R0039-20144 · q-20261014-advance-20261107-r01-finite-group-action-0044
完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-identity-audit；資料={"group_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"identity":0,"point_count":8,"action_table":[[0,1,2,3,4,5,6,7],[1,2,3,4,0,5,6,7],[2,3,4,0,1,5,6,7],[3,4,0,1,2,5,6,7],[4,0,1,2,3,5,6,7]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。
G001 — 完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-identity-audit；資料={"group_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"identity":0,"point_count":8,"action_table":[[0,1,2,3,4,5,6,7],[1,2,3,4,0,5,6,7],[2,3,4,0,1,5,6,7],[3,4,0,1,2,5,6,7],[4,0,1,2,3,5,6,7]]}。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 operation=groupaction-identity-audit;original snapshot and mathematics retained
操作後
explicit ordered source output fields=[declared_identity, point_checks, identity_action]
條件：無
原題快照保留歷史介面標記；本頁直接列來源欄位順序，不需JavaScript或外部表。欄位標籤不是額外數學證據，No保持原題的聲稱範圍。
規則：source_ordered_interface_explicit_without_javascript · 依賴：G001
操作前
ORIGINAL complete group-label carrier Fin(5);declared identity label=0;Cayley table row g,column h gives g*h=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]]
操作後
ORIGINAL complete point carrier Fin(8);TOTAL action table row g,column x gives g*x=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 0, 5, 6, 7], [2, 3, 4, 0, 1, 5, 6, 7], [3, 4, 0, 1, 2, 5, 6, 7], [4, 0, 1, 2, 3, 5, 6, 7]];all entries in range,so structural closure/totality checked;GROUP AXIOMS not assumed from shape
條件：無
兩張表的column角色不同，一張h是群標籤，一張x是作用點。in-range完整表只保closed／total，群候選可不滿足公理，不能因叫group_table就先填True。
規則：groupaction_two_table_roles_complete_candidate_not_group_assumption · 依賴：G001, S001
操作前
ACTUAL declared identity label=0;ORIGINAL identity action row=[0, 1, 2, 3, 4, 5, 6, 7];complete point carrier Fin(8)
操作後
ALL ordered [point,identity_image,equality] rows=[[0, 0, 是], [1, 1, 是], [2, 2, 是], [3, 3, 是], [4, 4, 是], [5, 5, 是], [6, 6, 是], [7, 7, 是]];identity-action axiom=是
條件：無
逐作用點驗e*x=x，e是來源宣告的真標籤，不預設0；此axiom與群乘法表的雙邊identity責任分開。
規則：groupaction_every_point_identity_action_not_group_identity · 依賴：G001, S002
操作前
this component audited ONLY the complete identity action row
操作後
identity-action flag alone does NOT establish group qualification or action compatibility
條件：無
本題輸出只回答identity-row責任；其他公理沒有被此旗標自動取代，不把一個component通過升格valid_action。
規則：groupaction_identity_component_not_full_action_claim · 依賴：G001, S003
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。(0, [[0, 0, 是], [1, 1, 是], [2, 2, 是], [3, 3, 是], [4, 4, 是], [5, 5, 是], [6, 6, 是], [7, 7, 是]], 是)
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: fe35ba957e886cbcbb1d3cddc868c36ed62d8a571549fae42223660b7297422fcandidate record SHA-256: 3a42a262a7ba6cd8fed00e9cdcaf9431a037a208819e142c509b762ad6927b09semantic review receipt SHA-256: 6e11e35fafe32f16900ce9b522d01acb7b2ee6b463ab179392076cfbe1af8034question-first witness SHA-256: cc0fa361fc25151729e70264340d2eae94f179f6f2a2edfe80ba5704fbbd0682Lean: NOT_RUN · human review: NOT_PERFORMED