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第 24933 題：有限群同態的核

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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

R0048-24933 · q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0008

原題與已知條件

完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-homomorphism-qualification；資料={"source_table":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8]],"target_identity":0,"mapping":[0,9,8,7,6,5,4,3,2,1]}。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-homomorphism-qualification；資料={"source_table":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8]],"target_identity":0,"mapping":[0,9,8,7,6,5,4,3,2,1]}。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 = [kernel-homomorphism-qualification]

操作後

derived = [[source_order, target_order, mapping, product_law_rows, homomorphism, first_failed_pair_or_null, 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 = [Kernel ORIGINAL SOURCE group, 64, [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 5, 6, 7, 8, 9, 0], [2, 3, 4, 5, 6, 7, 8, 9, 0, 1], [3, 4, 5, 6, 7, 8, 9, 0, 1, 2], [4, 5, 6, 7, 8, 9, 0, 1, 2, 3], [5, 6, 7, 8, 9, 0, 1, 2, 3, 4], [6, 7, 8, 9, 0, 1, 2, 3, 4, 5], [7, 8, 9, 0, 1, 2, 3, 4, 5, 6], [8, 9, 0, 1, 2, 3, 4, 5, 6, 7], [9, 0, 1, 2, 3, 4, 5, 6, 7, 8]], 0]

操作後

derived = [10, [[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]], 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 = [Kernel ORIGINAL SOURCE group, 10, 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]], 1000, 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 = [Kernel ORIGINAL SOURCE group, 0]

操作後

derived = [[0, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0], [1, 9, 0, 0], [2, 8, 0, 0], [3, 7, 0, 0], [4, 6, 0, 0], [5, 5, 0, 0], [6, 4, 0, 0], [7, 3, 0, 0], [8, 2, 0, 0], [9, 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。

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 = [Kernel ORIGINAL TARGET group, 64, [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 5, 6, 7, 8, 9, 0], [2, 3, 4, 5, 6, 7, 8, 9, 0, 1], [3, 4, 5, 6, 7, 8, 9, 0, 1, 2], [4, 5, 6, 7, 8, 9, 0, 1, 2, 3], [5, 6, 7, 8, 9, 0, 1, 2, 3, 4], [6, 7, 8, 9, 0, 1, 2, 3, 4, 5], [7, 8, 9, 0, 1, 2, 3, 4, 5, 6], [8, 9, 0, 1, 2, 3, 4, 5, 6, 7], [9, 0, 1, 2, 3, 4, 5, 6, 7, 8]], 0]

操作後

derived = [10, [[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]], 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, S004

S006 · implies

操作前

source roles = [Kernel ORIGINAL TARGET group, 10, 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]], 1000, 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, S005

S007 · implies

操作前

source roles = [Kernel ORIGINAL TARGET group, 0]

操作後

derived = [[0, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0], [1, 9, 0, 0], [2, 8, 0, 0], [3, 7, 0, 0], [4, 6, 0, 0], [5, 5, 0, 0], [6, 4, 0, 0], [7, 3, 0, 0], [8, 2, 0, 0], [9, 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。

步驟識別與依賴

規則：Group_ALL_two_sided_inverses · 依賴：G001, S006

S008 · implies

操作前

source roles = [[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 5, 6, 7, 8, 9, 0], [2, 3, 4, 5, 6, 7, 8, 9, 0, 1], [3, 4, 5, 6, 7, 8, 9, 0, 1, 2], [4, 5, 6, 7, 8, 9, 0, 1, 2, 3], [5, 6, 7, 8, 9, 0, 1, 2, 3, 4], [6, 7, 8, 9, 0, 1, 2, 3, 4, 5], [7, 8, 9, 0, 1, 2, 3, 4, 5, 6], [8, 9, 0, 1, 2, 3, 4, 5, 6, 7], [9, 0, 1, 2, 3, 4, 5, 6, 7, 8]], [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 5, 6, 7, 8, 9, 0], [2, 3, 4, 5, 6, 7, 8, 9, 0, 1], [3, 4, 5, 6, 7, 8, 9, 0, 1, 2], [4, 5, 6, 7, 8, 9, 0, 1, 2, 3], [5, 6, 7, 8, 9, 0, 1, 2, 3, 4], [6, 7, 8, 9, 0, 1, 2, 3, 4, 5], [7, 8, 9, 0, 1, 2, 3, 4, 5, 6], [8, 9, 0, 1, 2, 3, 4, 5, 6, 7], [9, 0, 1, 2, 3, 4, 5, 6, 7, 8]], [0, 9, 8, 7, 6, 5, 4, 3, 2, 1]]

