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第 24859 題：有限群共軛類

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  1. 原題與已知條件
  2. 定義與適用範圍
  3. 逐步推導
  4. 逐步反向核對
  5. 結論
  6. 本題覆核與來源綁定

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← 逐題詳解目錄 · 原題批次

R0048-24859 · q-20261015-advance-20261114-r03-finite-group-conjugacy-class-0059

原題與已知條件

由完整Cayley table導出全部conjugation action與orbits。操作=conjugacy-partition-audit；資料={"group_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[1,2,3,4,5,6,7,8,9,10,0,12,13,14,15,16,17,18,19,20,21,11],[2,3,4,5,6,7,8,9,10,0,1,13,14,15,16,17,18,19,20,21,11,12],[3,4,5,6,7,8,9,10,0,1,2,14,15,16,17,18,19,20,21,11,12,13],[4,5,6,7,8,9,10,0,1,2,3,15,16,17,18,19,20,21,11,12,13,14],[5,6,7,8,9,10,0,1,2,3,4,16,17,18,19,20,21,11,12,13,14,15],[6,7,8,9,10,0,1,2,3,4,5,17,18,19,20,21,11,12,13,14,15,16],[7,8,9,10,0,1,2,3,4,5,6,18,19,20,21,11,12,13,14,15,16,17],[8,9,10,0,1,2,3,4,5,6,7,19,20,21,11,12,13,14,15,16,17,18],[9,10,0,1,2,3,4,5,6,7,8,20,21,11,12,13,14,15,16,17,18,19],[10,0,1,2,3,4,5,6,7,8,9,21,11,12,13,14,15,16,17,18,19,20],[11,21,20,19,18,17,16,15,14,13,12,0,10,9,8,7,6,5,4,3,2,1],[12,11,21,20,19,18,17,16,15,14,13,1,0,10,9,8,7,6,5,4,3,2],[13,12,11,21,20,19,18,17,16,15,14,2,1,0,10,9,8,7,6,5,4,3],[14,13,12,11,21,20,19,18,17,16,15,3,2,1,0,10,9,8,7,6,5,4],[15,14,13,12,11,21,20,19,18,17,16,4,3,2,1,0,10,9,8,7,6,5],[16,15,14,13,12,11,21,20,19,18,17,5,4,3,2,1,0,10,9,8,7,6],[17,16,15,14,13,12,11,21,20,19,18,6,5,4,3,2,1,0,10,9,8,7],[18,17,16,15,14,13,12,11,21,20,19,7,6,5,4,3,2,1,0,10,9,8],[19,18,17,16,15,14,13,12,11,21,20,8,7,6,5,4,3,2,1,0,10,9],[20,19,18,17,16,15,14,13,12,11,21,9,8,7,6,5,4,3,2,1,0,10],[21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

