R0023-12146 · q-20261013-advance-20261026-r03-cauchy-sequence-0021
保留forall epsilon exists N forall m,n>=N。操作=cauchy-finite-tail-audit；資料={"sequence":[[1,22],[1,23],[1,24],[1,25],[1,26]],"epsilon":[1,3],"N":2}。請依 [sequence,epsilon,N,bad_pairs,finite_tail_pass,general_cauchy,boundary] 欄序回答。
G001 — 保留forall epsilon exists N forall m,n>=N。操作=cauchy-finite-tail-audit；資料={"sequence":[[1,22],[1,23],[1,24],[1,25],[1,26]],"epsilon":[1,3],"N":2}。請依 [sequence,epsilon,N,bad_pairs,finite_tail_pass,general_cauchy,boundary] 欄序回答。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。
操作前
raw sequence=[[1, 22], [1, 23], [1, 24], [1, 25], [1, 26]];raw epsilon=[1, 3]
操作後
normalized sequence=[1/22, 1/23, 1/24, 1/25, 1/26];positive epsilon=1/3;cutoff=2
條件：無
原分母非零，正規化後epsilon必須正；只核紀錄中的索引，N可等於長度而得到空尾段。
規則：finite_Cauchy_exact_data_and_cutoff · 依賴：G001
操作前
recorded pair2,2 signed difference := sub(1/24, 1/24)
操作後
recorded pair2,2 signed difference = 0
條件：無
依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。
規則：exact_rational_arithmetic · 依賴：G001, S001
操作前
ordered recorded indices=[2, 2];signed difference=0;epsilon=1/3
操作後
absolute distance=0;strictly below epsilon=是
條件：無
用絕對距離與嚴格小於，等於epsilon也失敗；按完整有序索引對輸出，不只比相鄰項。
規則：finite_Cauchy_each_ordered_tail_pair · 依賴：G001, S002
操作前
recorded pair2,3 signed difference := sub(1/24, 1/25)
操作後
recorded pair2,3 signed difference = 1/600
條件：無
依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。
規則：exact_rational_arithmetic · 依賴：G001, S003
操作前
ordered recorded indices=[2, 3];signed difference=1/600;epsilon=1/3
操作後
absolute distance=1/600;strictly below epsilon=是
條件：無
用絕對距離與嚴格小於，等於epsilon也失敗；按完整有序索引對輸出，不只比相鄰項。
規則：finite_Cauchy_each_ordered_tail_pair · 依賴：G001, S004
操作前
recorded pair2,4 signed difference := sub(1/24, 1/26)
操作後
recorded pair2,4 signed difference = 1/312
條件：無
依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。
規則：exact_rational_arithmetic · 依賴：G001, S005
操作前
ordered recorded indices=[2, 4];signed difference=1/312;epsilon=1/3
操作後
absolute distance=1/312;strictly below epsilon=是
條件：無
用絕對距離與嚴格小於，等於epsilon也失敗；按完整有序索引對輸出，不只比相鄰項。
規則：finite_Cauchy_each_ordered_tail_pair · 依賴：G001, S006
操作前
recorded pair3,2 signed difference := sub(1/25, 1/24)
操作後
recorded pair3,2 signed difference = -1/600
條件：無
依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。
規則：exact_rational_arithmetic · 依賴：G001, S007
操作前
ordered recorded indices=[3, 2];signed difference=-1/600;epsilon=1/3
操作後
absolute distance=1/600;strictly below epsilon=是
條件：無
用絕對距離與嚴格小於，等於epsilon也失敗；按完整有序索引對輸出，不只比相鄰項。
規則：finite_Cauchy_each_ordered_tail_pair · 依賴：G001, S008
操作前
recorded pair3,3 signed difference := sub(1/25, 1/25)
操作後
recorded pair3,3 signed difference = 0
條件：無
依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。
規則：exact_rational_arithmetic · 依賴：G001, S009
操作前
ordered recorded indices=[3, 3];signed difference=0;epsilon=1/3
操作後
absolute distance=0;strictly below epsilon=是
條件：無
用絕對距離與嚴格小於，等於epsilon也失敗；按完整有序索引對輸出，不只比相鄰項。
規則：finite_Cauchy_each_ordered_tail_pair · 依賴：G001, S010
操作前
recorded pair3,4 signed difference := sub(1/25, 1/26)
操作後
recorded pair3,4 signed difference = 1/650
條件：無
依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。
規則：exact_rational_arithmetic · 依賴：G001, S011
操作前
ordered recorded indices=[3, 4];signed difference=1/650;epsilon=1/3
操作後
