第 12868 題：完備度量空間

原題與已知條件

完備性量化空間內每個Cauchy序列。操作=complete-finite-certificate；資料={"n":4,"metric":[[0,8,12,14],[8,0,14,16],[12,14,0,20],[14,16,20,0]]}。請依 [n,min_positive_distance,universal_witness,complete] 欄序回答。

• G001 — 完備性量化空間內每個Cauchy序列。操作=complete-finite-certificate；資料={"n":4,"metric":[[0,8,12,14],[8,0,14,16],[12,14,0,20],[14,16,20,0]]}。請依 [n,min_positive_distance,universal_witness,complete] 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-complete-metric-space

逐步推導

carrier Fin(4);complete nonnegative integer metric candidate=[[0, 8, 12, 14], [8, 0, 14, 16], [12, 14, 0, 20], [14, 16, 20, 0]]

zero iff equal=是;symmetric=是

endpoints=[0, 0];direct distance=0;all intermediate labels=[0, 1, 2, 3]

all two-leg distances=[0, 16, 24, 28];direct<=every two-leg=是

endpoints=[0, 1];direct distance=8;all intermediate labels=[0, 1, 2, 3]

all two-leg distances=[8, 8, 26, 30];direct<=every two-leg=是

endpoints=[0, 2];direct distance=12;all intermediate labels=[0, 1, 2, 3]

all two-leg distances=[12, 22, 12, 34];direct<=every two-leg=是

endpoints=[0, 3];direct distance=14;all intermediate labels=[0, 1, 2, 3]

all two-leg distances=[14, 24, 32, 14];direct<=every two-leg=是

endpoints=[1, 0];direct distance=8;all intermediate labels=[0, 1, 2, 3]

all two-leg distances=[8, 8, 26, 30];direct<=every two-leg=是

endpoints=[1, 1];direct distance=0;all intermediate labels=[0, 1, 2, 3]

all two-leg distances=[16, 0, 28, 32];direct<=every two-leg=是

endpoints=[1, 2];direct distance=14;all intermediate labels=[0, 1, 2, 3]

all two-leg distances=[20, 14, 14, 36];direct<=every two-leg=是

endpoints=[1, 3];direct distance=16;all intermediate labels=[0, 1, 2, 3]

all two-leg distances=[22, 16, 34, 16];direct<=every two-leg=是

endpoints=[2, 0];direct distance=12;all intermediate labels=[0, 1, 2, 3]

all two-leg distances=[12, 22, 12, 34];direct<=every two-leg=是

endpoints=[2, 1];direct distance=14;all intermediate labels=[0, 1, 2, 3]

all two-leg distances=[20, 14, 14, 36];direct<=every two-leg=是

endpoints=[2, 2];direct distance=0;all intermediate labels=[0, 1, 2, 3]

all two-leg distances=[24, 28, 0, 40];direct<=every two-leg=是

endpoints=[2, 3];direct distance=20;all intermediate labels=[0, 1, 2, 3]

all two-leg distances=[26, 30, 20, 20];direct<=every two-leg=是

endpoints=[3, 0];direct distance=14;all intermediate labels=[0, 1, 2, 3]

all two-leg distances=[14, 24, 32, 14];direct<=every two-leg=是

endpoints=[3, 1];direct distance=16;all intermediate labels=[0, 1, 2, 3]

all two-leg distances=[22, 16, 34, 16];direct<=every two-leg=是

endpoints=[3, 2];direct distance=20;all intermediate labels=[0, 1, 2, 3]

all two-leg distances=[26, 30, 20, 20];direct<=every two-leg=是

endpoints=[3, 3];direct distance=0;all intermediate labels=[0, 1, 2, 3]

all two-leg distances=[28, 32, 40, 0];direct<=every two-leg=是

all off-diagonal positive distances=[8, 12, 14, 8, 14, 16, 12, 14, 20, 14, 16, 20]

minimum positive distance delta=8>0

arbitrary Cauchy sequence in Fin(4);choose epsilon=delta=8>0

some N has every m,k>=N with d(x_m,x_k)<8;minimum separation forces x_m=x_k

all tail terms equal x_N;typed sequence guarantees x_N in Fin(4)

for every positive eta,all k>=N have distance(x_k,x_N)=0<eta;limit belongs to same space

逐步反向核對

• S020 · 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。
• 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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S016 · 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。
• S015 · 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。
• S014 · 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。
• S013 · 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。
• S012 · 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。
• S011 · 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。
• 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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S008 · 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。
• S007 · 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。
• S006 · 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。
• S005 · 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。
• S004 · 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。
• S003 · 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。
• 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

(4, 8, epsilon=min_positive_distance forces every Cauchy tail to be constant, 是)

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。