  • G001 — 由完整Cayley table導出全部conjugation action與orbits。操作=conjugacy-partition-audit；資料={"group_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[1,2,3,4,5,6,7,8,9,10,0,12,13,14,15,16,17,18,19,20,21,11],[2,3,4,5,6,7,8,9,10,0,1,13,14,15,16,17,18,19,20,21,11,12],[3,4,5,6,7,8,9,10,0,1,2,14,15,16,17,18,19,20,21,11,12,13],[4,5,6,7,8,9,10,0,1,2,3,15,16,17,18,19,20,21,11,12,13,14],[5,6,7,8,9,10,0,1,2,3,4,16,17,18,19,20,21,11,12,13,14,15],[6,7,8,9,10,0,1,2,3,4,5,17,18,19,20,21,11,12,13,14,15,16],[7,8,9,10,0,1,2,3,4,5,6,18,19,20,21,11,12,13,14,15,16,17],[8,9,10,0,1,2,3,4,5,6,7,19,20,21,11,12,13,14,15,16,17,18],[9,10,0,1,2,3,4,5,6,7,8,20,21,11,12,13,14,15,16,17,18,19],[10,0,1,2,3,4,5,6,7,8,9,21,11,12,13,14,15,16,17,18,19,20],[11,21,20,19,18,17,16,15,14,13,12,0,10,9,8,7,6,5,4,3,2,1],[12,11,21,20,19,18,17,16,15,14,13,1,0,10,9,8,7,6,5,4,3,2],[13,12,11,21,20,19,18,17,16,15,14,2,1,0,10,9,8,7,6,5,4,3],[14,13,12,11,21,20,19,18,17,16,15,3,2,1,0,10,9,8,7,6,5,4],[15,14,13,12,11,21,20,19,18,17,16,4,3,2,1,0,10,9,8,7,6,5],[16,15,14,13,12,11,21,20,19,18,17,5,4,3,2,1,0,10,9,8,7,6],[17,16,15,14,13,12,11,21,20,19,18,6,5,4,3,2,1,0,10,9,8,7],[18,17,16,15,14,13,12,11,21,20,19,7,6,5,4,3,2,1,0,10,9,8],[19,18,17,16,15,14,13,12,11,21,20,8,7,6,5,4,3,2,1,0,10,9],[20,19,18,17,16,15,14,13,12,11,21,9,8,7,6,5,4,3,2,1,0,10],[21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0}。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。

逐步推導

S001 · implies

操作前

source roles = [conjugacy-partition-audit]

操作後

derived = [[classes, partition_certificate, covered_element_count, group_order, 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。

步驟識別與依賴

規則：source_ordered_fields_without_JS · 依賴：G001

S002 · implies

操作前

source roles = [Conjugacy ORIGINAL group, 64, [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 11], [2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 1, 13, 14, 15, 16, 17, 18, 19, 20, 21, 11, 12], [3, 4, 5, 6, 7, 8, 9, 10, 0, 1, 2, 14, 15, 16, 17, 18, 19, 20, 21, 11, 12, 13], [4, 5, 6, 7, 8, 9, 10, 0, 1, 2, 3, 15, 16, 17, 18, 19, 20, 21, 11, 12, 13, 14], [5, 6, 7, 8, 9, 10, 0, 1, 2, 3, 4, 16, 17, 18, 19, 20, 21, 11, 12, 13, 14, 15], [6, 7, 8, 9, 10, 0, 1, 2, 3, 4, 5, 17, 18, 19, 20, 21, 11, 12, 13, 14, 15, 16], [7, 8, 9, 10, 0, 1, 2, 3, 4, 5, 6, 18, 19, 20, 21, 11, 12, 13, 14, 15, 16, 17], [8, 9, 10, 0, 1, 2, 3, 4, 5, 6, 7, 19, 20, 21, 11, 12, 13, 14, 15, 16, 17, 18], [9, 10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 20, 21, 11, 12, 13, 14, 15, 16, 17, 18, 19], [10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 21, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [11, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [12, 11, 21, 20, 19, 18, 17, 16, 15, 14, 13, 1, 0, 10, 9, 8, 7, 6, 5, 4, 3, 2], [13, 12, 11, 21, 20, 19, 18, 17, 16, 15, 14, 2, 1, 0, 10, 9, 8, 7, 6, 5, 4, 3], [14, 13, 12, 11, 21, 20, 19, 18, 17, 16, 15, 3, 2, 1, 0, 10, 9, 8, 7, 6, 5, 4], [15, 14, 13, 12, 11, 21, 20, 19, 18, 17, 16, 4, 3, 2, 1, 0, 10, 9, 8, 7, 6, 5], [16, 15, 14, 13, 12, 11, 21, 20, 19, 18, 17, 5, 4, 3, 2, 1, 0, 10, 9, 8, 7, 6], [17, 16, 15, 14, 13, 12, 11, 21, 20, 19, 18, 6, 5, 4, 3, 2, 1, 0, 10, 9, 8, 7], [18, 17, 16, 15, 14, 13, 12, 11, 21, 20, 19, 7, 6, 5, 4, 3, 2, 1, 0, 10, 9, 8], [19, 18, 17, 16, 15, 14, 13, 12, 11, 21, 20, 8, 7, 6, 5, 4, 3, 2, 1, 0, 10, 9], [20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 21, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 10], [21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], 0]

