R0048-24907 · q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0057
完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-fiber-coset-equivalence；資料={"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14]],"source_identity":0,"target_table":[[0,1],[1,0]],"target_identity":0,"mapping":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],"a":6,"b":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。
G001 — 完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-fiber-coset-equivalence；資料={"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14]],"source_identity":0,"target_table":[[0,1],[1,0]],"target_identity":0,"mapping":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],"a":6,"b":8}。請依 exact_ordered_tuple_declared_by_interface 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
source roles = [kernel-fiber-coset-equivalence]
操作後
derived = [[a, b, image_a, image_b, equal_images, left_division_a_inverse_b, difference_in_kernel, fiber, left_coset, fiber_equals_coset, equivalence_holds, certificate]]
條件：無
直接列來源完整欄序，不需JavaScript；原題、布林旗標、數學物件與非聲稱保持分層。
規則：source_ordered_fields_without_JS · 依賴：G001
操作前
source roles = [Kernel ORIGINAL SOURCE group, 64, [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]], 0]
操作後
derived = [16, [[0, 0, 0, 0], [1, 1, 1, 1], [2, 2, 2, 2], [3, 3, 3, 3], [4, 4, 4, 4], [5, 5, 5, 5], [6, 6, 6, 6], [7, 7, 7, 7], [8, 8, 8, 8], [9, 9, 9, 9], [10, 10, 10, 10], [11, 11, 11, 11], [12, 12, 12, 12], [13, 13, 13, 13], [14, 14, 14, 14], [15, 15, 15, 15]], 是]
條件：無
先驗完整n×n閉合表與實際identity標籤，非預設0。mod44上限32，mod45上限64；同一carrier不能混入另一表。
規則：Group_complete_Cayley_signature_actual_identity · 依賴：G001, S001
操作前
source roles = [Kernel ORIGINAL SOURCE group, 16, FOR EVERY x,y,z in0..n-1]
操作後
derived = [[[是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是]], 4096, 是]
條件：無
矩陣(x,y)逐全部z核(xy)z=x(yz)，故涵蓋全部n³三元組，不是抽樣。表已在前步完整列出，不重印數百萬同義查表項。
規則：Group_ALL_associativity_triples_compact_complete_pair_matrix · 依賴：G001, S002
操作前
source roles = [Kernel ORIGINAL SOURCE group, 0]
操作後
derived = [[0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0], [1, 15, 0, 0], [2, 14, 0, 0], [3, 13, 0, 0], [4, 12, 0, 0], [5, 11, 0, 0], [6, 10, 0, 0], [7, 9, 0, 0], [8, 8, 0, 0], [9, 7, 0, 0], [10, 6, 0, 0], [11, 5, 0, 0], [12, 4, 0, 0], [13, 3, 0, 0], [14, 2, 0, 0], [15, 1, 0, 0]], 是]
