{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0013", "pilot_number": "R0048-24938", "number": 24938, "family_id": "qf-mod45-kernel-homomorphism-qualification", "title": { "zh-Hant": "第 24938 題：有限群同態的核", "en": "Question 24938: Qualify Both Groups and Every Product Pair before Applying Kernel Theorems" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0013", "batch_id": "batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-finite-group-homomorphism-kernel" ], "title_zh": "有限群同態的核", "stem_zh": "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\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,0,1,2,3,4,5,6,7,8,9,10,11,12,13]],\"source_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,0,1,2,3,4,5,6,7,8,9,10,11,12,13]],\"target_identity\":0,\"mapping\":[0,14,13,12,11,10,9,8,7,6,5,4,3,2,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "(15, 15, [0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, 是], [0, 1, 14, 14, 是], [0, 2, 13, 13, 是], [0, 3, 12, 12, 是], [0, 4, 11, 11, 是], [0, 5, 10, 10, 是], [0, 6, 9, 9, 是], [0, 7, 8, 8, 是], [0, 8, 7, 7, 是], [0, 9, 6, 6, 是], [0, 10, 5, 5, 是], [0, 11, 4, 4, 是], [0, 12, 3, 3, 是], [0, 13, 2, 2, 是], [0, 14, 1, 1, 是], [1, 0, 14, 14, 是], [1, 1, 13, 13, 是], [1, 2, 12, 12, 是], [1, 3, 11, 11, 是], [1, 4, 10, 10, 是], [1, 5, 9, 9, 是], [1, 6, 8, 8, 是], [1, 7, 7, 7, 是], [1, 8, 6, 6, 是], [1, 9, 5, 5, 是], [1, 10, 4, 4, 是], [1, 11, 3, 3, 是], [1, 12, 2, 2, 是], [1, 13, 1, 1, 是], [1, 14, 0, 0, 是], [2, 0, 13, 13, 是], [2, 1, 12, 12, 是], [2, 2, 11, 11, 是], [2, 3, 10, 10, 是], [2, 4, 9, 9, 是], [2, 5, 8, 8, 是], [2, 6, 7, 7, 是], [2, 7, 6, 6, 是], [2, 8, 5, 5, 是], [2, 9, 4, 4, 是], [2, 10, 3, 3, 是], [2, 11, 2, 2, 是], [2, 12, 1, 1, 是], [2, 13, 0, 0, 是], [2, 14, 14, 14, 是], [3, 0, 12, 12, 是], [3, 1, 11, 11, 是], [3, 2, 10, 10, 是], [3, 3, 9, 9, 是], [3, 4, 8, 8, 是], [3, 5, 7, 7, 是], [3, 6, 6, 6, 是], [3, 7, 5, 5, 是], [3, 8, 4, 4, 是], [3, 9, 3, 3, 是], [3, 10, 2, 2, 是], [3, 11, 1, 1, 是], [3, 12, 0, 0, 是], [3, 13, 14, 14, 是], [3, 14, 13, 13, 是], [4, 0, 11, 11, 是], [4, 1, 10, 10, 是], [4, 2, 9, 9, 是], [4, 3, 8, 8, 是], [4, 4, 7, 7, 是], [4, 5, 6, 6, 是], [4, 6, 5, 5, 是], [4, 7, 4, 4, 是], [4, 8, 3, 3, 是], [4, 9, 2, 2, 是], [4, 10, 1, 1, 是], [4, 11, 0, 0, 是], [4, 12, 14, 14, 是], [4, 13, 13, 13, 是], [4, 14, 12, 12, 是], [5, 0, 10, 10, 是], [5, 1, 9, 9, 是], [5, 2, 8, 8, 是], [5, 3, 7, 7, 是], [5, 4, 6, 6, 是], [5, 5, 5, 5, 是], [5, 6, 4, 4, 是], [5, 7, 3, 3, 是], [5, 8, 2, 2, 是], [5, 9, 1, 1, 是], [5, 