操作後

derived = [22, [[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], [16, 16, 16, 16], [17, 17, 17, 17], [18, 18, 18, 18], [19, 19, 19, 19], [20, 20, 20, 20], [21, 21, 21, 21]], 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。

步驟識別與依賴

規則：Group_complete_Cayley_signature_actual_identity · 依賴：G001, S001

S003 · implies

操作前

source roles = [Conjugacy ORIGINAL group, 22, 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, 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]], 10648, 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。

步驟識別與依賴

規則：Group_ALL_associativity_triples_compact_complete_pair_matrix · 依賴：G001, S002

S004 · implies

操作前

source roles = [Conjugacy ORIGINAL group, 0]

操作後

derived = [[0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [[0, 0, 0, 0], [1, 10, 0, 0], [2, 9, 0, 0], [3, 8, 0, 0], [4, 7, 0, 0], [5, 6, 0, 0], [6, 5, 0, 0], [7, 4, 0, 0], [8, 3, 0, 0], [9, 2, 0, 0], [10, 1, 0, 0], [11, 11, 0, 0], [12, 12, 0, 0], [13, 13, 0, 0], [14, 14, 0, 0], [15, 15, 0, 0], [16, 16, 0, 0], [17, 17, 0, 0], [18, 18, 0, 0], [19, 19, 0, 0], [20, 20, 0, 0], [21, 21, 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。

步驟識別與依賴

規則：Group_ALL_two_sided_inverses · 依賴：G001, S003

S005 · implies

操作前

source roles = [0, 0, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]]

操作後

derived = [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]]

條件：無

對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。

步驟識別與依賴

規則：Conjugacy_EVERY_g_EVERY_element_exact_action_row · 依賴：G001, S004

S006 · implies

操作前

source roles = [1, 10, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]]

操作後

derived = [[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13, 14, 15, 16, 17, 18, 19, 20, 21, 11, 12]]

條件：無

對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。

步驟識別與依賴

規則：Conjugacy_EVERY_g_EVERY_element_exact_action_row · 依賴：G001, S005

S007 · implies

操作前

source roles = [2, 9, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]]

操作後

derived = [[2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 1, 13, 14, 15, 16, 17, 18, 19, 20, 21, 11, 12], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 15, 16, 17, 18, 19, 20, 21, 11, 12, 13, 14]]

條件：無

對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。

步驟識別與依賴

規則：Conjugacy_EVERY_g_EVERY_element_exact_action_row · 依賴：G001, S006

S008 · implies

操作前

source roles = [3, 8, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]]

操作後

derived = [[3, 4, 5, 6, 7, 8, 9, 10, 0, 1, 2, 14, 15, 16, 17, 18, 19, 20, 21, 11, 12, 13], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 17, 18, 19, 20, 21, 11, 12, 13, 14, 15, 16]]

條件：無

對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。

步驟識別與依賴

規則：Conjugacy_EVERY_g_EVERY_element_exact_action_row · 依賴：G001, S007

S009 · implies

操作前

source roles = [4, 7, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]]

操作後

derived = [[4, 5, 6, 7, 8, 9, 10, 0, 1, 2, 3, 15, 16, 17, 18, 19, 20, 21, 11, 12, 13, 14], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 19, 20, 21, 11, 12, 13, 14, 15, 16, 17, 18]]

條件：無

對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。

步驟識別與依賴

規則：Conjugacy_EVERY_g_EVERY_element_exact_action_row · 依賴：G001, S008

S010 · implies

操作前

source roles = [5, 6, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]]

操作後

derived = [[5, 6, 7, 8, 9, 10, 0, 1, 2, 3, 4, 16, 17, 18, 19, 20, 21, 11, 12, 13, 14, 15], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 21, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]]

