{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0055",
  "pilot_number": "R0036-18555",
  "number": 18555,
  "family_id": "qf-mod35-uniform-pointwise-max",
  "title": {
    "zh-Hant": "第 18555 題：一致收斂",
    "en": "Question 18555: A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0055",
    "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-pointwise-max；資料={\"limit\":[4,5],\"coefficients\":[5,-6],\"shift\":2,\"epsilon\":[1,14]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([[0, 69], [1, 83]], 83, 是, finite_maximum_only_not_infinite_domain_implication)",
      "explanation_zh": "Wave5第四層核對=([[0, 69], [1, 83]], 83, 是, finite_maximum_only_not_infinite_domain_implication)；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": 1,
      "skills": [
        "一致收斂"
      ]
    },
    "variant": {
      "family_id": "qf-mod35-uniform-pointwise-max",
      "seed": 202611042055,
      "parameters": {
        "limit": [
          4,
          5
        ],
        "coefficients": [
          5,
          -6
        ],
        "shift": 2,
        "epsilon": [
          1,
          14
        ],
        "domain": "Finite_domain_symbolic_uniform_tail",
        "operation": "uniform-pointwise-max",
        "semantic_template": "qf-mod35-uniform-pointwise-max/{\"coefficients\":[5,-6],\"epsilon\":[1,14],\"limit\":[4,5],\"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": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0055",
    "batch_id": "batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "46a6854611f0d02a067106039f7965ddbfe70aff5e6695d7905604a1fcbbe8b6",
      "family_id": "qf-mod35-uniform-pointwise-max",
      "seed": 202611042055,
      "semantic_template": "qf-mod35-uniform-pointwise-max/{\"coefficients\":[5,-6],\"epsilon\":[1,14],\"limit\":[4,5],\"shift\":2}"
    },
    "fields": {
      "title": "A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange",
      "stem": "Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[4, 5], coefficients=[5, -6], 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. Positive epsilon=1/14. Give (pointwise_witnesses, N, uniform_on_this_finite_domain, boundary).",
      "answer": {
        "kind": "expression",
        "value": "([[0, 69], [1, 83]], 83, Yes, finite_maximum_only_not_infinite_domain_implication)",
        "explanation": "List pointwise strict witnesses in domain order and take their maximum. That supplies a common N on this complete finite domain only, not a general implication from pointwise to uniform convergence. The exact result, retaining the requested order, is ([[0, 69], [1, 83]], 83, Yes, finite_maximum_only_not_infinite_domain_implication)."
      },
      "skills": [
        "A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange"
      ],
      "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-pointwise-max；資料={\"limit\":[4,5],\"coefficients\":[5,-6],\"shift\":2,\"epsilon\":[1,14]}。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=[4, 5], coefficients=[5, -6], 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. Positive epsilon=1/14. Give (pointwise_witnesses, N, uniform_on_this_finite_domain, boundary)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source operation=uniform-pointwise-max;original snapshot and mathematics retained",
      "after": "explicit ordered source output fields=[pointwise_witnesses, N, uniform_on_this_finite_domain, boundary]",
      "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=[4, 5];signed coefficients=[5, -6];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 point0 strict numerator threshold := div(5, 1/14)",
      "after": "uniform point0 strict numerator threshold = 70",
      "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 strict numerator threshold，不以小數近似替代。",
          "en": "Recompute uniform point0 strict numerator threshold with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "5",
          "1/14"
        ],
        "output": "70",
        "label": "uniform point0 strict numerator threshold"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "uniform point0 floor threshold minus shift := sub(70, 2)",
      "after": "uniform point0 floor threshold minus shift = 68",
      "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 floor threshold minus shift，不以小數近似替代。",
          "en": "Recompute uniform point0 floor threshold minus shift with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "70",
          "2"
        ],
        "output": "68",
        "label": "uniform point0 floor threshold minus shift"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "uniform point0 first strict integer index := sum(68, 1)",
      "after": "uniform point0 first strict integer index = 69",
      "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 first strict integer index，不以小數近似替代。",
          "en": "Recompute uniform point0 first strict integer index with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "68",
          "1"
        ],
        "output": "69",
        "label": "uniform point0 first strict integer index"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "uniform point0 cutoff denominator := sum(69, 2)",
