{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261007-advance-20261009-r03-function-limit-0041",
  "pilot_number": "R0006-03916",
  "number": 3916,
  "family_id": "qf-density-limit-center-versus-limit",
  "title": {
    "zh-Hant": "第 3916 題：明示點值與去心極限成對",
    "en": "Question 3916: Ordered Center-Value and Punctured-Limit Pair"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261007-advance-20261009-r03-function-limit-0041",
    "batch_id": "batch-20261007-advance-20261009-r03-function-limit-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-function-limit"
    ],
    "title_zh": "明示點值與去心極限成對",
    "stem_zh": "實數模型；點值與去心極限分別解讀。x≠a時 f(x)=3x+(9)，a=-2，另明示 f(a)=3。求 (中心點值, x→a的去心極限)，用整數字串。",
    "answer": {
      "kind": "expression",
      "value": "(3, 3)",
      "explanation_zh": "點值與去心極限=(3, 3)。點值獨立明示，去心極限只讀仿射分支；兩者可相等或不同，不擅自要求連續。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "明示點值與去心極限成對"
      ]
    },
    "variant": {
      "family_id": "qf-density-limit-center-versus-limit",
      "seed": 202610093041,
      "parameters": {
        "m": 3,
        "a": -2,
        "b": 9,
        "value": 3,
        "operation": "density_limit_center_versus_limit",
        "domain": "reals_with_punctured_limit",
        "semantic_template": "qf-density-limit-center-versus-limit/[3,\"3\"]"
      }
    },
    "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-0041",
    "batch_id": "batch-20261007-advance-20261009-r03-function-limit-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "9c70670f77f2b3aee05f62b6abc62c81e5e06184f346125303d1e8119ed2ce54",
      "family_id": "qf-density-limit-center-versus-limit",
      "seed": 202610093041,
      "semantic_template": "qf-density-limit-center-versus-limit/[3,\"3\"]"
    },
    "fields": {
      "title": "Ordered Center-Value and Punctured-Limit Pair",
      "stem": "The full-real function has affine expression (3)x+(9) away from a=-2, and separately defined center value 3. Give (center value,punctured two-sided limit).",
      "answer": {
        "kind": "expression",
        "value": "(3, 3)",
        "explanation": "The explicit point value and affine approach value give (3, 3), preserving order. Changing the center does not alter the punctured limit; unequal values prevent center continuity."
      },
      "skills": [
        "Ordered Center-Value and Punctured-Limit Pair"
      ],
      "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。x≠a時 f(x)=3x+(9)，a=-2，另明示 f(a)=3。求 (中心點值, x→a的去心極限)，用整數字串。",
        "en": "The full-real function has affine expression (3)x+(9) away from a=-2, and separately defined center value 3. Give (center value,punctured two-sided limit)."
      }
    }
  ],
  "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": "punctured branch slope times center := mul(3, -2)",
      "after": "punctured branch slope times center = -6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "取宣告分支係數乘中心。",
        "en": "Multiply the branch slope by the center."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 punctured branch slope times center，不以小數近似替代。",
          "en": "Recompute punctured branch slope times center with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3",
          "-2"
        ],
        "output": "-6",
        "label": "punctured branch slope times center"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "punctured branch limit value := sum(-6, 9)",
      "after": "punctured branch limit value = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加該分支常數；仿射差m(x−a)可用ε−δ控制。",
        "en": "Add that branch's constant;the affine difference m(x−a) admits ε−δ control."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 punctured branch limit value，不以小數近似替代。",
          "en": "Recompute punctured branch limit value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-6",
          "9"
        ],
        "output": "3",
        "label": "punctured branch limit value"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "given f(a)=3;punctured affine limit=3",
      "after": "ordered pair=(3,3)",
      "relation": "implies",
      "rule_id": "point_value_independent_of_limit",
      "conditions": [],
      "explanation": {
        "zh-Hant": "中心值是獨立給定；去心處f(x)−L=m(x−a)，取δ=ε/(1+|m|)可證極限，無須改中心值或假設連續。",
        "en": "The center value is given separately.Off-center f(x)−L=m(x−a),so δ=ε/(1+|m|) proves the limit without changing the point value or assuming continuity."
      },
      "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": [
      3,
      "3"
    ]
  },
  "conclusion": {
    "canonical_value": "(3, 3)",
    "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_candidate_line_sha256": "c7885ad7f8126084fd41c5052710de26c53be0dcb8c5806c353204d88c0697bf"
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      "formalization": "not_formalized",
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      "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": {
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    "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.",
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    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
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}