10, 0, 0, 是], [5, 11, 14, 14, 是], [5, 12, 13, 13, 是], [5, 13, 12, 12, 是], [5, 14, 11, 11, 是], [6, 0, 9, 9, 是], [6, 1, 8, 8, 是], [6, 2, 7, 7, 是], [6, 3, 6, 6, 是], [6, 4, 5, 5, 是], [6, 5, 4, 4, 是], [6, 6, 3, 3, 是], [6, 7, 2, 2, 是], [6, 8, 1, 1, 是], [6, 9, 0, 0, 是], [6, 10, 14, 14, 是], [6, 11, 13, 13, 是], [6, 12, 12, 12, 是], [6, 13, 11, 11, 是], [6, 14, 10, 10, 是], [7, 0, 8, 8, 是], [7, 1, 7, 7, 是], [7, 2, 6, 6, 是], [7, 3, 5, 5, 是], [7, 4, 4, 4, 是], [7, 5, 3, 3, 是], [7, 6, 2, 2, 是], [7, 7, 1, 1, 是], [7, 8, 0, 0, 是], [7, 9, 14, 14, 是], [7, 10, 13, 13, 是], [7, 11, 12, 12, 是], [7, 12, 11, 11, 是], [7, 13, 10, 10, 是], [7, 14, 9, 9, 是], [8, 0, 7, 7, 是], [8, 1, 6, 6, 是], [8, 2, 5, 5, 是], [8, 3, 4, 4, 是], [8, 4, 3, 3, 是], [8, 5, 2, 2, 是], [8, 6, 1, 1, 是], [8, 7, 0, 0, 是], [8, 8, 14, 14, 是], [8, 9, 13, 13, 是], [8, 10, 12, 12, 是], [8, 11, 11, 11, 是], [8, 12, 10, 10, 是], [8, 13, 9, 9, 是], [8, 14, 8, 8, 是], [9, 0, 6, 6, 是], [9, 1, 5, 5, 是], [9, 2, 4, 4, 是], [9, 3, 3, 3, 是], [9, 4, 2, 2, 是], [9, 5, 1, 1, 是], [9, 6, 0, 0, 是], [9, 7, 14, 14, 是], [9, 8, 13, 13, 是], [9, 9, 12, 12, 是], [9, 10, 11, 11, 是], [9, 11, 10, 10, 是], [9, 12, 9, 9, 是], [9, 13, 8, 8, 是], [9, 14, 7, 7, 是], [10, 0, 5, 5, 是], [10, 1, 4, 4, 是], [10, 2, 3, 3, 是], [10, 3, 2, 2, 是], [10, 4, 1, 1, 是], [10, 5, 0, 0, 是], [10, 6, 14, 14, 是], [10, 7, 13, 13, 是], [10, 8, 12, 12, 是], [10, 9, 11, 11, 是], [10, 10, 10, 10, 是], [10, 11, 9, 9, 是], [10, 12, 8, 8, 是], [10, 13, 7, 7, 是], [10, 14, 6, 6, 是], [11, 0, 4, 4, 是], [11, 1, 3, 3, 是], [11, 2, 2, 2, 是], [11, 3, 1, 1, 是], [11, 4, 0, 0, 是], [11, 5, 14, 14, 是], [11, 6, 13, 13, 是], [11, 7, 12, 12, 是], [11, 8, 11, 11, 是], [11, 9, 10, 10, 是], [11, 10, 9, 9, 是], [11, 11, 8, 8, 是], [11, 12, 7, 7, 是], [11, 13, 6, 6, 是], [11, 14, 5, 5, 是], [12, 0, 3, 3, 是], [12, 1, 2, 2, 是], [12, 2, 1, 1, 是], [12, 3, 0, 0, 是], [12, 4, 14, 14, 是], [12, 5, 13, 13, 是], [12, 6, 12, 12, 是], [12, 7, 11, 11, 是], [12, 8, 10, 10, 是], [12, 9, 9, 9, 是], [12, 10, 8, 8, 是], [12, 11, 7, 7, 是], [12, 12, 6, 6, 是], [12, 13, 5, 5, 是], [12, 14, 4, 4, 是], [13, 0, 2, 2, 是], [13, 1, 1, 1, 是], [13, 2, 0, 0, 是], [13, 3, 14, 14, 是], [13, 4, 13, 13, 是], [13, 5, 12, 12, 是], [13, 6, 11, 11, 是], [13, 7, 10, 10, 是], [13, 8, 9, 9, 是], [13, 9, 8, 8, 是], [13, 10, 7, 7, 是], [13, 11, 6, 6, 是], [13, 12, 5, 5, 是], [13, 13, 4, 4, 是], [13, 14, 3, 3, 是], [14, 0, 1, 1, 是], [14, 1, 0, 0, 是], [14, 2, 14, 14, 是], [14, 3, 13, 13, 是], [14, 4, 12, 12, 是], [14, 5, 11, 11, 是], [14, 6, 10, 10, 是], [14, 