{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0042",
  "pilot_number": "R0034-17517",
  "number": 17517,
  "family_id": "qf-mod34-pointwise-epsilon-witness",
  "title": {
    "zh-Hant": "第 17517 題：函數列逐點收斂",
    "en": "Question 17517: A Strict Pointwise Tail Witness Can Depend on the Point"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0042",
    "batch_id": "batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-pointwise-convergence-of-functions"
    ],
    "title_zh": "函數列逐點收斂",
    "stem_zh": "同一有限domain上的symbolic rational function tail。操作=pointwise-epsilon-witness；資料={\"limit\":[16,17],\"coefficients\":[17,-18],\"shift\":2,\"epsilon\":[1,26],\"point\":0}。請依 [point,epsilon,N,error_at_N,strict_tail,certificate] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(0, 1/26, 441, 17/443, 是, forall_n_ge_N)",
      "explanation_zh": "Wave5第三層核對=(0, 1/26, 441, 17/443, 是, forall_n_ge_N)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "函數列逐點收斂"
      ]
    },
    "variant": {
      "family_id": "qf-mod34-pointwise-epsilon-witness",
      "seed": 202611032042,
      "parameters": {
        "limit": [
          16,
          17
        ],
        "coefficients": [
          17,
          -18
        ],
        "shift": 2,
        "epsilon": [
          1,
          26
        ],
        "point": 0,
        "domain": "Finite_domain_symbolic_pointwise_tail",
        "operation": "pointwise-epsilon-witness",
        "semantic_template": "qf-mod34-pointwise-epsilon-witness/{\"coefficients\":[17,-18],\"epsilon\":[1,26],\"limit\":[16,17],\"point\":0,\"shift\":2}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_or_symbolic_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-14",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0042",
    "batch_id": "batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "f0f44dac600d9216be36f36254ef0dc48cee9ac2442582561e9d45ec31ac4676",
      "family_id": "qf-mod34-pointwise-epsilon-witness",
      "seed": 202611032042,
      "semantic_template": "qf-mod34-pointwise-epsilon-witness/{\"coefficients\":[17,-18],\"epsilon\":[1,26],\"limit\":[16,17],\"point\":0,\"shift\":2}"
    },
    "fields": {
      "title": "A Strict Pointwise Tail Witness Can Depend on the Point",
      "stem": "Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[16, 17], coefficients=[17, -18], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. Positive epsilon=1/26, point=0. Give (point,epsilon,N,error_at_N,strict_tail?,forall_n_ge_N), with error_at_N the absolute error bound.",
      "answer": {
        "kind": "expression",
        "value": "(0, 1/26, 441, 17/443, Yes, forall_n_ge_N)",
        "explanation": "Choose the least nonnegative N with |coefficient[point]|/(N+shift)<epsilon. The denominator increases thereafter, so every n≥N satisfies the strict bound. Equality at N is insufficient; a zero coefficient permits N=0. The exact result, retaining the requested order, is (0, 1/26, 441, 17/443, Yes, forall_n_ge_N)."
      },
      "skills": [
        "A Strict Pointwise Tail Witness Can Depend on the 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": "同一有限domain上的symbolic rational function tail。操作=pointwise-epsilon-witness；資料={\"limit\":[16,17],\"coefficients\":[17,-18],\"shift\":2,\"epsilon\":[1,26],\"point\":0}。請依 [point,epsilon,N,error_at_N,strict_tail,certificate] 欄序回答。",
        "en": "Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[16, 17], coefficients=[17, -18], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. Positive epsilon=1/26, point=0. Give (point,epsilon,N,error_at_N,strict_tail?,forall_n_ge_N), with error_at_N the absolute error bound."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "same COMPLETE finite domain Fin(2);original aligned limits=[16, 17];signed coefficients=[17, -18];positive shift=2",
      "after": "declared for EVERY n>=0:f_n(x)=limit[x]+coefficient[x]/(n+shift);signed error=f_n(x)-limit[x]=coefficient[x]/(n+shift)",
      "relation": "implies",
      "rule_id": "pointwise_explicit_common_domain_all_n_tail_not_probability",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整共同有限domain與全尾公式是題目明示前提，不從樣本猜出。函數值與誤差不是機率，可帶負號或超過1；不截斷、不平移索引。",
        "en": "The complete common finite domain and all-n formula are declared premises,not extrapolated from samples.Function values and errors are not probabilities;they may be signed or exceed1.Do not clip or shift indices."
