{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0001",
  "pilot_number": "R0034-17526",
  "number": 17526,
  "family_id": "qf-mod34-pointwise-finite-domain-evaluate",
  "title": {
    "zh-Hant": "第 17526 題：函數列逐點收斂",
    "en": "Question 17526: Keep the Signed Function Error Separate from Its Absolute Bound"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0001",
    "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-finite-domain-evaluate；資料={\"limit\":[0,1],\"coefficients\":[1,-2],\"shift\":2,\"n\":1,\"point\":0}。請依 [n,point,f_n_x,limit_x,error] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(1, 0, 1/3, 0, 1/3)",
      "explanation_zh": "Wave5第三層核對=(1, 0, 1/3, 0, 1/3)；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": 1,
      "skills": [
        "函數列逐點收斂"
      ]
    },
    "variant": {
      "family_id": "qf-mod34-pointwise-finite-domain-evaluate",
      "seed": 202611032001,
      "parameters": {
        "limit": [
          0,
          1
        ],
        "coefficients": [
          1,
          -2
        ],
        "shift": 2,
        "n": 1,
        "point": 0,
        "domain": "Finite_domain_symbolic_pointwise_tail",
        "operation": "pointwise-finite-domain-evaluate",
        "semantic_template": "qf-mod34-pointwise-finite-domain-evaluate/{\"coefficients\":[1,-2],\"limit\":[0,1],\"n\":1,\"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-0001",
    "batch_id": "batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "5747e4f0ae8091fb322f7bd8731adc856a799001aac8d106d7ef92dd70807a96",
      "family_id": "qf-mod34-pointwise-finite-domain-evaluate",
      "seed": 202611032001,
      "semantic_template": "qf-mod34-pointwise-finite-domain-evaluate/{\"coefficients\":[1,-2],\"limit\":[0,1],\"n\":1,\"point\":0,\"shift\":2}"
    },
    "fields": {
      "title": "Keep the Signed Function Error Separate from Its Absolute Bound",
      "stem": "Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[0, 1], coefficients=[1, -2], 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. n=1, point=0. Give (n,point,f_n_x,limit_x,error), where error=f_n(x)−limit[x] is signed.",
      "answer": {
        "kind": "expression",
        "value": "(1, 0, 1/3, 0, 1/3)",
        "explanation": "Evaluate the exact rational tail at the selected point. A negative coefficient gives a negative signed error; do not replace that field with its absolute value. One value is not the full pointwise-limit proof. The exact result, retaining the requested order, is (1, 0, 1/3, 0, 1/3)."
      },
      "skills": [
        "Keep the Signed Function Error Separate from Its Absolute Bound"
      ],
      "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-finite-domain-evaluate；資料={\"limit\":[0,1],\"coefficients\":[1,-2],\"shift\":2,\"n\":1,\"point\":0}。請依 [n,point,f_n_x,limit_x,error] 欄序回答。",
        "en": "Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[0, 1], coefficients=[1, -2], 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. n=1, point=0. Give (n,point,f_n_x,limit_x,error), where error=f_n(x)−limit[x] is signed."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "same COMPLETE finite domain Fin(2);original aligned limits=[0, 1];signed coefficients=[1, -2];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 function evaluation denominator := sum(1, 2)",
      "after": "selected function evaluation denominator = 3",
      "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 function evaluation denominator，不以小數近似替代。",
          "en": "Recompute selected function evaluation denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "2"
        ],
        "output": "3",
        "label": "selected function evaluation denominator"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "selected function evaluation signed correction := div(1, 3)",
      "after": "selected function evaluation signed correction = 1/3",
      "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 function evaluation signed correction，不以小數近似替代。",
          "en": "Recompute selected function evaluation signed correction with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "1",
          "3"
        ],
        "output": "1/3",
        "label": "selected function evaluation signed correction"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "selected function evaluation function value := sum(0, 1/3)",
      "after": "selected function evaluation function value = 1/3",
      "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 function evaluation function value，不以小數近似替代。",
          "en": "Recompute selected function evaluation function value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "1/3"
        ],
        "output": "1/3",
        "label": "selected function evaluation function value"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "selected function evaluation reverse signed error := sub(1/3, 0)",
      "after": "selected function evaluation reverse signed error = 1/3",
      "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 function evaluation reverse signed error，不以小數近似替代。",
          "en": "Recompute selected function evaluation reverse signed error with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "1/3",
          "0"
        ],
        "output": "1/3",
        "label": "selected function evaluation reverse signed error"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "selected function evaluation:original point=0;n=1;limit=0;signed coefficient=1",
      "after": "f_n(x)=1/3;f_n(x)-limit=1/3;this error field remains SIGNED",
      "relation": "implies",
      "rule_id": "pointwise_selected_value_reverse_error_not_absolute_value",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先加修正項得到函數值，再同減limit反向核誤差；原evaluate欄位保留正負，只有epsilon界才取絕對值。",
        "en": "Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute 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": [
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      0,
      "1/3",
      "0",
      "1/3"
    ]
  },
  "conclusion": {
    "canonical_value": "(1, 0, 1/3, 0, 1/3)",
    "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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      "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."
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}