條件：無

對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。

步驟識別與依賴

規則：Conjugacy_EVERY_g_EVERY_element_exact_action_row · 依賴：G001, S009

S011 · implies

操作前

source roles = [6, 5, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]]

操作後

derived = [[6, 7, 8, 9, 10, 0, 1, 2, 3, 4, 5, 17, 18, 19, 20, 21, 11, 12, 13, 14, 15, 16], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 11]]

條件：無

對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。

步驟識別與依賴

規則：Conjugacy_EVERY_g_EVERY_element_exact_action_row · 依賴：G001, S010

S012 · implies

操作前

source roles = [7, 4, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]]

操作後

derived = [[7, 8, 9, 10, 0, 1, 2, 3, 4, 5, 6, 18, 19, 20, 21, 11, 12, 13, 14, 15, 16, 17], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 15, 16, 17, 18, 19, 20, 21, 11, 12, 13]]

條件：無

對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。

步驟識別與依賴

規則：Conjugacy_EVERY_g_EVERY_element_exact_action_row · 依賴：G001, S011

S013 · implies

操作前

source roles = [8, 3, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]]

操作後

derived = [[8, 9, 10, 0, 1, 2, 3, 4, 5, 6, 7, 19, 20, 21, 11, 12, 13, 14, 15, 16, 17, 18], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 16, 17, 18, 19, 20, 21, 11, 12, 13, 14, 15]]

條件：無

對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。

步驟識別與依賴

規則：Conjugacy_EVERY_g_EVERY_element_exact_action_row · 依賴：G001, S012

S014 · implies

操作前

source roles = [9, 2, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]]

操作後

derived = [[9, 10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 20, 21, 11, 12, 13, 14, 15, 16, 17, 18, 19], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 18, 19, 20, 21, 11, 12, 13, 14, 15, 16, 17]]

條件：無

對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。

步驟識別與依賴

規則：Conjugacy_EVERY_g_EVERY_element_exact_action_row · 依賴：G001, S013

S015 · implies

操作前

source roles = [10, 1, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]]

操作後

derived = [[10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 21, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 21, 11, 12, 13, 14, 15, 16, 17, 18, 19]]

條件：無

對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。

步驟識別與依賴

規則：Conjugacy_EVERY_g_EVERY_element_exact_action_row · 依賴：G001, S014

S016 · implies

操作前

source roles = [11, 11, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]]

操作後

derived = [[11, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 11, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12]]

條件：無

對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。

步驟識別與依賴

規則：Conjugacy_EVERY_g_EVERY_element_exact_action_row · 依賴：G001, S015

S017 · implies

操作前

source roles = [12, 12, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]]

操作後

derived = [[12, 11, 21, 20, 19, 18, 17, 16, 15, 14, 13, 1, 0, 10, 9, 8, 7, 6, 5, 4, 3, 2], [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 13, 12, 11, 21, 20, 19, 18, 17, 16, 15, 14]]

條件：無

對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。

步驟識別與依賴

規則：Conjugacy_EVERY_g_EVERY_element_exact_action_row · 依賴：G001, S016

S018 · implies

操作前

source roles = [13, 13, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]]

操作後

derived = [[13, 12, 11, 21, 20, 19, 18, 17, 16, 15, 14, 2, 1, 0, 10, 9, 8, 7, 6, 5, 4, 3], [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 15, 14, 13, 12, 11, 21, 20, 19, 18, 17, 16]]

條件：無

對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。

步驟識別與依賴

規則：Conjugacy_EVERY_g_EVERY_element_exact_action_row · 依賴：G001, S017

S019 · implies

操作前

source roles = [14, 14, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]]

操作後

derived = [[14, 13, 12, 11, 21, 20, 19, 18, 17, 16, 15, 3, 2, 1, 0, 10, 9, 8, 7, 6, 5, 4], [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 17, 16, 15, 14, 13, 12, 11, 21, 20, 19, 18]]