條件：無
每個元素的左、右逆都要回到同一實際identity；結合律使兩側逆唯一。這才完成群前提。
規則：Group_ALL_two_sided_inverses · 依賴：G001, S003
操作前
source roles = [Kernel ORIGINAL TARGET group, 64, [[0, 1], [1, 0]], 0]
操作後
derived = [2, [[0, 0, 0, 0], [1, 1, 1, 1]], 是]
條件：無
先驗完整n×n閉合表與實際identity標籤，非預設0。mod44上限32，mod45上限64；同一carrier不能混入另一表。
規則：Group_complete_Cayley_signature_actual_identity · 依賴：G001, S004
操作前
source roles = [Kernel ORIGINAL TARGET group, 2, FOR EVERY x,y,z in0..n-1]
操作後
derived = [[[是, 是], [是, 是]], 8, 是]
條件：無
矩陣(x,y)逐全部z核(xy)z=x(yz)，故涵蓋全部n³三元組，不是抽樣。表已在前步完整列出，不重印數百萬同義查表項。
規則：Group_ALL_associativity_triples_compact_complete_pair_matrix · 依賴：G001, S005
操作前
source roles = [Kernel ORIGINAL TARGET group, 0]
操作後
derived = [[0, 1], [[0, 0, 0, 0], [1, 1, 0, 0]], 是]
條件：無
每個元素的左、右逆都要回到同一實際identity；結合律使兩側逆唯一。這才完成群前提。
規則：Group_ALL_two_sided_inverses · 依賴：G001, S006
操作前
source roles = [[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]], [[0, 1], [1, 0]], [0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]]
操作後
derived = [[[0, 0, 0, 0, 是], [0, 1, 1, 1, 是], [0, 2, 0, 0, 是], [0, 3, 1, 1, 是], [0, 4, 0, 0, 是], [0, 5, 1, 1, 是], [0, 6, 0, 0, 是], [0, 7, 1, 1, 是], [0, 8, 0, 0, 是], [0, 9, 1, 1, 是], [0, 10, 0, 0, 是], [0, 11, 1, 1, 是], [0, 12, 0, 0, 是], [0, 13, 1, 1, 是], [0, 14, 0, 0, 是], [0, 15, 1, 1, 是], [1, 0, 1, 1, 是], [1, 1, 0, 0, 是], [1, 2, 1, 1, 是], [1, 3, 0, 0, 是], [1, 4, 1, 1, 是], [1, 5, 0, 0, 是], [1, 6, 1, 1, 是], [1, 7, 0, 0, 是], [1, 8, 1, 1, 是], [1, 9, 0, 0, 是], [1, 10, 1, 1, 是], [1, 11, 0, 0, 是], [1, 12, 1, 1, 是], [1, 13, 0, 0, 是], [1, 14, 1, 1, 是], [1, 15, 0, 0, 是], [2, 0, 0, 0, 是], [2, 1, 1, 1, 是], [2, 2, 0, 0, 是], [2, 3, 1, 1, 是], [2, 4, 0, 0, 是], [2, 5, 1, 1, 是], [2, 6, 0, 0, 是], [2, 7, 1, 1, 是], [2, 8, 0, 0, 是], [2, 9, 1, 1, 是], [2, 10, 0, 0, 是], [2, 11, 1, 1, 是], [2, 12, 0, 0, 是], [2, 13, 1, 1, 是], [2, 14, 0, 0, 是], [2, 15, 1, 1, 是], [3, 0, 1, 1, 是], [3, 1, 0, 0, 是], [3, 2, 1, 1, 是], [3, 3, 0, 0, 是], [3, 4, 1, 1, 是], [3, 5, 0, 0, 是], [3, 6, 1, 1, 是], [3, 7, 0, 0, 是], [3, 8, 1, 1, 是], [3, 9, 0, 0, 是], [3, 10, 1, 1, 是], [3, 11, 0, 0, 是], [3, 12, 1, 1, 