      "after": "uniform point0 cutoff denominator = 71",
      "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 cutoff denominator，不以小數近似替代。",
          "en": "Recompute uniform point0 cutoff denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "69",
          "2"
        ],
        "output": "71",
        "label": "uniform point0 cutoff denominator"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "uniform point0 cutoff gap := div(5, 71)",
      "after": "uniform point0 cutoff gap = 5/71",
      "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 cutoff gap，不以小數近似替代。",
          "en": "Recompute uniform point0 cutoff gap with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "5",
          "71"
        ],
        "output": "5/71",
        "label": "uniform point0 cutoff gap"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "uniform point0 previous denominator := sub(71, 1)",
      "after": "uniform point0 previous denominator = 70",
      "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 previous denominator，不以小數近似替代。",
          "en": "Recompute uniform point0 previous denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "71",
          "1"
        ],
        "output": "70",
        "label": "uniform point0 previous denominator"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "uniform point0 previous gap := div(5, 70)",
      "after": "uniform point0 previous gap = 1/14",
      "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 previous gap，不以小數近似替代。",
          "en": "Recompute uniform point0 previous gap with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "5",
          "70"
        ],
        "output": "1/14",
        "label": "uniform point0 previous gap"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "original point=0;epsilon=1/14;abs coefficient=5;exact threshold=70;floor=70;unclamped index=69",
      "after": "least allowed N_x=69;gap_at_Nx=5/71<epsilon;previous gap=1/14;all later denominators only increase",
      "relation": "implies",
      "rule_id": "uniform_least_pointwise_strict_index_not_float_rounding",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正epsilon下，條件等價n+shift>|c|/epsilon，最小整數為floor門檻加1再扣shift並與0取max。若N_x>0還核前一項失敗，零係數自然給0，不執行負索引。",
        "en": "For positive epsilon the condition is n+shift>|c|/epsilon.The first strict integer is floor(threshold)+1 minus the shift,clamped at0.When N_x>0 verify previous failure;zero coefficients naturally give0 without evaluating negative 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": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "uniform point1 strict numerator threshold := div(6, 1/14)",
      "after": "uniform point1 strict numerator threshold = 84",
      "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 strict numerator threshold，不以小數近似替代。",
          "en": "Recompute uniform point1 strict numerator threshold with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "6",
          "1/14"
        ],
        "output": "84",
        "label": "uniform point1 strict numerator threshold"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "uniform point1 floor threshold minus shift := sub(84, 2)",
      "after": "uniform point1 floor threshold minus shift = 82",
      "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 floor threshold minus shift，不以小數近似替代。",
          "en": "Recompute uniform point1 floor threshold minus shift with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "84",
          "2"
        ],
        "output": "82",
        "label": "uniform point1 floor threshold minus shift"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "uniform point1 first strict integer index := sum(82, 1)",
      "after": "uniform point1 first strict integer index = 83",
      "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 first strict integer index，不以小數近似替代。",
          "en": "Recompute uniform point1 first strict integer index with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "82",
          "1"
        ],
        "output": "83",
        "label": "uniform point1 first strict integer index"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "uniform point1 cutoff denominator := sum(83, 2)",
      "after": "uniform point1 cutoff denominator = 85",
      "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 cutoff denominator，不以小數近似替代。",
          "en": "Recompute uniform point1 cutoff denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "83",
          "2"
        ],
        "output": "85",
        "label": "uniform point1 cutoff denominator"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "uniform point1 cutoff gap := div(6, 85)",
      "after": "uniform point1 cutoff gap = 6/85",
      "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 cutoff gap，不以小數近似替代。",
          "en": "Recompute uniform point1 cutoff gap with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "6",
          "85"
        ],
        "output": "6/85",
