第 18570 題：一致收斂

原題與已知條件

共同有限domain的symbolic tail與同一N(epsilon)。操作=uniform-pointwise-max；資料={"limit":[19,20],"coefficients":[20,-21],"shift":2,"epsilon":[1,29]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — 共同有限domain的symbolic tail與同一N(epsilon)。操作=uniform-pointwise-max；資料={"limit":[19,20],"coefficients":[20,-21],"shift":2,"epsilon":[1,29]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

定義與適用範圍

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。

• mko-uniform-convergence-of-functions

逐步推導

source operation=uniform-pointwise-max;original snapshot and mathematics retained

explicit ordered source output fields=[pointwise_witnesses, N, uniform_on_this_finite_domain, boundary]

original complete domain Fin(2);aligned limit=[19, 20];signed coefficients=[20, -21];shift=2

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

uniform point0 strict numerator threshold := div(20, 1/29)

uniform point0 strict numerator threshold = 580

uniform point0 floor threshold minus shift := sub(580, 2)

uniform point0 floor threshold minus shift = 578

uniform point0 first strict integer index := sum(578, 1)

uniform point0 first strict integer index = 579

uniform point0 cutoff denominator := sum(579, 2)

uniform point0 cutoff denominator = 581

uniform point0 cutoff gap := div(20, 581)

uniform point0 cutoff gap = 20/581

uniform point0 previous denominator := sub(581, 1)

uniform point0 previous denominator = 580

uniform point0 previous gap := div(20, 580)

uniform point0 previous gap = 1/29

original point=0;epsilon=1/29;abs coefficient=20;exact threshold=580;floor=580;unclamped index=579

least allowed N_x=579;gap_at_Nx=20/581<epsilon;previous gap=1/29;all later denominators only increase

uniform point1 strict numerator threshold := div(21, 1/29)

uniform point1 strict numerator threshold = 609

uniform point1 floor threshold minus shift := sub(609, 2)

uniform point1 floor threshold minus shift = 607

uniform point1 first strict integer index := sum(607, 1)

uniform point1 first strict integer index = 608

uniform point1 cutoff denominator := sum(608, 2)

uniform point1 cutoff denominator = 610

uniform point1 cutoff gap := div(21, 610)

uniform point1 cutoff gap = 21/610

uniform point1 previous denominator := sub(610, 1)

uniform point1 previous denominator = 609

uniform point1 previous gap := div(21, 609)

uniform point1 previous gap = 1/29

original point=1;epsilon=1/29;abs coefficient=21;exact threshold=609;floor=609;unclamped index=608

least allowed N_x=608;gap_at_Nx=21/610<epsilon;previous gap=1/29;all later denominators only increase

ALL original ordered point witnesses=[[0, 579], [1, 608]];finite domain size=2;maximum attained at point=1

common N=608;for EVERY x and EVERY n>=N,n>=N_x and gap_n(x)<=gap_Nx(x)<epsilon

逐步反向核對

• S019 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S018 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S017 · premise_and_result_audit — 以精確有理數重算 uniform point1 previous gap，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 uniform point1 previous denominator，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 uniform point1 cutoff gap，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 uniform point1 cutoff denominator，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 uniform point1 first strict integer index，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 uniform point1 floor threshold minus shift，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 uniform point1 strict numerator threshold，不以小數近似替代。
• S010 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S009 · premise_and_result_audit — 以精確有理數重算 uniform point0 previous gap，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 uniform point0 previous denominator，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 uniform point0 cutoff gap，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 uniform point0 cutoff denominator，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 uniform point0 first strict integer index，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 uniform point0 floor threshold minus shift，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 uniform point0 strict numerator threshold，不以小數近似替代。
• S002 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S001 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。

結論

([[0, 579], [1, 608]], 608, 是, finite_maximum_only_not_infinite_domain_implication)

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。

本題覆核與來源綁定

覆核日期：2026-10-26

原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。