是], [3, 13, 0, 0, 是], [3, 14, 1, 1, 是], [3, 15, 0, 0, 是], [4, 0, 0, 0, 是], [4, 1, 1, 1, 是], [4, 2, 0, 0, 是], [4, 3, 1, 1, 是], [4, 4, 0, 0, 是], [4, 5, 1, 1, 是], [4, 6, 0, 0, 是], [4, 7, 1, 1, 是], [4, 8, 0, 0, 是], [4, 9, 1, 1, 是], [4, 10, 0, 0, 是], [4, 11, 1, 1, 是], [4, 12, 0, 0, 是], [4, 13, 1, 1, 是], [4, 14, 0, 0, 是], [4, 15, 1, 1, 是], [5, 0, 1, 1, 是], [5, 1, 0, 0, 是], [5, 2, 1, 1, 是], [5, 3, 0, 0, 是], [5, 4, 1, 1, 是], [5, 5, 0, 0, 是], [5, 6, 1, 1, 是], [5, 7, 0, 0, 是], [5, 8, 1, 1, 是], [5, 9, 0, 0, 是], [5, 10, 1, 1, 是], [5, 11, 0, 0, 是], [5, 12, 1, 1, 是], [5, 13, 0, 0, 是], [5, 14, 1, 1, 是], [5, 15, 0, 0, 是], [6, 0, 0, 0, 是], [6, 1, 1, 1, 是], [6, 2, 0, 0, 是], [6, 3, 1, 1, 是], [6, 4, 0, 0, 是], [6, 5, 1, 1, 是], [6, 6, 0, 0, 是], [6, 7, 1, 1, 是], [6, 8, 0, 0, 是], [6, 9, 1, 1, 是], [6, 10, 0, 0, 是], [6, 11, 1, 1, 是], [6, 12, 0, 0, 是], [6, 13, 1, 1, 是], [6, 14, 0, 0, 是], [6, 15, 1, 1, 是], [7, 0, 1, 1, 是], [7, 1, 0, 0, 是], [7, 2, 1, 1, 是], [7, 3, 0, 0, 是], [7, 4, 1, 1, 是], [7, 5, 0, 0, 是], [7, 6, 1, 1, 是], [7, 7, 0, 0, 是], [7, 8, 1, 1, 是], [7, 9, 0, 0, 是], [7, 10, 1, 1, 是], [7, 11, 0, 0, 是], [7, 12, 1, 1, 是], [7, 13, 0, 0, 是], [7, 14, 1, 1, 是], [7, 15, 0, 0, 是], [8, 0, 0, 0, 是], [8, 1, 1, 1, 是], [8, 2, 0, 0, 是], [8, 3, 1, 1, 是], [8, 4, 0, 0, 是], [8, 5, 1, 1, 是], [8, 6, 0, 0, 是], [8, 7, 1, 1, 是], [8, 8, 0, 0, 是], [8, 9, 1, 1, 是], [8, 10, 0, 0, 是], [8, 11, 1, 1, 是], [8, 12, 0, 0, 是], [8, 13, 1, 1, 是], [8, 14, 0, 0, 是], [8, 15, 1, 1, 是], [9, 0, 1, 1, 是], [9, 1, 0, 0, 是], [9, 2, 1, 1, 是], [9, 3, 0, 0, 是], [9, 4, 1, 1, 是], [9, 5, 0, 0, 是], [9, 6, 1, 1, 是], [9, 7, 0, 0, 是], [9, 8, 1, 1, 是], [9, 9, 0, 0, 是], [9, 10, 1, 1, 是], [9, 11, 0, 0, 是], [9, 12, 1, 1, 是], [9, 13, 0, 0, 是], [9, 14, 1, 1, 是], [9, 15, 0, 0, 是], [10, 0, 0, 0, 是], [10, 1, 1, 1, 是], [10, 2, 0, 0, 是], [10, 3, 1, 1, 是], [10, 4, 0, 0, 是], [10, 5, 1, 1, 是], [10, 6, 0, 0, 是], [10, 7, 1, 1, 是], [10, 8, 0, 0, 是], [10, 9, 1, 1, 是], [10, 10, 0, 0, 是], [10, 11, 1, 1, 是], [10, 12, 0, 0, 是], [10, 13, 1, 1, 是], [10, 14, 0, 0, 是], [10, 15, 1, 1, 是], [11, 0, 1, 1, 是], [11, 1, 0, 0, 是], [11, 2, 1, 1, 是], [11, 3, 0, 0, 是], [11, 4, 1, 1, 是], [11, 5, 0, 0, 是], [11, 6, 1, 1, 是], [11, 7, 0, 0, 是], [11, 8, 1, 1, 是], [11, 9, 0, 