7, 9, 9, 是], [14, 8, 8, 8, 是], [14, 9, 7, 7, 是], [14, 10, 6, 6, 是], [14, 11, 5, 5, 是], [14, 12, 4, 4, 是], [14, 13, 3, 3, 是], [14, 14, 2, 2, 是]], 是, null, complete_total_map_satisfies_every_product_pair)", "explanation_zh": "Wave6第八層核對=(15, 15, [0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, 是], [0, 1, 14, 14, 是], [0, 2, 13, 13, 是], [0, 3, 12, 12, 是], [0, 4, 11, 11, 是], [0, 5, 10, 10, 是], [0, 6, 9, 9, 是], [0, 7, 8, 8, 是], [0, 8, 7, 7, 是], [0, 9, 6, 6, 是], [0, 10, 5, 5, 是], [0, 11, 4, 4, 是], [0, 12, 3, 3, 是], [0, 13, 2, 2, 是], [0, 14, 1, 1, 是], [1, 0, 14, 14, 是], [1, 1, 13, 13, 是], [1, 2, 12, 12, 是], [1, 3, 11, 11, 是], [1, 4, 10, 10, 是], [1, 5, 9, 9, 是], [1, 6, 8, 8, 是], [1, 7, 7, 7, 是], [1, 8, 6, 6, 是], [1, 9, 5, 5, 是], [1, 10, 4, 4, 是], [1, 11, 3, 3, 是], [1, 12, 2, 2, 是], [1, 13, 1, 1, 是], [1, 14, 0, 0, 是], [2, 0, 13, 13, 是], [2, 1, 12, 12, 是], [2, 2, 11, 11, 是], [2, 3, 10, 10, 是], [2, 4, 9, 9, 是], [2, 5, 8, 8, 是], [2, 6, 7, 7, 是], [2, 7, 6, 6, 是], [2, 8, 5, 5, 是], [2, 9, 4, 4, 是], [2, 10, 3, 3, 是], [2, 11, 2, 2, 是], [2, 12, 1, 1, 是], [2, 13, 0, 0, 是], [2, 14, 14, 14, 是], [3, 0, 12, 12, 是], [3, 1, 11, 11, 是], [3, 2, 10, 10, 是], [3, 3, 9, 9, 是], [3, 4, 8, 8, 是], [3, 5, 7, 7, 是], [3, 6, 6, 6, 是], [3, 7, 5, 5, 是], [3, 8, 4, 4, 是], [3, 9, 3, 3, 是], [3, 10, 2, 2, 是], [3, 11, 1, 1, 是], [3, 12, 0, 0, 是], [3, 13, 14, 14, 是], [3, 14, 13, 13, 是], [4, 0, 11, 11, 是], [4, 1, 10, 10, 是], [4, 2, 9, 9, 是], [4, 3, 8, 8, 是], [4, 4, 7, 7, 是], [4, 5, 6, 6, 是], [4, 6, 5, 5, 是], [4, 7, 4, 4, 是], [4, 8, 3, 3, 是], [4, 9, 2, 2, 是], [4, 10, 1, 1, 是], [4, 11, 0, 0, 是], [4, 12, 14, 14, 是], [4, 13, 13, 13, 是], [4, 14, 12, 12, 是], [5, 0, 10, 10, 是], [5, 1, 9, 9, 是], [5, 2, 8, 8, 是], [5, 3, 7, 7, 是], [5, 4, 6, 6, 是], [5, 5, 5, 5, 是], [5, 6, 4, 4, 是], [5, 7, 3, 3, 是], [5, 8, 2, 2, 是], [5, 9, 1, 1, 是], [5, 10, 0, 0, 是], [5, 11, 14, 14, 是], [5, 12, 13, 13, 是], [5, 13, 12, 12, 是], [5, 14, 11, 11, 是], [6, 0, 9, 9, 是], [6, 1, 8, 8, 是], [6, 2, 7, 7, 是], [6, 3, 6, 6, 是], [6, 4, 5, 5, 是], [6, 5, 4, 4, 是], [6, 6, 3, 3, 是], [6, 7, 2, 2, 是], [6, 8, 1, 1, 是], [6, 9, 0, 0, 是], [6, 10, 14, 14, 是], [6, 11, 13, 13, 是], [6, 12, 12, 12, 是], [6, 13, 11, 11, 是], [6, 14, 10, 10, 是], [7, 0, 8, 8, 是], [7, 1, 7, 7, 是], [7, 2, 6, 6, 是], [7, 3, 5, 5, 是], [7, 4, 4, 4, 是], [7, 5, 3, 3, 是], [7, 6, 2, 2, 是], [7, 7, 1, 1, 是], [7, 8, 0, 0, 是], [7, 9, 14, 14, 是], [7, 10, 13, 13, 是], [7, 11, 12, 12, 是], [7, 12, 11, 11, 是], [7, 13, 10, 10, 是], [7, 14, 9, 9, 是], [8, 