{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261007-advance-20261009-r03-function-limit-0023",
  "pilot_number": "R0006-03948",
  "number": 3948,
  "family_id": "qf-density-limit-invalid-delta-witness",
  "title": {
    "zh-Hant": "第 3948 題：失敗δ的精確去心反例",
    "en": "Question 3948: Exact Punctured Counterexample to a Failed δ"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261007-advance-20261009-r03-function-limit-0023",
    "batch_id": "batch-20261007-advance-20261009-r03-function-limit-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-function-limit"
    ],
    "title_zh": "失敗δ的精確去心反例",
    "stem_zh": "實數模型；點值與去心極限分別解讀。固定實數A、L，x≠A時 f(x)=L+(23)(x−A)。ε=23/3、δ=69/69 且 |m|δ>ε。取 h=ε/|m|，列 (h, |f(A+h)−L|)，使0<h<δ卻不滿足 |f(x)−L|<ε。",
    "answer": {
      "kind": "expression",
      "value": "(1/3, 23/3)",
      "explanation_zh": "去心反例距離與誤差=(1/3, 23/3)。反例在去心內部且誤差等於ε，等號已違反嚴格小於；不用未定義的中心點值當比較目標。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 3,
      "skills": [
        "失敗δ的精確去心反例"
      ]
    },
    "variant": {
      "family_id": "qf-density-limit-invalid-delta-witness",
      "seed": 202610093023,
      "parameters": {
        "m": 23,
        "eps_n": 23,
        "eps_d": 3,
        "delta_n": 69,
        "delta_d": 69,
        "operation": "density_limit_invalid_delta_witness",
        "domain": "punctured_reals",
        "semantic_template": "qf-density-limit-invalid-delta-witness/[23,\"23/3\",\"1/1\"]"
      }
    },
    "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": "2026-10-07",
      "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-0023",
    "batch_id": "batch-20261007-advance-20261009-r03-function-limit-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "31f30c91b9f7d12b66bbe7c6814f2203f03db23748f6e35533e1eccd614b0205",
      "family_id": "qf-density-limit-invalid-delta-witness",
      "seed": 202610093023,
      "semantic_template": "qf-density-limit-invalid-delta-witness/[23,\"23/3\",\"1/1\"]"
    },
    "fields": {
      "title": "Exact Punctured Counterexample to a Failed δ",
      "stem": "On a punctured real domain, f(x)=L+(23)(x−A) for x≠A. Given ε=23/3>0 and failed δ=69/69>0 with |m|δ>ε, give (h,error) for the specified positive witness h=ε/|m|. It must satisfy 0<h<δ but fail |f(A+h)−L|<ε. Use reduced positive-denominator fractions, retaining /1 for integers.",
      "answer": {
        "kind": "expression",
        "value": "(1/3, 23/3)",
        "explanation": "The witness pair is (1/3, 23/3): h lies in the strict punctured interior, but the error equals ε rather than being smaller. The target is L, not an undefined f(A), and the nonzero-slope premise is required."
      },
      "skills": [
        "Exact Punctured Counterexample to a Failed δ"
      ],
      "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。固定實數A、L，x≠A時 f(x)=L+(23)(x−A)。ε=23/3、δ=69/69 且 |m|δ>ε。取 h=ε/|m|，列 (h, |f(A+h)−L|)，使0<h<δ卻不滿足 |f(x)−L|<ε。",
        "en": "On a punctured real domain, f(x)=L+(23)(x−A) for x≠A. Given ε=23/3>0 and failed δ=69/69>0 with |m|δ>ε, give (h,error) for the specified positive witness h=ε/|m|. It must satisfy 0<h<δ but fail |f(A+h)−L|<ε. 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": "counterexample displacement := div(23/3, 23)",
      "after": "counterexample displacement = 1/3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "取h=ε/|m|>0，非零斜率才可除。",
        "en": "Choose h=ε/|m|>0;division requires the nonzero slope."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 counterexample displacement，不以小數近似替代。",
          "en": "Recompute counterexample displacement with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "23/3",
          "23"
        ],
        "output": "1/3",
        "label": "counterexample displacement"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "punctured output error := mul(23, 1/3)",
      "after": "punctured output error = 23/3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "在x=A+h處，|f(x)−L|=|m|h恰等ε。",
        "en": "At x=A+h,the output error |f(x)−L|=|m|h equals ε exactly."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 punctured output error，不以小數近似替代。",
          "en": "Recompute punctured output error with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "23",
          "1/3"
        ],
        "output": "23/3",
        "label": "punctured output error"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "h=1/3>0;δ=1;h<δ=是;error=23/3;ε=23/3",
      "after": "legal punctured input but strict output inequality fails",
      "relation": "implies",
      "rule_id": "strict_delta_counterexample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "0<h<δ保證反例位於去心內部；誤差等於ε已違反嚴格小於ε，不能把≤混入定義。",
        "en": "The condition0<h<δ places the witness inside the punctured neighborhood;error equal to ε already violates strict error<ε,not merely error≤ε."
      },
      "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": "this particular proposed δ is invalid",
      "after": "not a proof that the affine limit fails to exist",
      "relation": "implies",
      "rule_id": "bad_delta_not_no_limit",
      "conditions": [],
      "explanation": {
        "zh-Hant": "否定的是這個δ的資格；該仿射模型仍可取更小δ=ε/(1+|m|)，不把單一壞選擇當作極限不存在。",
        "en": "Only this proposed δ fails;the affine model still allows the smaller δ=ε/(1+|m|).One bad choice does not disprove existence of the limit."
      },
      "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": [
      "1/3",
      "23/3"
    ],
    "epsilon": "23/3",
    "delta": "1",
    "legal_punctured_distance": true,
    "strict_error_bound_satisfied": false
  },
  "conclusion": {
    "canonical_value": "(1/3, 23/3)",
    "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."
  },
  "source": {
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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": "2f6fce51b4dd99cd981f299fad0af1ae4d1275a1139f04d23806aadec64ddd30",
    "original_candidate_line_sha256": "2f6fce51b4dd99cd981f299fad0af1ae4d1275a1139f04d23806aadec64ddd30"
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  "historical_source_state": {
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      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
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    "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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    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 5,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
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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."
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}