0, 是], [11, 10, 1, 1, 是], [11, 11, 0, 0, 是], [11, 12, 1, 1, 是], [11, 13, 0, 0, 是], [11, 14, 1, 1, 是], [11, 15, 0, 0, 是], [12, 0, 0, 0, 是], [12, 1, 1, 1, 是], [12, 2, 0, 0, 是], [12, 3, 1, 1, 是], [12, 4, 0, 0, 是], [12, 5, 1, 1, 是], [12, 6, 0, 0, 是], [12, 7, 1, 1, 是], [12, 8, 0, 0, 是], [12, 9, 1, 1, 是], [12, 10, 0, 0, 是], [12, 11, 1, 1, 是], [12, 12, 0, 0, 是], [12, 13, 1, 1, 是], [12, 14, 0, 0, 是], [12, 15, 1, 1, 是], [13, 0, 1, 1, 是], [13, 1, 0, 0, 是], [13, 2, 1, 1, 是], [13, 3, 0, 0, 是], [13, 4, 1, 1, 是], [13, 5, 0, 0, 是], [13, 6, 1, 1, 是], [13, 7, 0, 0, 是], [13, 8, 1, 1, 是], [13, 9, 0, 0, 是], [13, 10, 1, 1, 是], [13, 11, 0, 0, 是], [13, 12, 1, 1, 是], [13, 13, 0, 0, 是], [13, 14, 1, 1, 是], [13, 15, 0, 0, 是], [14, 0, 0, 0, 是], [14, 1, 1, 1, 是], [14, 2, 0, 0, 是], [14, 3, 1, 1, 是], [14, 4, 0, 0, 是], [14, 5, 1, 1, 是], [14, 6, 0, 0, 是], [14, 7, 1, 1, 是], [14, 8, 0, 0, 是], [14, 9, 1, 1, 是], [14, 10, 0, 0, 是], [14, 11, 1, 1, 是], [14, 12, 0, 0, 是], [14, 13, 1, 1, 是], [14, 14, 0, 0, 是], [14, 15, 1, 1, 是], [15, 0, 1, 1, 是], [15, 1, 0, 0, 是], [15, 2, 1, 1, 是], [15, 3, 0, 0, 是], [15, 4, 1, 1, 是], [15, 5, 0, 0, 是], [15, 6, 1, 1, 是], [15, 7, 0, 0, 是], [15, 8, 1, 1, 是], [15, 9, 0, 0, 是], [15, 10, 1, 1, 是], [15, 11, 0, 0, 是], [15, 12, 1, 1, 是], [15, 13, 0, 0, 是], [15, 14, 1, 1, 是], [15, 15, 0, 0, 是]], 是, null]
條件：無
先對同一source/target每對(x,y)驗f(xy)=f(x)f(y)，保首failed row/null。total map不是homomorphism；失敗即擋所有kernel定理，不把內部fallback當真正identity preimage。
規則：Kernel_complete_TOTAL_map_ALL_product_pairs_before_theorems · 依賴：G001, S007
操作前
source roles = [[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], 0, 0]
操作後
derived = [[0, 2, 4, 6, 8, 10, 12, 14], 0, 0]
條件：無
kernel逐全部source元素取目標實際identity的preimage。f(e)=f(e)^2在target群消去得f(e)=e_target，故真kernel非空；不能假設target identity=0。
規則：Kernel_actual_TARGET_identity_complete_preimage · 依賴：G001, S008
操作前
source roles = [Kernel ACTUAL identity preimage, [0, 2, 4, 6, 8, 10, 12, 14], 0]
操作後
