{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0035",
  "pilot_number": "R0036-18585",
  "number": 18585,
  "family_id": "qf-mod35-uniform-sup-gap",
  "title": {
    "zh-Hant": "第 18585 題：一致收斂",
    "en": "Question 18585: Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0035",
    "batch_id": "batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-uniform-convergence-of-functions"
    ],
    "title_zh": "一致收斂",
    "stem_zh": "共同有限domain的symbolic tail與同一N(epsilon)。操作=uniform-sup-gap；資料={\"limit\":[9,10],\"coefficients\":[10,-11],\"shift\":2,\"n\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(10, 11/12, exact_finite_domain_supremum)",
      "explanation_zh": "Wave5第四層核對=(10, 11/12, exact_finite_domain_supremum)；依完整明示資料重算；同一N控制所有x且有限domain可取pointwise witnesses最大值；不可外推無限domain且finite prefix不足。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "一致收斂"
      ]
    },
    "variant": {
      "family_id": "qf-mod35-uniform-sup-gap",
      "seed": 202611042035,
      "parameters": {
        "limit": [
          9,
          10
        ],
        "coefficients": [
          10,
          -11
        ],
        "shift": 2,
        "n": 10,
        "domain": "Finite_domain_symbolic_uniform_tail",
        "operation": "uniform-sup-gap",
        "semantic_template": "qf-mod35-uniform-sup-gap/{\"coefficients\":[10,-11],\"limit\":[9,10],\"n\":10,\"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": "候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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": {
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    "question_id": "q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0035",
    "batch_id": "batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "38182227513742bf0633c9c833e4f0b409ec19c85d26e461ceba4701ac68aea6",
      "family_id": "qf-mod35-uniform-sup-gap",
      "seed": 202611042035,
      "semantic_template": "qf-mod35-uniform-sup-gap/{\"coefficients\":[10,-11],\"limit\":[9,10],\"n\":10,\"shift\":2}"
    },
    "fields": {
      "title": "Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio",
      "stem": "Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[9, 10], coefficients=[10, -11], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. n=10. Give (n, sup_gap, certificate).",
      "answer": {
        "kind": "expression",
        "value": "(10, 11/12, exact_finite_domain_supremum)",
        "explanation": "The exact supremum is max_x |coefficient[x]|/(n+shift). Retain the source direct-ratio text, which may be unreduced, instead of changing its numerical bytes during translation. The exact result, retaining the requested order, is (10, 11/12, exact_finite_domain_supremum)."
      },
      "skills": [
        "Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio"
      ],
      "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": "共同有限domain的symbolic tail與同一N(epsilon)。操作=uniform-sup-gap；資料={\"limit\":[9,10],\"coefficients\":[10,-11],\"shift\":2,\"n\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[9, 10], coefficients=[10, -11], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. n=10. Give (n, sup_gap, certificate)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source operation=uniform-sup-gap;original snapshot and mathematics retained",
      "after": "explicit ordered source output fields=[n, sup_gap, certificate]",
      "relation": "implies",
      "rule_id": "source_ordered_interface_explicit_without_javascript",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原題歷史介面標記仍在快照；這裡直接給原始欄位順序，主文不需JavaScript或外部表。描述欄位不自動證成任何數學聲稱。",
        "en": "Keep the historical interface marker in the original snapshot while listing the original ordered fields here.No JavaScript or external table is needed for the main text,and field labels are not mathematical evidence."
      },
      "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": "original complete domain Fin(2);aligned limit=[9, 10];signed coefficients=[10, -11];shift=2",
      "after": "for EVERY n>=0 and EVERY original point x,f_n(x)-limit[x]=coefficient[x]/(n+shift);same positive denominator on the whole domain",
      "relation": "implies",
      "rule_id": "uniform_declared_full_family_error_cancellation",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先從明示的全序列定義消去同一limit[x]，不是只看已觀察數值；limit與coefficient保原點序，係數與limit不是機率值。",
        "en": "Cancel the same limit[x] from the declared whole-sequence formula,not merely observed samples.Retain original point order;these coefficients and limits are not probabilities."
      },
      "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": "uniform selected denominator := sum(10, 2)",
      "after": "uniform selected denominator = 12",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 uniform selected denominator，不以小數近似替代。",
          "en": "Recompute uniform selected denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "10",
          "2"
        ],
        "output": "12",
        "label": "uniform selected denominator"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "uniform point0 absolute gap := div(10, 12)",
      "after": "uniform point0 absolute gap = 5/6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 uniform point0 absolute gap，不以小數近似替代。",
          "en": "Recompute uniform point0 absolute gap with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "10",
          "12"
        ],
        "output": "5/6",
        "label": "uniform point0 absolute gap"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "uniform point1 absolute gap := div(11, 12)",
      "after": "uniform point1 absolute gap = 11/12",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 uniform point1 absolute gap，不以小數近似替代。",
          "en": "Recompute uniform point1 absolute gap with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "11",
          "12"
        ],
        "output": "11/12",
        "label": "uniform point1 absolute gap"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "uniform finite maximum gap := div(11, 12)",
      "after": "uniform finite maximum gap = 11/12",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 uniform finite maximum gap，不以小數近似替代。",
          "en": "Recompute uniform finite maximum gap with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "11",
          "12"
        ],
        "output": "11/12",
        "label": "uniform finite maximum gap"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "original n=10;all exact point gaps=[5/6, 11/12];maximum absolute coefficient=11;attaining original point=1",
      "after": "EVERY gap<=maximum and selected point attains it;source direct ratio text=11/12;exact reduced value=11/12",
      "relation": "implies",
      "rule_id": "uniform_finite_sup_attained_source_ratio_bytes_retained",
      "conditions": [],
      "explanation": {
        "zh-Hant": "有限非空domain的最大值同時給上界與達到點，所以是sup而非僅upper bound。來源直接ratio可能未約分，輸出字節照留，另核其精確值，不替換為另一字串。",
        "en": "The nonempty finite domain supplies both an upper bound and an attaining point,hence the supremum rather than merely an upper bound.Keep potentially unreduced source ratio bytes while checking their exact value separately."
      },
      "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,
      "11/12",
      "exact_finite_domain_supremum"
    ]
  },
  "conclusion": {
    "canonical_value": "(10, 11/12, exact_finite_domain_supremum)",
    "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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}