操作後

derived = [[[0, 0, 0, 0, 是], [0, 1, 9, 9, 是], [0, 2, 8, 8, 是], [0, 3, 7, 7, 是], [0, 4, 6, 6, 是], [0, 5, 5, 5, 是], [0, 6, 4, 4, 是], [0, 7, 3, 3, 是], [0, 8, 2, 2, 是], [0, 9, 1, 1, 是], [1, 0, 9, 9, 是], [1, 1, 8, 8, 是], [1, 2, 7, 7, 是], [1, 3, 6, 6, 是], [1, 4, 5, 5, 是], [1, 5, 4, 4, 是], [1, 6, 3, 3, 是], [1, 7, 2, 2, 是], [1, 8, 1, 1, 是], [1, 9, 0, 0, 是], [2, 0, 8, 8, 是], [2, 1, 7, 7, 是], [2, 2, 6, 6, 是], [2, 3, 5, 5, 是], [2, 4, 4, 4, 是], [2, 5, 3, 3, 是], [2, 6, 2, 2, 是], [2, 7, 1, 1, 是], [2, 8, 0, 0, 是], [2, 9, 9, 9, 是], [3, 0, 7, 7, 是], [3, 1, 6, 6, 是], [3, 2, 5, 5, 是], [3, 3, 4, 4, 是], [3, 4, 3, 3, 是], [3, 5, 2, 2, 是], [3, 6, 1, 1, 是], [3, 7, 0, 0, 是], [3, 8, 9, 9, 是], [3, 9, 8, 8, 是], [4, 0, 6, 6, 是], [4, 1, 5, 5, 是], [4, 2, 4, 4, 是], [4, 3, 3, 3, 是], [4, 4, 2, 2, 是], [4, 5, 1, 1, 是], [4, 6, 0, 0, 是], [4, 7, 9, 9, 是], [4, 8, 8, 8, 是], [4, 9, 7, 7, 是], [5, 0, 5, 5, 是], [5, 1, 4, 4, 是], [5, 2, 3, 3, 是], [5, 3, 2, 2, 是], [5, 4, 1, 1, 是], [5, 5, 0, 0, 是], [5, 6, 9, 9, 是], [5, 7, 8, 8, 是], [5, 8, 7, 7, 是], [5, 9, 6, 6, 是], [6, 0, 4, 4, 是], [6, 1, 3, 3, 是], [6, 2, 2, 2, 是], [6, 3, 1, 1, 是], [6, 4, 0, 0, 是], [6, 5, 9, 9, 是], [6, 6, 8, 8, 是], [6, 7, 7, 7, 是], [6, 8, 6, 6, 是], [6, 9, 5, 5, 是], [7, 0, 3, 3, 是], [7, 1, 2, 2, 是], [7, 2, 1, 1, 是], [7, 3, 0, 0, 是], [7, 4, 9, 9, 是], [7, 5, 8, 8, 是], [7, 6, 7, 7, 是], [7, 7, 6, 6, 是], [7, 8, 5, 5, 是], [7, 9, 4, 4, 是], [8, 0, 2, 2, 是], [8, 1, 1, 1, 是], [8, 2, 0, 0, 是], [8, 3, 9, 9, 是], [8, 4, 8, 8, 是], [8, 5, 7, 7, 是], [8, 6, 6, 6, 是], [8, 7, 5, 5, 是], [8, 8, 4, 4, 是], [8, 9, 3, 3, 是], [9, 0, 1, 1, 是], [9, 1, 0, 0, 是], [9, 2, 9, 9, 是], [9, 3, 8, 8, 是], [9, 4, 7, 7, 是], [9, 5, 6, 6, 是], [9, 6, 5, 5, 是], [9, 7, 4, 4, 是], [9, 8, 3, 3, 是], [9, 9, 2, 2, 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。