derived = [是, [[0, 0, 0, 是], [0, 2, 2, 是], [0, 4, 4, 是], [0, 6, 6, 是], [0, 8, 8, 是], [0, 10, 10, 是], [0, 12, 12, 是], [0, 14, 14, 是], [2, 0, 2, 是], [2, 2, 4, 是], [2, 4, 6, 是], [2, 6, 8, 是], [2, 8, 10, 是], [2, 10, 12, 是], [2, 12, 14, 是], [2, 14, 0, 是], [4, 0, 4, 是], [4, 2, 6, 是], [4, 4, 8, 是], [4, 6, 10, 是], [4, 8, 12, 是], [4, 10, 14, 是], [4, 12, 0, 是], [4, 14, 2, 是], [6, 0, 6, 是], [6, 2, 8, 是], [6, 4, 10, 是], [6, 6, 12, 是], [6, 8, 14, 是], [6, 10, 0, 是], [6, 12, 2, 是], [6, 14, 4, 是], [8, 0, 8, 是], [8, 2, 10, 是], [8, 4, 12, 是], [8, 6, 14, 是], [8, 8, 0, 是], [8, 10, 2, 是], [8, 12, 4, 是], [8, 14, 6, 是], [10, 0, 10, 是], [10, 2, 12, 是], [10, 4, 14, 是], [10, 6, 0, 是], [10, 8, 2, 是], [10, 10, 4, 是], [10, 12, 6, 是], [10, 14, 8, 是], [12, 0, 12, 是], [12, 2, 14, 是], [12, 4, 0, 是], [12, 6, 2, 是], [12, 8, 4, 是], [12, 10, 6, 是], [12, 12, 8, 是], [12, 14, 10, 是], [14, 0, 14, 是], [14, 2, 0, 是], [14, 4, 2, 是], [14, 6, 4, 是], [14, 8, 6, 是], [14, 10, 8, 是], [14, 12, 10, 是], [14, 14, 12, 是]], 是, [[0, 0, 是], [2, 14, 是], [4, 12, 是], [6, 10, 是], [8, 8, 是], [10, 6, 是], [12, 4, 是], [14, 2, 是]], 是, 是]
條件：無
H非空且無重複；含identity、任兩元素乘積、每個逆元分別核全域，三者合取才可稱subgroup。order divisibility或有限性不足。
規則：Subgroup_identity_ALL_products_ALL_inverses_separate · 依賴：G001, S009
操作前
source roles = [Kernel ACTUAL identity preimage, 0, [0, 2, 4, 6, 8, 10, 12, 14], 0]
操作後
derived = [[[0, 0, 0, 0], [2, 2, 0, 2], [4, 4, 0, 4], [6, 6, 0, 6], [8, 8, 0, 8], [10, 10, 0, 10], [12, 12, 0, 12], [14, 14, 0, 14]], [0, 2, 4, 6, 8, 10, 12, 14], 是]
條件：無
按同一表先乘g*h，再乘g的右逆，遍歷全部h；再對全部g核集合等式，不由取樣或生成元猜normal。
規則：Subgroup_EVERY_g_entire_conjugate_set · 依賴：G001, S010
操作前
source roles = [Kernel ACTUAL identity preimage, 1, [0, 2, 4, 6, 8, 10, 12, 14], 15]
操作後
derived = [[[0, 1, 15, 0], [2, 3, 15, 2], [4, 5, 15, 4], [6, 7, 15, 6], [8, 9, 15, 8], [10, 11, 15, 10], [12, 13, 15, 12], [14, 15, 15, 14]], [0, 2, 4, 6, 8, 10, 12, 14], 是]
條件：無
按同一表先乘g*h，再乘g的右逆，遍歷全部h；再對全部g核集合等式，不由取樣或生成元猜normal。
規則：Subgroup_EVERY_g_entire_conjugate_set · 依賴：G001, S011
操作前
source roles = [Kernel ACTUAL identity preimage, 2, [0, 2, 4, 6, 8, 10, 12, 14], 14]
操作後
derived = [[[0, 2, 14, 0], [2, 4, 14, 2], [4, 6, 14, 4], [6, 8, 14, 6], [8, 10, 14, 8], [10, 12, 14, 10], [12, 14, 14, 12], [14, 0, 14, 14]], [0, 2, 4, 6, 8, 10, 12, 14], 是]
條件：無
按同一表先乘g*h，再乘g的右逆，遍歷全部h；再對全部g核集合等式，不由取樣或生成元猜normal。
規則：Subgroup_EVERY_g_entire_conjugate_set · 依賴：G001, S012
操作前
source roles = [Kernel ACTUAL identity preimage, 3, [0, 2, 4, 6, 8, 10, 12, 14], 13]
