第 12831 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-evidence-mode-audit；資料={"n":4,"metric":[[0,4,7,9],[4,0,9,11],[7,9,0,14],[9,11,14,0]],"evidence_mode":"finite_minimum_separation"}。請依 [n,min_positive_distance,evidence_mode,sufficient,boundary] 欄序回答。

• G001 — 完備性量化空間內每個Cauchy序列。操作=complete-evidence-mode-audit；資料={"n":4,"metric":[[0,4,7,9],[4,0,9,11],[7,9,0,14],[9,11,14,0]],"evidence_mode":"finite_minimum_separation"}。請依 [n,min_positive_distance,evidence_mode,sufficient,boundary] 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, 4, 7, 9], [4, 0, 9, 11], [7, 9, 0, 14], [9, 11, 14, 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, 8, 14, 18];direct<=every two-leg=是

endpoints=[0, 1];direct distance=4;all intermediate labels=[0, 1, 2, 3]

all two-leg distances=[4, 4, 16, 20];direct<=every two-leg=是

endpoints=[0, 2];direct distance=7;all intermediate labels=[0, 1, 2, 3]

all two-leg distances=[7, 13, 7, 23];direct<=every two-leg=是

endpoints=[0, 3];direct distance=9;all intermediate labels=[0, 1, 2, 3]

all two-leg distances=[9, 15, 21, 9];direct<=every two-leg=是

endpoints=[1, 0];direct distance=4;all intermediate labels=[0, 1, 2, 3]

all two-leg distances=[4, 4, 16, 20];direct<=every two-leg=是

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

all two-leg distances=[8, 0, 18, 22];direct<=every two-leg=是

endpoints=[1, 2];direct distance=9;all intermediate labels=[0, 1, 2, 3]

all two-leg distances=[11, 9, 9, 25];direct<=every two-leg=是

endpoints=[1, 3];direct distance=11;all intermediate labels=[0, 1, 2, 3]

all two-leg distances=[13, 11, 23, 11];direct<=every two-leg=是

endpoints=[2, 0];direct distance=7;all intermediate labels=[0, 1, 2, 3]

all two-leg distances=[7, 13, 7, 23];direct<=every two-leg=是

endpoints=[2, 1];direct distance=9;all intermediate labels=[0, 1, 2, 3]

all two-leg distances=[11, 9, 9, 25];direct<=every two-leg=是

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

all two-leg distances=[14, 18, 0, 28];direct<=every two-leg=是

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

all two-leg distances=[16, 20, 14, 14];direct<=every two-leg=是

endpoints=[3, 0];direct distance=9;all intermediate labels=[0, 1, 2, 3]

all two-leg distances=[9, 15, 21, 9];direct<=every two-leg=是

endpoints=[3, 1];direct distance=11;all intermediate labels=[0, 1, 2, 3]

all two-leg distances=[13, 11, 23, 11];direct<=every two-leg=是

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

all two-leg distances=[16, 20, 14, 14];direct<=every two-leg=是

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

all two-leg distances=[18, 22, 28, 0];direct<=every two-leg=是

all off-diagonal positive distances=[4, 7, 9, 4, 9, 11, 7, 9, 14, 9, 11, 14]

minimum positive distance delta=4>0

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

some N has every m,k>=N with d(x_m,x_k)<4;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

qualified finite metric;delta=4;selected evidence mode=finite_minimum_separation

selected evidence sufficient=是;space completeness is not denied by sample insufficiency

逐步反向核對

• S021 · 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。
• 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, 4, finite_minimum_separation, 是, universal_finite_certificate)

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。