    "canonical_value": "-10",
    "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."
  },
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        "original_path": "public/data/mko/continuity.json",
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    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
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    "original_candidate_sha256": "23eb20f03c865a16c346080b9477f83e97edb552c34c72babd61ed9768596d4c",
    "original_candidate_line_sha256": "23eb20f03c865a16c346080b9477f83e97edb552c34c72babd61ed9768596d4c"
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  "historical_source_state": {
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      "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-p6",
    "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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    "bilingual_semantics": true,
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    "human_review": "NOT_PERFORMED",
    "formalization": "not_formalized",
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    "Canonical_truth_promotion": false,
    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
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}