操作後
derived = [[[0, 3, 13, 0], [2, 5, 13, 2], [4, 7, 13, 4], [6, 9, 13, 6], [8, 11, 13, 8], [10, 13, 13, 10], [12, 15, 13, 12], [14, 1, 13, 14]], [0, 2, 4, 6, 8, 10, 12, 14], 是]
條件：無
按同一表先乘g*h，再乘g的右逆，遍歷全部h；再對全部g核集合等式，不由取樣或生成元猜normal。
規則：Subgroup_EVERY_g_entire_conjugate_set · 依賴：G001, S013
操作前
source roles = [Kernel ACTUAL identity preimage, 4, [0, 2, 4, 6, 8, 10, 12, 14], 12]
操作後
derived = [[[0, 4, 12, 0], [2, 6, 12, 2], [4, 8, 12, 4], [6, 10, 12, 6], [8, 12, 12, 8], [10, 14, 12, 10], [12, 0, 12, 12], [14, 2, 12, 14]], [0, 2, 4, 6, 8, 10, 12, 14], 是]
條件：無
按同一表先乘g*h，再乘g的右逆，遍歷全部h；再對全部g核集合等式，不由取樣或生成元猜normal。
規則：Subgroup_EVERY_g_entire_conjugate_set · 依賴：G001, S014
操作前
source roles = [Kernel ACTUAL identity preimage, 5, [0, 2, 4, 6, 8, 10, 12, 14], 11]
操作後
derived = [[[0, 5, 11, 0], [2, 7, 11, 2], [4, 9, 11, 4], [6, 11, 11, 6], [8, 13, 11, 8], [10, 15, 11, 10], [12, 1, 11, 12], [14, 3, 11, 14]], [0, 2, 4, 6, 8, 10, 12, 14], 是]
條件：無
按同一表先乘g*h，再乘g的右逆，遍歷全部h；再對全部g核集合等式，不由取樣或生成元猜normal。
規則：Subgroup_EVERY_g_entire_conjugate_set · 依賴：G001, S015
操作前
source roles = [Kernel ACTUAL identity preimage, 6, [0, 2, 4, 6, 8, 10, 12, 14], 10]
操作後
derived = [[[0, 6, 10, 0], [2, 8, 10, 2], [4, 10, 10, 4], [6, 12, 10, 6], [8, 14, 10, 8], [10, 0, 10, 10], [12, 2, 10, 12], [14, 4, 10, 14]], [0, 2, 4, 6, 8, 10, 12, 14], 是]
條件：無
按同一表先乘g*h，再乘g的右逆，遍歷全部h；再對全部g核集合等式，不由取樣或生成元猜normal。
規則：Subgroup_EVERY_g_entire_conjugate_set · 依賴：G001, S016
操作前
source roles = [Kernel ACTUAL identity preimage, 7, [0, 2, 4, 6, 8, 10, 12, 14], 9]
操作後
derived = [[[0, 7, 9, 0], [2, 9, 9, 2], [4, 11, 9, 4], [6, 13, 9, 6], [8, 15, 9, 8], [10, 1, 9, 10], [12, 3, 9, 12], [14, 5, 9, 14]], [0, 2, 4, 6, 8, 10, 12, 14], 是]
條件：無
按同一表先乘g*h，再乘g的右逆，遍歷全部h；再對全部g核集合等式，不由取樣或生成元猜normal。
規則：Subgroup_EVERY_g_entire_conjugate_set · 依賴：G001, S017
操作前
source roles = [Kernel ACTUAL identity preimage, 8, [0, 2, 4, 6, 8, 10, 12, 14], 8]
操作後
derived = [[[0, 8, 8, 0], [2, 10, 8, 2], [4, 12, 8, 4], [6, 14, 8, 6], [8, 0, 8, 8], [10, 2, 8, 10], [12, 4, 8, 12], [14, 6, 8, 14]], [0, 2, 4, 6, 8, 10, 12, 14], 是]
條件：無
按同一表先乘g*h，再乘g的右逆，遍歷全部h；再對全部g核集合等式，不由取樣或生成元猜normal。
規則：Subgroup_EVERY_g_entire_conjugate_set · 依賴：G001, S018
操作前