步驟識別與依賴

規則：Kernel_complete_TOTAL_map_ALL_product_pairs_before_theorems · 依賴：G001, S007

逐步反向核對

  • 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

(10, 10, [0, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, 是], [0, 1, 9, 9, 是], [0, 2, 8, 8, 是], [0, 3, 7, 7, 是], [0, 4, 6, 6, 是], [0, 5, 5, 5, 是], [0, 6, 4, 4, 是], [0, 7, 3, 3, 是], [0, 8, 2, 2, 是], [0, 9, 1, 1, 是], [1, 0, 9, 9, 是], [1, 1, 8, 8, 是], [1, 2, 7, 7, 是], [1, 3, 6, 6, 是], [1, 4, 5, 5, 是], [1, 5, 4, 4, 是], [1, 6, 3, 3, 是], [1, 7, 2, 2, 是], [1, 8, 1, 1, 是], [1, 9, 0, 0, 是], [2, 0, 8, 8, 是], [2, 1, 7, 7, 是], [2, 2, 6, 6, 是], [2, 3, 5, 5, 是], [2, 4, 4, 4, 是], [2, 5, 3, 3, 是], [2, 6, 2, 2, 是], [2, 7, 1, 1, 是], [2, 8, 0, 0, 是], [2, 9, 9, 9, 是], [3, 0, 7, 7, 是], [3, 1, 6, 6, 是], [3, 2, 5, 5, 是], [3, 3, 4, 4, 是], [3, 4, 3, 3, 是], [3, 5, 2, 2, 是], [3, 6, 1, 1, 是], [3, 7, 0, 0, 是], [3, 8, 9, 9, 是], [3, 9, 8, 8, 是], [4, 0, 6, 6, 是], [4, 1, 5, 5, 是], [4, 2, 4, 4, 是], [4, 3, 3, 3, 是], [4, 4, 2, 2, 是], [4, 5, 1, 1, 是], [4, 6, 0, 0, 是], [4, 7, 9, 9, 是], [4, 8, 8, 8, 是], [4, 9, 7, 7, 是], [5, 0, 5, 5, 是], [5, 1, 4, 4, 是], [5, 2, 3, 3, 是], [5, 3, 2, 2, 是], [5, 4, 1, 1, 是], [5, 5, 0, 0, 是], [5, 6, 9, 9, 是], [5, 7, 8, 8, 是], [5, 8, 7, 7, 是], [5, 9, 6, 6, 是], [6, 0, 4, 4, 是], [6, 1, 3, 3, 是], [6, 2, 2, 2, 是], [6, 3, 1, 1, 是], [6, 4, 0, 0, 是], [6, 5, 9, 9, 是], [6, 6, 8, 8, 是], [6, 7, 7, 7, 是], [6, 8, 6, 6, 是], [6, 9, 5, 5, 是], [7, 0, 3, 3, 是], [7, 1, 2, 2, 是], [7, 2, 1, 1, 是], [7, 3, 0, 0, 是], [7, 4, 9, 9, 是], [7, 5, 8, 8, 是], [7, 6, 7, 7, 是], [7, 7, 6, 6, 是], [7, 8, 5, 5, 是], [7, 9, 4, 4, 是], [8, 0, 2, 2, 是], [8, 1, 1, 1, 是], [8, 2, 0, 0, 是], [8, 3, 9, 9, 是], [8, 4, 8, 8, 是], [8, 5, 7, 7, 是], [8, 6, 6, 6, 是], [8, 7, 5, 5, 是], [8, 8, 4, 4, 是], [8, 9, 3, 3, 是], [9, 0, 1, 1, 是], [9, 1, 0, 0, 是], [9, 2, 9, 9, 是], [9, 3, 8, 8, 是], [9, 4, 7, 7, 是], [9, 5, 6, 6, 是], [9, 6, 5, 5, 是], [9, 7, 4, 4, 是], [9, 8, 3, 3, 是], [9, 9, 2, 2, 是]], 是, null, complete_total_map_satisfies_every_product_pair)

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: 629f53adb33a060ebadfcc4f467f2a09a550237e4fd2b90df007f416886617a9
  • candidate record SHA-256: f180da1f42e60af89b9693f8d664a1013927e50fea8f17a679654d757aaa4a15
  • semantic review receipt SHA-256: 049a9e07e69fead88d93b34f0937b9470a06d71c1a8d097058fc83554529bdd0
  • question-first witness SHA-256: 0c71ebb418f3de347d232d159fc25e93036f4ab9166bcfb5ee5b4be86d5c55b9
  • Lean: NOT_RUN · human review: NOT_PERFORMED
同源完整 JSON