{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0037",
  "pilot_number": "R0053-27037",
  "number": 27037,
  "family_id": "qf-mod48-uniform-epsilon-index",
  "title": {
    "zh-Hant": "第 27037 題：一致極限保持連續",
    "en": "Question 27037: One Positive Index Must Give Strict Error Control for Every Point"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0037",
    "batch_id": "batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-uniform-limit-continuity-theorem"
    ],
    "title_zh": "一致極限保持連續",
    "stem_zh": "共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-epsilon-index；資料={\"error_scale\":12,\"epsilon\":[2,8]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(12, 1/4, 48, 12/49, 是, same_N_controls_every_x_in_[0,1])",
      "explanation_zh": "Wave6第十一層核對=(12, 1/4, 48, 12/49, 是, same_N_controls_every_x_in_[0,1])；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "一致極限保持連續"
      ]
    },
    "variant": {
      "family_id": "qf-mod48-uniform-epsilon-index",
      "seed": 202611173037,
      "parameters": {
        "error_scale": 12,
        "epsilon": [
          2,
          8
        ],
        "domain": "Exact_uniform_limit_continuity",
        "operation": "uniform-epsilon-index",
        "semantic_template": "uniform-epsilon-index|{epsilon:[1,4],error_scale:[12,1]}"
      }
    },
    "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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-15",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0037",
    "batch_id": "batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "89c0a0c3a6566d5acfe48998bab975b27e8710cbae7ae2de8f1365f41d18c123",
      "family_id": "qf-mod48-uniform-epsilon-index",
      "seed": 202611173037,
      "semantic_template": "uniform-epsilon-index|{epsilon:[1,4],error_scale:[12,1]}"
    },
    "fields": {
      "title": "One Positive Index Must Give Strict Error Control for Every Point",
      "stem": "Use exact E=12>0 and epsilon=2/8>0 for sup error E/(n+1), n≥1. Give (error_scale, epsilon, N, error_at_N, strict_error_bound, certificate).",
      "answer": {
        "kind": "expression",
        "value": "(12, 1/4, 48, 12/49, Yes, same_N_controls_every_x_in_[0,1])",
        "explanation": "Choose the first positive integer N with E/(N+1)<epsilon and retain the strict bound, including exact integer E/epsilon cases. The same N then works for all x and every later n; it is not chosen separately for each point. The exact result, retaining the requested order, is (12, 1/4, 48, 12/49, Yes, same_N_controls_every_x_in_[0,1])."
      },
      "skills": [
        "One Positive Index Must Give Strict Error Control for Every 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": "共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-epsilon-index；資料={\"error_scale\":12,\"epsilon\":[2,8]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use exact E=12>0 and epsilon=2/8>0 for sup error E/(n+1), n≥1. Give (error_scale, epsilon, N, error_at_N, strict_error_bound, certificate)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [uniform-epsilon-index]",
      "after": "derived = [[error_scale, epsilon, N, error_at_N, strict_error_bound, certificate]]",
      "relation": "implies",
      "rule_id": "source_ordered_fields_without_JS",
      "conditions": [],
      "explanation": {
        "zh-Hant": "此頁直接列完整來源欄序與mode特定角色，不靠JavaScript或由答案外形猜介面；No、undefined、infinity各保原意。",
        "en": "List the complete source field order and mode-specific roles without JavaScript or answer-shape inference;No,undefined and infinity retain their distinct meanings."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "Uniform ORIGINAL error divided by epsilon := div(12, 1/4)",
      "after": "Uniform ORIGINAL error divided by epsilon = 48",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Uniform ORIGINAL error divided by epsilon，不以小數近似替代。",
          "en": "Recompute Uniform ORIGINAL error divided by epsilon with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "12",
          "1/4"
        ],
        "output": "48",
        "label": "Uniform ORIGINAL error divided by epsilon"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "Uniform ORIGINAL error at FIRST positive N := div(12, 49)",
      "after": "Uniform ORIGINAL error at FIRST positive N = 12/49",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Uniform ORIGINAL error at FIRST positive N，不以小數近似替代。",
          "en": "Recompute Uniform ORIGINAL error at FIRST positive N with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "12",
          "49"
        ],
        "output": "12/49",
        "label": "Uniform ORIGINAL error at FIRST positive N"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "Uniform previous positive index error := div(12, 48)",
      "after": "Uniform previous positive index error = 1/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Uniform previous positive index error，不以小數近似替代。",
          "en": "Recompute Uniform previous positive index error with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "12",
          "48"
        ],
        "output": "1/4",
        "label": "Uniform previous positive index error"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "source roles = [12, 1/4, 48, 48]",
      "after": "derived = [12/49, 1/4, 是, 是]",
      "relation": "implies",
      "rule_id": "Uniform_FIRST_positive_N_strict_all_x_all_later_n",
      "conditions": [],
      "explanation": {
        "zh-Hant": "E/(n+1)<epsilon iff n+1>E/epsilon。取正整數N=max(1,floor(E/epsilon))使N+1嚴格大於比值，整除邊界亦成立；N>1時前一正整數的誤差E/N>=epsilon，故最小。所有n>=N的sup誤差只減少，同一N同時控制所有x，不用ceil或N=0。",
        "en": "E/(n+1)<epsilon iff n+1>E/epsilon.The positive integer N=max(1,floor(E/epsilon)) makes N+1 strictly larger,including exact-division cases.If N>1,the previous positive index has error E/N>=epsilon,so N is minimal.For every n>=N the sup error decreases;the same N controls all x.No ceil or N=0 substitution."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      "12",
      "1/4",
      48,
      "12/49",
      true,
      "same_N_controls_every_x_in_[0,1]"
    ]
  },
  "conclusion": {
    "canonical_value": "(12, 1/4, 48, 12/49, 是, same_N_controls_every_x_in_[0,1])",
    "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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      "english_manifest_path": "public/data/i18n/editions/batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations.en.jsonl"
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        "original_path": "public/data/mko/uniform-limit-continuity-theorem.json",
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    "selected_part_sha256": "a54409c0bc3f68730e1072c6d3d772970462c1bc7ffc203b4030a6136a610283",
    "original_candidate_sha256": "8c95cadf6c9111d8293ac630e2fda1e9fa2112ef22c5a04f5f29aa7bd50dd9cb",
    "original_candidate_line_sha256": "8c95cadf6c9111d8293ac630e2fda1e9fa2112ef22c5a04f5f29aa7bd50dd9cb"
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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."
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  "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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    "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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}