source roles = [Kernel ACTUAL identity preimage, 9, [0, 2, 4, 6, 8, 10, 12, 14], 7]
操作後
derived = [[[0, 9, 7, 0], [2, 11, 7, 2], [4, 13, 7, 4], [6, 15, 7, 6], [8, 1, 7, 8], [10, 3, 7, 10], [12, 5, 7, 12], [14, 7, 7, 14]], [0, 2, 4, 6, 8, 10, 12, 14], 是]
條件：無
按同一表先乘g*h，再乘g的右逆，遍歷全部h；再對全部g核集合等式，不由取樣或生成元猜normal。
規則：Subgroup_EVERY_g_entire_conjugate_set · 依賴：G001, S019
操作前
source roles = [Kernel ACTUAL identity preimage, 10, [0, 2, 4, 6, 8, 10, 12, 14], 6]
操作後
derived = [[[0, 10, 6, 0], [2, 12, 6, 2], [4, 14, 6, 4], [6, 0, 6, 6], [8, 2, 6, 8], [10, 4, 6, 10], [12, 6, 6, 12], [14, 8, 6, 14]], [0, 2, 4, 6, 8, 10, 12, 14], 是]
條件：無
按同一表先乘g*h，再乘g的右逆，遍歷全部h；再對全部g核集合等式，不由取樣或生成元猜normal。
規則：Subgroup_EVERY_g_entire_conjugate_set · 依賴：G001, S020
操作前
source roles = [Kernel ACTUAL identity preimage, 11, [0, 2, 4, 6, 8, 10, 12, 14], 5]
操作後
derived = [[[0, 11, 5, 0], [2, 13, 5, 2], [4, 15, 5, 4], [6, 1, 5, 6], [8, 3, 5, 8], [10, 5, 5, 10], [12, 7, 5, 12], [14, 9, 5, 14]], [0, 2, 4, 6, 8, 10, 12, 14], 是]
條件：無
按同一表先乘g*h，再乘g的右逆，遍歷全部h；再對全部g核集合等式，不由取樣或生成元猜normal。
規則：Subgroup_EVERY_g_entire_conjugate_set · 依賴：G001, S021
操作前
source roles = [Kernel ACTUAL identity preimage, 12, [0, 2, 4, 6, 8, 10, 12, 14], 4]
操作後
derived = [[[0, 12, 4, 0], [2, 14, 4, 2], [4, 0, 4, 4], [6, 2, 4, 6], [8, 4, 4, 8], [10, 6, 4, 10], [12, 8, 4, 12], [14, 10, 4, 14]], [0, 2, 4, 6, 8, 10, 12, 14], 是]
條件：無
按同一表先乘g*h，再乘g的右逆，遍歷全部h；再對全部g核集合等式，不由取樣或生成元猜normal。
規則：Subgroup_EVERY_g_entire_conjugate_set · 依賴：G001, S022
操作前
source roles = [Kernel ACTUAL identity preimage, 13, [0, 2, 4, 6, 8, 10, 12, 14], 3]
操作後
derived = [[[0, 13, 3, 0], [2, 15, 3, 2], [4, 1, 3, 4], [6, 3, 3, 6], [8, 5, 3, 8], [10, 7, 3, 10], [12, 9, 3, 12], [14, 11, 3, 14]], [0, 2, 4, 6, 8, 10, 12, 14], 是]
條件：無
按同一表先乘g*h，再乘g的右逆，遍歷全部h；再對全部g核集合等式，不由取樣或生成元猜normal。
規則：Subgroup_EVERY_g_entire_conjugate_set · 依賴：G001, S023
操作前
source roles = [Kernel ACTUAL identity preimage, 14, [0, 2, 4, 6, 8, 10, 12, 14], 2]
操作後
derived = [[[0, 14, 2, 0], [2, 0, 2, 2], [4, 2, 2, 4], [6, 4, 2, 6], [8, 6, 2, 8], [10, 8, 2, 10], [12, 10, 2, 12], [14, 12, 2, 14]], [0, 2, 4, 6, 8, 10, 12, 14], 是]
條件：無
按同一表先乘g*h，再乘g的右逆，遍歷全部h；再對全部g核集合等式，不由取樣或生成元猜normal。