0, 7, 7, 是], [8, 1, 6, 6, 是], [8, 2, 5, 5, 是], [8, 3, 4, 4, 是], [8, 4, 3, 3, 是], [8, 5, 2, 2, 是], [8, 6, 1, 1, 是], [8, 7, 0, 0, 是], [8, 8, 14, 14, 是], [8, 9, 13, 13, 是], [8, 10, 12, 12, 是], [8, 11, 11, 11, 是], [8, 12, 10, 10, 是], [8, 13, 9, 9, 是], [8, 14, 8, 8, 是], [9, 0, 6, 6, 是], [9, 1, 5, 5, 是], [9, 2, 4, 4, 是], [9, 3, 3, 3, 是], [9, 4, 2, 2, 是], [9, 5, 1, 1, 是], [9, 6, 0, 0, 是], [9, 7, 14, 14, 是], [9, 8, 13, 13, 是], [9, 9, 12, 12, 是], [9, 10, 11, 11, 是], [9, 11, 10, 10, 是], [9, 12, 9, 9, 是], [9, 13, 8, 8, 是], [9, 14, 7, 7, 是], [10, 0, 5, 5, 是], [10, 1, 4, 4, 是], [10, 2, 3, 3, 是], [10, 3, 2, 2, 是], [10, 4, 1, 1, 是], [10, 5, 0, 0, 是], [10, 6, 14, 14, 是], [10, 7, 13, 13, 是], [10, 8, 12, 12, 是], [10, 9, 11, 11, 是], [10, 10, 10, 10, 是], [10, 11, 9, 9, 是], [10, 12, 8, 8, 是], [10, 13, 7, 7, 是], [10, 14, 6, 6, 是], [11, 0, 4, 4, 是], [11, 1, 3, 3, 是], [11, 2, 2, 2, 是], [11, 3, 1, 1, 是], [11, 4, 0, 0, 是], [11, 5, 14, 14, 是], [11, 6, 13, 13, 是], [11, 7, 12, 12, 是], [11, 8, 11, 11, 是], [11, 9, 10, 10, 是], [11, 10, 9, 9, 是], [11, 11, 8, 8, 是], [11, 12, 7, 7, 是], [11, 13, 6, 6, 是], [11, 14, 5, 5, 是], [12, 0, 3, 3, 是], [12, 1, 2, 2, 是], [12, 2, 1, 1, 是], [12, 3, 0, 0, 是], [12, 4, 14, 14, 是], [12, 5, 13, 13, 是], [12, 6, 12, 12, 是], [12, 7, 11, 11, 是], [12, 8, 10, 10, 是], [12, 9, 9, 9, 是], [12, 10, 8, 8, 是], [12, 11, 7, 7, 是], [12, 12, 6, 6, 是], [12, 13, 5, 5, 是], [12, 14, 4, 4, 是], [13, 0, 2, 2, 是], [13, 1, 1, 1, 是], [13, 2, 0, 0, 是], [13, 3, 14, 14, 是], [13, 4, 13, 13, 是], [13, 5, 12, 12, 是], [13, 6, 11, 11, 是], [13, 7, 10, 10, 是], [13, 8, 9, 9, 是], [13, 9, 8, 8, 是], [13, 10, 7, 7, 是], [13, 11, 6, 6, 是], [13, 12, 5, 5, 是], [13, 13, 4, 4, 是], [13, 14, 3, 3, 是], [14, 0, 1, 1, 是], [14, 1, 0, 0, 是], [14, 2, 14, 14, 是], [14, 3, 13, 13, 是], [14, 4, 12, 12, 是], [14, 5, 11, 11, 是], [14, 6, 10, 10, 是], [14, 7, 9, 9, 是], [14, 8, 8, 8, 是], [14, 9, 7, 7, 是], [14, 10, 6, 6, 是], [14, 11, 5, 5, 是], [14, 12, 4, 4, 是], [14, 13, 3, 3, 是], [14, 14, 2, 2, 是]], 是, null, complete_total_map_satisfies_every_product_pair)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。" }, "difficulty": { "band": "undergraduate_core", "level": 2, "skills": [ "有限群同態的核" ] }, "variant": { "family_id": "qf-mod45-kernel-homomorphism-qualification", "seed": 202611141013, "parameters": { "source_table": [ [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0 ], [ 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1 ], [ 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2 ], [ 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3 ], [ 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4 ], [ 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5 ], [ 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6 ], [ 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7 ], [ 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8 ], [ 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 ], [ 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ], [ 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 ], [ 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ], [ 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 ] ], "source_identity": 0, "target_table": [ [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0 ], [ 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1 ], [ 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2 ], [ 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3 ], [ 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4 ], [ 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5 ], [ 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6 ], [ 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7 ], [ 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8 ], [ 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 ], [ 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ], [ 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 ], [ 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ], [ 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 ] ], "target_identity": 0, "mapping": [ 0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1 ], "domain": "Complete_finite_group_homomorphism_kernel", "operation": "kernel-homomorphism-qualification", "semantic_template": "qf-mod45-kernel-homomorphism-qualification/{\"mapping\":[0,14,13,12,11,10,9,8,7,6,5,4,3,2,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,0,1,2,3,4,5,6,7,8,9,10,11,12,13]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,0,1,2,3,4,5,6,7,8,9,10,11,12,13]]}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_or_symbolic_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-15", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0013", "batch_id": "batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "5eaa9d545a11ecf939a939d7ab7f21bdafa6ea16d4d706c373de5b9aa6bc909c", "family_id": "qf-mod45-kernel-homomorphism-qualification", "seed": 202611141013, "semantic_template": "qf-mod45-kernel-homomorphism-qualification/{\"mapping\":[0,14,13,12,11,10,9,8,7,6,5,4,3,2,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,0,1,2,3,4,5,6,7,8,9,10,11,12,13]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,0,1,2,3,4,5,6,7,8,9,10,11,12,13]]}" }, "fields": { "title": "Qualify Both Groups and Every Product Pair before Applying Kernel Theorems", "stem": "Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]], target identity=0, and total mapping=[0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (source_order, target_order, mapping, product_law_rows, homomorphism, first_failed_pair_or_null, certificate).", "answer": { "kind": "expression", "value": "(15, 15, [0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 14, 14, Yes], [0, 2, 13, 13, Yes], [0, 3, 12, 12, Yes], [0, 4, 11, 11, Yes], [0, 5, 10, 10, Yes], [0, 6, 9, 9, Yes], [0, 7, 8, 8, Yes], [0, 8, 7, 7, Yes], [0, 9, 6, 6, Yes], [0, 10, 5, 5, Yes], [0, 11, 4, 4, Yes], [0, 12, 3, 3, Yes], [0, 13, 2, 2, Yes], [0, 14, 1, 1, Yes], [1, 0, 14, 14, Yes], [1, 1, 13, 13, Yes], [1, 2, 12, 12, Yes], [1, 3, 11, 11, Yes], [1, 4, 10, 10, Yes], [1, 5, 9, 9, Yes], [1, 6, 8, 8, Yes], [1, 7, 7, 7, Yes], [1, 8, 6, 6, Yes], [1, 9, 5, 5, Yes], [1, 10, 4, 4, Yes], [1, 11, 3, 3, Yes], [1, 12, 2, 2, Yes], [1, 13, 1, 1, Yes], [1, 14, 0, 0, Yes], [2, 0, 13, 13, Yes], [2, 1, 12, 12, Yes], [2, 2, 11, 11, Yes], [2, 3, 10, 10, Yes], [2, 4, 9, 9, Yes], [2, 5, 8, 8, Yes], [2, 6, 7, 7, Yes], [2, 7, 6, 6, Yes], [2, 8, 5, 5, Yes], [2, 9, 4, 4, Yes], [2, 10, 3, 3, Yes], [2, 11, 2, 2, Yes], [2, 12, 1, 1, Yes], [2, 13, 0, 0, Yes], [2, 14, 14, 14, Yes], [3, 0, 12, 12, Yes], [3, 1, 11, 11, Yes], [3, 2, 10, 10, Yes], [3, 3, 9, 9, Yes], [3, 4, 8, 8, Yes], [3, 5, 7, 7, Yes], [3, 6, 6, 6, Yes], [3, 7, 5, 5, Yes], [3, 8, 4, 4, Yes], [3, 9, 3, 3, Yes], [3, 10, 2, 2, Yes], [3, 11, 1, 1, Yes], [3, 12, 0, 0, Yes], [3, 13, 14, 14, Yes], [3, 14, 13, 13, Yes], [4, 0, 11, 11, Yes], [4, 1, 10, 10, Yes], [4, 2, 9, 9, Yes], [4, 3, 8, 8, Yes], [4, 4, 7, 7, Yes], [4, 5, 6, 6, Yes], [4, 6, 5, 5, Yes], [4, 7, 4, 4, Yes], [4, 8, 3, 3, Yes], [4, 9, 2, 2, Yes], [4, 10, 1, 1, Yes], [4, 11, 0, 0, Yes], [4, 12, 14, 14, Yes], [4, 13, 13, 13, Yes], [4, 14, 12, 12, Yes], [5, 0, 10, 10, Yes], [5, 1, 9, 9, Yes], [5, 2, 8, 8, Yes], [5, 3, 7, 7, Yes], [5, 4, 6, 6, Yes], [5, 5, 5, 5, Yes], [5, 6, 4, 4, Yes], [5, 7, 3, 3, Yes], [5, 8, 2, 2, Yes], [5, 9, 1, 1, Yes], [5, 10, 0, 0, Yes], [5, 11, 14, 