第 18512 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-finite-domain-witness；資料={"limit":[11,12],"coefficients":[12,-13],"shift":2,"epsilon":[1,21]}。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-finite-domain-witness；資料={"limit":[11,12],"coefficients":[12,-13],"shift":2,"epsilon":[1,21]}。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-finite-domain-witness;original snapshot and mathematics retained

explicit ordered source output fields=[epsilon, pointwise_witnesses, N, same_N_for_all_points, certificate]

original complete domain Fin(2);aligned limit=[11, 12];signed coefficients=[12, -13];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(12, 1/21)

uniform point0 strict numerator threshold = 252

uniform point0 floor threshold minus shift := sub(252, 2)

uniform point0 floor threshold minus shift = 250

uniform point0 first strict integer index := sum(250, 1)

uniform point0 first strict integer index = 251

uniform point0 cutoff denominator := sum(251, 2)

uniform point0 cutoff denominator = 253

uniform point0 cutoff gap := div(12, 253)

uniform point0 cutoff gap = 12/253

uniform point0 previous denominator := sub(253, 1)

uniform point0 previous denominator = 252

uniform point0 previous gap := div(12, 252)

uniform point0 previous gap = 1/21

original point=0;epsilon=1/21;abs coefficient=12;exact threshold=252;floor=252;unclamped index=251

least allowed N_x=251;gap_at_Nx=12/253<epsilon;previous gap=1/21;all later denominators only increase

uniform point1 strict numerator threshold := div(13, 1/21)

uniform point1 strict numerator threshold = 273

uniform point1 floor threshold minus shift := sub(273, 2)

uniform point1 floor threshold minus shift = 271

uniform point1 first strict integer index := sum(271, 1)

uniform point1 first strict integer index = 272

uniform point1 cutoff denominator := sum(272, 2)

uniform point1 cutoff denominator = 274

uniform point1 cutoff gap := div(13, 274)

uniform point1 cutoff gap = 13/274

uniform point1 previous denominator := sub(274, 1)

uniform point1 previous denominator = 273

uniform point1 previous gap := div(13, 273)

uniform point1 previous gap = 1/21

original point=1;epsilon=1/21;abs coefficient=13;exact threshold=273;floor=273;unclamped index=272

least allowed N_x=272;gap_at_Nx=13/274<epsilon;previous gap=1/21;all later denominators only increase

ALL original ordered point witnesses=[[0, 251], [1, 272]];finite domain size=2;maximum attained at point=1

common N=272;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

(1/21, [[0, 251], [1, 272]], 272, 是, same_N_for_all_finite_domain_points)

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。