規則：Subgroup_EVERY_g_entire_conjugate_set · 依賴：G001, S024
操作前
source roles = [Kernel ACTUAL identity preimage, 15, [0, 2, 4, 6, 8, 10, 12, 14], 1]
操作後
derived = [[[0, 15, 1, 0], [2, 1, 1, 2], [4, 3, 1, 4], [6, 5, 1, 6], [8, 7, 1, 8], [10, 9, 1, 10], [12, 11, 1, 12], [14, 13, 1, 14]], [0, 2, 4, 6, 8, 10, 12, 14], 是]
條件：無
按同一表先乘g*h，再乘g的右逆，遍歷全部h；再對全部g核集合等式，不由取樣或生成元猜normal。
規則：Subgroup_EVERY_g_entire_conjugate_set · 依賴：G001, S025
操作前
source roles = [[0, 2, 4, 6, 8, 10, 12, 14], 是, 0]
操作後
derived = [f(kl)=f(k)f(l)=e, f(k^-1)=f(k)^-1=e, f(gkg^-1)=f(g)e*f(g)^-1=e]
條件：無
乘積、逆元與全g共軛的像都回到target identity，這給一般closure/normality理由，和本題完整枚舉互相核對；不宣稱quotient/isomorphism。
規則：Kernel_GENERAL_closure_normality_from_homomorphism · 依賴：G001, S026
操作前
source roles = [6, 8, 10, [0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], [0, 2, 4, 6, 8, 10, 12, 14]]
操作後
derived = [2, 是, 是, [0, 2, 4, 6, 8, 10, 12, 14], [0, 2, 4, 6, 8, 10, 12, 14], 是, 是]
條件：無
f(a)=f(b)等價f(a)^-1f(b)=e，故用a^-1*b而非b*a^-1。任意y在fiber iff a^-1y∈K；y↦a^-1y與k↦ak互逆，給完整fiber=aK。fiber含a，不假設target滿射。False/False的iff仍True。
規則：Kernel_actual_LEFT_division_fiber_LEFT_coset_two_directions · 依賴：G001, S027
S028 · 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。S027 · 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。S026 · 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。S025 · 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。S024 · 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。S023 · 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。S022 · 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。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。(6, 8, 0, 0, 是, 2, 是, [0, 2, 4, 6, 8, 10, 12, 14], [0, 2, 4, 6, 8, 10, 12, 14], 是, 是, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)
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: b2e012fb581538dce3e296f278237c14d2aa4d4775e53df840ae8d9fcba182a6candidate record SHA-256: 7bb92d753c601cca8624b9892db77b39092a5f8a9789ddc59acd8132e149d423semantic review receipt SHA-256: 049a9e07e69fead88d93b34f0937b9470a06d71c1a8d097058fc83554529bdd0question-first witness SHA-256: 0c71ebb418f3de347d232d159fc25e93036f4ab9166bcfb5ee5b4be86d5c55b9Lean: NOT_RUN · human review: NOT_PERFORMED