      },
      "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": "selected pointwise witness absolute coefficient cross product := mul(17, 26)",
      "after": "selected pointwise witness absolute coefficient cross product = 442",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 selected pointwise witness absolute coefficient cross product，不以小數近似替代。",
          "en": "Recompute selected pointwise witness absolute coefficient cross product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "17",
          "26"
        ],
        "output": "442",
        "label": "selected pointwise witness absolute coefficient cross product"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "selected pointwise witness strict reciprocal threshold := div(442, 1)",
      "after": "selected pointwise witness strict reciprocal threshold = 442",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 selected pointwise witness strict reciprocal threshold，不以小數近似替代。",
          "en": "Recompute selected pointwise witness strict reciprocal threshold with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "442",
          "1"
        ],
        "output": "442",
        "label": "selected pointwise witness strict reciprocal threshold"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "selected pointwise witness raw strict integer cutoff := sum(442, -2, 1)",
      "after": "selected pointwise witness raw strict integer cutoff = 441",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 selected pointwise witness raw strict integer cutoff，不以小數近似替代。",
          "en": "Recompute selected pointwise witness raw strict integer cutoff with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "442",
          "-2",
          "1"
        ],
        "output": "441",
        "label": "selected pointwise witness raw strict integer cutoff"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "selected pointwise witness:point=0;SIGNED coefficient=17;absolute coefficient=17;positive epsilon=1/26;shift=2",
      "after": "threshold=442;floor=442;least nonnegative strict cutoff N=441",
      "relation": "implies",
      "rule_id": "pointwise_least_strict_cutoff_floor_not_probability_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "|c|/(N+s)<a/b等價N+s>|c|b/a，故floor後加1再減shift，僅與0取max。等號不夠；零係數N=0。這裡沒有機率≤1的額外guard，不改原答案索引。",
        "en": "The strict inequality is N+s>|c|b/a,so take its floor plus1 minus shift and clamp only at0.Equality is insufficient and a zero coefficient permits N=0.No probability≤1 guard belongs here."
      },
      "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": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "selected pointwise witness cutoff denominator := sum(441, 2)",
      "after": "selected pointwise witness cutoff denominator = 443",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 selected pointwise witness cutoff denominator，不以小數近似替代。",
          "en": "Recompute selected pointwise witness cutoff denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "441",
          "2"
        ],
        "output": "443",
        "label": "selected pointwise witness cutoff denominator"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "selected pointwise witness absolute error at cutoff := div(17, 443)",
      "after": "selected pointwise witness absolute error at cutoff = 17/443",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 selected pointwise witness absolute error at cutoff，不以小數近似替代。",
          "en": "Recompute selected pointwise witness absolute error at cutoff with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "17",
          "443"
        ],
        "output": "17/443",
        "label": "selected pointwise witness absolute error at cutoff"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "selected pointwise witness previous denominator := sum(440, 2)",
      "after": "selected pointwise witness previous denominator = 442",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 selected pointwise witness previous denominator，不以小數近似替代。",
          "en": "Recompute selected pointwise witness previous denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "440",
          "2"
        ],
        "output": "442",
        "label": "selected pointwise witness previous denominator"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "selected pointwise witness previous absolute error := div(17, 442)",
      "after": "selected pointwise witness previous absolute error = 1/26",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 selected pointwise witness previous absolute error，不以小數近似替代。",
          "en": "Recompute selected pointwise witness previous absolute error with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "17",
          "442"
        ],
        "output": "1/26",
        "label": "selected pointwise witness previous absolute error"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "selected pointwise witness:current N=441;current error=17/443;previous error=1/26;epsilon=1/26",
      "after": "current strict bound holds;previous fails;all smaller indices fail by monotonicity",
      "relation": "implies",
      "rule_id": "pointwise_minimality_previous_index_strict_threshold",
      "conditions": [],
      "explanation": {
        "zh-Hant": "前一項仍≥epsilon，分母隨n增加，故所有更小非負n亦不合嚴格界。",
        "en": "The preceding error is still≥epsilon.As denominators increase with n,every smaller nonnegative index also fails the strict bound."
      },
      "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": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "selected pointwise witness:declared exact tail;positive denominators increase for all n>=N=441",
      "after": "for EVERY later n,absolute error<=bound=17/443<epsilon=1/26",
      "relation": "implies",
      "rule_id": "pointwise_all_later_tail_not_single_numeric_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一明示公式讓所有後續分母不小於N處，才給全尾段界，不把一個代值成功當成未知序列收斂證明。",
        "en": "The same declared formula bounds every later denominator,which supplies an all-tail bound rather than a conclusion about an unknown sequence from one successful value."
      },
      "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": [
      0,
      "1/26",
      441,
      "17/443",
      true,
      "forall_n_ge_N"
    ]
  },
  "conclusion": {
    "canonical_value": "(0, 1/26, 441, 17/443, 是, forall_n_ge_N)",
    "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": {
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      "batch_path": "public/data/questions/batches/batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations/questions.jsonl",
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    "mko_bindings": {
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        "original_path": "public/data/mko/pointwise-convergence-of-functions.json",
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    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "d5b0037cbaeef26e3828d04bf29e0e0f6e466926098b6b09e3237d1fdbdbd0e2",
    "original_candidate_sha256": "a23107a98ec6e85d06c7453b64ef65ae4e0c376f7177bfe7944a36a9ed52a371",
    "original_candidate_line_sha256": "a23107a98ec6e85d06c7453b64ef65ae4e0c376f7177bfe7944a36a9ed52a371"
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  "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-20261026-008-p35",
    "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": "f73d099ebe3ff1c69a5d96f08c75b11b9e5b05120af40cf3ef388fce3b5c5336",
      "question_first_witness_sha256": "22675ffb6209b4fea076032b29b96e3dc19aad4244c54493f5f65f5e5d875699"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 11,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "97e29ec65ede8eb0e33ae829893e04143e607722ebac1976c98e816bf4c1d321",
      "arithmetic_steps": 7
    },
    "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."
  }
}