        "label": "uniform point1 cutoff gap"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "uniform point1 previous denominator := sub(85, 1)",
      "after": "uniform point1 previous denominator = 84",
      "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 previous denominator，不以小數近似替代。",
          "en": "Recompute uniform point1 previous denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "85",
          "1"
        ],
        "output": "84",
        "label": "uniform point1 previous denominator"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "uniform point1 previous gap := div(6, 84)",
      "after": "uniform point1 previous gap = 1/14",
      "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 previous gap，不以小數近似替代。",
          "en": "Recompute uniform point1 previous gap with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "6",
          "84"
        ],
        "output": "1/14",
        "label": "uniform point1 previous gap"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "original point=1;epsilon=1/14;abs coefficient=6;exact threshold=84;floor=84;unclamped index=83",
      "after": "least allowed N_x=83;gap_at_Nx=6/85<epsilon;previous gap=1/14;all later denominators only increase",
      "relation": "implies",
      "rule_id": "uniform_least_pointwise_strict_index_not_float_rounding",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正epsilon下，條件等價n+shift>|c|/epsilon，最小整數為floor門檻加1再扣shift並與0取max。若N_x>0還核前一項失敗，零係數自然給0，不執行負索引。",
        "en": "For positive epsilon the condition is n+shift>|c|/epsilon.The first strict integer is floor(threshold)+1 minus the shift,clamped at0.When N_x>0 verify previous failure;zero coefficients naturally give0 without evaluating negative 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": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "ALL original ordered point witnesses=[[0, 69], [1, 83]];finite domain size=2;maximum attained at point=1",
      "after": "common N=83;for EVERY x and EVERY n>=N,n>=N_x and gap_n(x)<=gap_Nx(x)<epsilon",
      "relation": "implies",
      "rule_id": "uniform_one_N_full_finite_domain_all_later_quantifiers",
      "conditions": [],
      "explanation": {
        "zh-Hant": "有限點集可取最大值，同一N不依x。若N>0，達到最大N_x的點也給任何更小共同索引失敗；不是拿一點或一次n代替全稱量詞。無限domain的pointwise門檻可能無共同有限上界，不外推。",
        "en": "Take the maximum over the complete finite domain,so the same N is independent of x.If N>0,an attaining point also defeats every smaller common cutoff.This controls both quantifiers,not one point or one n.Pointwise cutoffs on an infinite domain need not share a finite 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
    }
  ],
  "math_model": {
    "result": [
      [
        [
          0,
          69
        ],
        [
          1,
          83
        ]
      ],
      83,
      true,
      "finite_maximum_only_not_infinite_domain_implication"
    ]
  },
  "conclusion": {
    "canonical_value": "([[0, 69], [1, 83]], 83, 是, finite_maximum_only_not_infinite_domain_implication)",
    "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": {
    "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db",
    "question_binding": {
      "batch_id": "batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations",
      "batch_path": "public/data/questions/batches/batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations/questions.jsonl",
      "batch_sha256": "c5e99643b18779ea8210be153283f9fac0853c289c3f16e39f8c065d3178ecea",
      "original_line": 55,
      "original_line_sha256": "e47ba355eb41bc0e80ef43d05d1a100f9c3bbcd9dedb009af204f53cf7716471",
      "question_compact_sha256": "46a6854611f0d02a067106039f7965ddbfe70aff5e6695d7905604a1fcbbe8b6",
      "english_row_compact_sha256": "e4e81e745eeda380bebf75e306f47ba56d888932567f101a51b559c97ce5b16e",
      "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-uniform-convergence-of-functions": {
        "sha256": "c28f03cdaf36d8b06ca1718e4253a1d705745496ed7c655d09f55c2a82148ac0",
        "english_sha256": "f3588cebfd521dd72a1db18c1ae10af42c4b9398d451eb7680dac158b73a4462",
        "original_path": "public/data/mko/uniform-convergence-of-functions.json",
        "english_path": "public/data/i18n/editions/mko-uniform-convergence-of-functions.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "49e5d7674cefa01298c2386543f777d2bdfc99446b4027156b80142f10031994",
    "original_candidate_sha256": "bb70fba4ee2931275acf36f7f0e6833d689ff123f411b8cfb4d15a84e2da9835",
    "original_candidate_line_sha256": "bb70fba4ee2931275acf36f7f0e6833d689ff123f411b8cfb4d15a84e2da9835"
  },
  "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-p37",
    "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": "efee5462991619d6ed9f64f11ddb085e5614ef33f9f8fcd8cf29b9329f8fd301",
      "question_first_witness_sha256": "acf91fa1622f01d2cf04b2c513294254323665351f660f2628aaafdb869c4ff2"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 19,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "83cbdfa033894bd5b6b15e396c6dd33d0a91c9dae8efa39149853c0929f6ed91",
      "arithmetic_steps": 14
    },
    "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."
  }
}