條件：無

對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。

步驟識別與依賴

規則：Conjugacy_EVERY_g_EVERY_element_exact_action_row · 依賴：G001, S018

S020 · implies

操作前

source roles = [15, 15, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]]

操作後

derived = [[15, 14, 13, 12, 11, 21, 20, 19, 18, 17, 16, 4, 3, 2, 1, 0, 10, 9, 8, 7, 6, 5], [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 19, 18, 17, 16, 15, 14, 13, 12, 11, 21, 20]]

條件：無

對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。

步驟識別與依賴

規則：Conjugacy_EVERY_g_EVERY_element_exact_action_row · 依賴：G001, S019

S021 · implies

操作前

source roles = [16, 16, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]]

操作後

derived = [[16, 15, 14, 13, 12, 11, 21, 20, 19, 18, 17, 5, 4, 3, 2, 1, 0, 10, 9, 8, 7, 6], [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11]]

條件：無

對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。

步驟識別與依賴

規則：Conjugacy_EVERY_g_EVERY_element_exact_action_row · 依賴：G001, S020

S022 · implies

操作前

source roles = [17, 17, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]]

操作後

derived = [[17, 16, 15, 14, 13, 12, 11, 21, 20, 19, 18, 6, 5, 4, 3, 2, 1, 0, 10, 9, 8, 7], [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 12, 11, 21, 20, 19, 18, 17, 16, 15, 14, 13]]

條件：無

對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。

步驟識別與依賴

規則：Conjugacy_EVERY_g_EVERY_element_exact_action_row · 依賴：G001, S021

S023 · implies

操作前

source roles = [18, 18, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]]

操作後

derived = [[18, 17, 16, 15, 14, 13, 12, 11, 21, 20, 19, 7, 6, 5, 4, 3, 2, 1, 0, 10, 9, 8], [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 14, 13, 12, 11, 21, 20, 19, 18, 17, 16, 15]]

條件：無

對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。

步驟識別與依賴

規則：Conjugacy_EVERY_g_EVERY_element_exact_action_row · 依賴：G001, S022

S024 · implies

操作前

source roles = [19, 19, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]]

操作後

derived = [[19, 18, 17, 16, 15, 14, 13, 12, 11, 21, 20, 8, 7, 6, 5, 4, 3, 2, 1, 0, 10, 9], [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 16, 15, 14, 13, 12, 11, 21, 20, 19, 18, 17]]

條件：無

對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。

步驟識別與依賴

規則：Conjugacy_EVERY_g_EVERY_element_exact_action_row · 依賴：G001, S023

S025 · implies

操作前

source roles = [20, 20, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]]

操作後

derived = [[20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 21, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 10], [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 18, 17, 16, 15, 14, 13, 12, 11, 21, 20, 19]]

條件：無

對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。

步驟識別與依賴

規則：Conjugacy_EVERY_g_EVERY_element_exact_action_row · 依賴：G001, S024

S026 · implies

操作前

source roles = [21, 21, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]]

操作後

derived = [[21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0], [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 21]]

條件：無

對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。

步驟識別與依賴

規則：Conjugacy_EVERY_g_EVERY_element_exact_action_row · 依賴：G001, S025

S027 · implies

操作前

source roles = [0, 22, [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13, 14, 15, 16, 17, 18, 19, 20, 21, 11, 12], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 15, 16, 17, 18, 19, 20, 21, 11, 12, 13, 14], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 17, 18, 19, 20, 21, 11, 12, 13, 14, 15, 16], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 19, 20, 21, 11, 12, 13, 14, 15, 16, 17, 18], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 21, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 15, 16, 17, 18, 19, 20, 21, 11, 12, 13], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 16, 17, 18, 19, 20, 21, 11, 12, 13, 14, 15], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 18, 19, 20, 21, 11, 12, 13, 14, 15, 16, 17], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 21, 11, 12, 13, 14, 15, 16, 17, 18, 19], [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 11, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12], [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 13, 12, 11, 21, 20, 19, 18, 17, 16, 15, 14], [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 15, 14, 13, 12, 11, 21, 20, 19, 18, 17, 16], [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 17, 16, 15, 14, 13, 12, 11, 21, 20, 19, 18], [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 19, 18, 17, 16, 15, 14, 13, 12, 11, 21, 20], [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 12, 11, 21, 20, 19, 18, 17, 16, 15, 14, 13], [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 14, 13, 12, 11, 21, 20, 19, 18, 17, 16, 15], [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 16, 15, 14, 13, 12, 11, 21, 20, 19, 18, 17], [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 18, 17, 16, 15, 14, 13, 12, 11, 21, 20, 19], [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 21]]]