absolute distance=1/650;strictly below epsilon=是
條件：無
用絕對距離與嚴格小於，等於epsilon也失敗；按完整有序索引對輸出，不只比相鄰項。
規則：finite_Cauchy_each_ordered_tail_pair · 依賴：G001, S012
操作前
recorded pair4,2 signed difference := sub(1/26, 1/24)
操作後
recorded pair4,2 signed difference = -1/312
條件：無
依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。
規則：exact_rational_arithmetic · 依賴：G001, S013
操作前
ordered recorded indices=[4, 2];signed difference=-1/312;epsilon=1/3
操作後
absolute distance=1/312;strictly below epsilon=是
條件：無
用絕對距離與嚴格小於，等於epsilon也失敗；按完整有序索引對輸出，不只比相鄰項。
規則：finite_Cauchy_each_ordered_tail_pair · 依賴：G001, S014
操作前
recorded pair4,3 signed difference := sub(1/26, 1/25)
操作後
recorded pair4,3 signed difference = -1/650
條件：無
依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。
規則：exact_rational_arithmetic · 依賴：G001, S015
操作前
ordered recorded indices=[4, 3];signed difference=-1/650;epsilon=1/3
操作後
absolute distance=1/650;strictly below epsilon=是
條件：無
用絕對距離與嚴格小於，等於epsilon也失敗；按完整有序索引對輸出，不只比相鄰項。
規則：finite_Cauchy_each_ordered_tail_pair · 依賴：G001, S016
操作前
recorded pair4,4 signed difference := sub(1/26, 1/26)
操作後
recorded pair4,4 signed difference = 0
條件：無
依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。
規則：exact_rational_arithmetic · 依賴：G001, S017
操作前
ordered recorded indices=[4, 4];signed difference=0;epsilon=1/3
操作後
absolute distance=0;strictly below epsilon=是
條件：無
用絕對距離與嚴格小於，等於epsilon也失敗；按完整有序索引對輸出，不只比相鄰項。
規則：finite_Cauchy_each_ordered_tail_pair · 依賴：G001, S018
操作前
all bad recorded pairs=[]
操作後
finite recorded tail passes=是;general Cauchy established=否
條件：無
空尾段可空泛通過，但有限紀錄沒有給∀epsilon存在N且所有無限尾對的義務。此No是未證，不是說每種無限接續都非Cauchy。
規則：finite_tail_not_quantified_infinite_Cauchy · 依賴：G001, S019
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 — 以精確有理數重算 recorded pair4,4 signed difference，不以小數近似替代。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 — 以精確有理數重算 recorded pair4,3 signed difference，不以小數近似替代。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 — 以精確有理數重算 recorded pair4,2 signed difference，不以小數近似替代。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 — 以精確有理數重算 recorded pair3,4 signed difference，不以小數近似替代。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 — 以精確有理數重算 recorded pair3,3 signed difference，不以小數近似替代。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 — 以精確有理數重算 recorded pair3,2 signed difference，不以小數近似替代。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 — 以精確有理數重算 recorded pair2,4 signed difference，不以小數近似替代。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 — 以精確有理數重算 recorded pair2,3 signed difference，不以小數近似替代。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 — 以精確有理數重算 recorded pair2,2 signed difference，不以小數近似替代。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。([1/22, 1/23, 1/24, 1/25, 1/26], 1/3, 2, [], 是, 否, 有限尾段不足以證一般Cauchy)
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。
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。
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 57f79c55e2156eb637b4cefcb4a8f884b74a6b16c1fd6441b290a55b35337f23candidate record SHA-256: 77f44de4d9b43607f6650a22969e3b86a942dbb9de8bbc8a57a2138aab343654semantic review receipt SHA-256: f783e81f44ff0faa13dbf948657e9187244a59a809ebdf96f22cca24966649ffquestion-first witness SHA-256: 49a50ed0550be55124d6562260c2ccf4fa6d3f316d7ad9d0d32174400e362e18Lean: NOT_RUN · human review: NOT_PERFORMED