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, 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, (gh)x(gh)^-1=g(hxh^-1)g^-1]

條件：無

逐全部g,h,x核composition，亦由(gh)^-1=h^-1g^-1及結合律推同一公式；identity另驗。不是僅列出一個action表就假定axioms。

步驟識別與依賴

規則：Conjugacy_ALL_action_axioms_and_GENERAL_algebra · 依賴：G001, S026

S028 · implies

操作前

source roles = [[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13, 14, 15, 16, 17, 18, 19, 20, 21, 11, 12], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 15, 16, 17, 18, 19, 20, 21, 11, 12, 13, 14], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 17, 18, 19, 20, 21, 11, 12, 13, 14, 15, 16], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 19, 20, 21, 11, 12, 13, 14, 15, 16, 17, 18], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 21, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 15, 16, 17, 18, 19, 20, 21, 11, 12, 13], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 16, 17, 18, 19, 20, 21, 11, 12, 13, 14, 15], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 18, 19, 20, 21, 11, 12, 13, 14, 15, 16, 17], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 21, 11, 12, 13, 14, 15, 16, 17, 18, 19], [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 11, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12], [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 13, 12, 11, 21, 20, 19, 18, 17, 16, 15, 14], [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 15, 14, 13, 12, 11, 21, 20, 19, 18, 17, 16], [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 17, 16, 15, 14, 13, 12, 11, 21, 20, 19, 18], [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 19, 18, 17, 16, 15, 14, 13, 12, 11, 21, 20], [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 12, 11, 21, 20, 19, 18, 17, 16, 15, 14, 13], [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 14, 13, 12, 11, 21, 20, 19, 18, 17, 16, 15], [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 16, 15, 14, 13, 12, 11, 21, 20, 19, 18, 17], [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 18, 17, 16, 15, 14, 13, 12, 11, 21, 20, 19], [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 21]]]

操作後

derived = [[[0], [1, 10], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [2, 9], [3, 8], [4, 7], [5, 6]], 22, 22, 是]

條件：無

e給自反、g^-1給反向、群乘積給傳遞；故所有完整orbits相等或不交並覆蓋G。按來源coset-like列表排序不代表它們是cosets；不引centralizer/class-size或類方程。

步驟識別與依賴

規則：Conjugacy_complete_orbits_equal_or_disjoint_partition · 依賴：G001, S027

逐步反向核對

  • 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。

結論

([[0], [1, 10], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [2, 9], [3, 8], [4, 7], [5, 6]], 是, 22, 22, all_conjugacy_classes_are_equal_or_disjoint_and_cover_G)

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。

來源雜湊與覆核憑證
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: d4b2683a226f4a5ae0b840202df0b317bfd17c4ca0fdf3baae84c70cc6d67783
  • candidate record SHA-256: 2a0dd9358b1eb57d9cd3ddacff47de11a79955378ece4ca72286cf8bbd054fb5
  • semantic review receipt SHA-256: 049a9e07e69fead88d93b34f0937b9470a06d71c1a8d097058fc83554529bdd0
  • question-first witness SHA-256: 0c71ebb418f3de347d232d159fc25e93036f4ab9166bcfb5ee5b4be86d5c55b9
  • Lean: NOT_RUN · human review: NOT_PERFORMED
同源完整 JSON