14, Yes], [5, 12, 13, 13, Yes], [5, 13, 12, 12, Yes], [5, 14, 11, 11, Yes], [6, 0, 9, 9, Yes], [6, 1, 8, 8, Yes], [6, 2, 7, 7, Yes], [6, 3, 6, 6, Yes], [6, 4, 5, 5, Yes], [6, 5, 4, 4, Yes], [6, 6, 3, 3, Yes], [6, 7, 2, 2, Yes], [6, 8, 1, 1, Yes], [6, 9, 0, 0, Yes], [6, 10, 14, 14, Yes], [6, 11, 13, 13, Yes], [6, 12, 12, 12, Yes], [6, 13, 11, 11, Yes], [6, 14, 10, 10, Yes], [7, 0, 8, 8, Yes], [7, 1, 7, 7, Yes], [7, 2, 6, 6, Yes], [7, 3, 5, 5, Yes], [7, 4, 4, 4, Yes], [7, 5, 3, 3, Yes], [7, 6, 2, 2, Yes], [7, 7, 1, 1, Yes], [7, 8, 0, 0, Yes], [7, 9, 14, 14, Yes], [7, 10, 13, 13, Yes], [7, 11, 12, 12, Yes], [7, 12, 11, 11, Yes], [7, 13, 10, 10, Yes], [7, 14, 9, 9, Yes], [8, 0, 7, 7, Yes], [8, 1, 6, 6, Yes], [8, 2, 5, 5, Yes], [8, 3, 4, 4, Yes], [8, 4, 3, 3, Yes], [8, 5, 2, 2, Yes], [8, 6, 1, 1, Yes], [8, 7, 0, 0, Yes], [8, 8, 14, 14, Yes], [8, 9, 13, 13, Yes], [8, 10, 12, 12, Yes], [8, 11, 11, 11, Yes], [8, 12, 10, 10, Yes], [8, 13, 9, 9, Yes], [8, 14, 8, 8, Yes], [9, 0, 6, 6, Yes], [9, 1, 5, 5, Yes], [9, 2, 4, 4, Yes], [9, 3, 3, 3, Yes], [9, 4, 2, 2, Yes], [9, 5, 1, 1, Yes], [9, 6, 0, 0, Yes], [9, 7, 14, 14, Yes], [9, 8, 13, 13, Yes], [9, 9, 12, 12, Yes], [9, 10, 11, 11, Yes], [9, 11, 10, 10, Yes], [9, 12, 9, 9, Yes], [9, 13, 8, 8, Yes], [9, 14, 7, 7, Yes], [10, 0, 5, 5, Yes], [10, 1, 4, 4, Yes], [10, 2, 3, 3, Yes], [10, 3, 2, 2, Yes], [10, 4, 1, 1, Yes], [10, 5, 0, 0, Yes], [10, 6, 14, 14, Yes], [10, 7, 13, 13, Yes], [10, 8, 12, 12, Yes], [10, 9, 11, 11, Yes], [10, 10, 10, 10, Yes], [10, 11, 9, 9, Yes], [10, 12, 8, 8, Yes], [10, 13, 7, 7, Yes], [10, 14, 6, 6, Yes], [11, 0, 4, 4, Yes], [11, 1, 3, 3, Yes], [11, 2, 2, 2, Yes], [11, 3, 1, 1, Yes], [11, 4, 0, 0, Yes], [11, 5, 14, 14, Yes], [11, 6, 13, 13, Yes], [11, 7, 12, 12, Yes], [11, 8, 11, 11, Yes], [11, 9, 10, 10, Yes], [11, 10, 9, 9, Yes], [11, 11, 8, 8, Yes], [11, 12, 7, 7, Yes], [11, 13, 6, 6, Yes], [11, 14, 5, 5, Yes], [12, 0, 3, 3, Yes], [12, 1, 2, 2, Yes], [12, 2, 1, 1, Yes], [12, 3, 0, 0, Yes], [12, 4, 14, 14, Yes], [12, 5, 13, 13, Yes], [12, 6, 12, 12, Yes], [12, 7, 11, 11, Yes], [12, 8, 10, 10, Yes], [12, 9, 9, 9, Yes], [12, 10, 8, 8, Yes], [12, 11, 7, 7, Yes], [12, 12, 6, 6, Yes], [12, 13, 5, 5, Yes], [12, 14, 4, 4, Yes], [13, 0, 2, 2, Yes], [13, 1, 1, 1, Yes], [13, 2, 0, 0, Yes], [13, 3, 14, 14, Yes], [13, 4, 13, 13, Yes], [13, 5, 12, 12, Yes], [13, 6, 11, 11, Yes], [13, 7, 10, 10, Yes], [13, 8, 9, 9, Yes], [13, 9, 8, 8, Yes], [13, 10, 7, 7, Yes], [13, 11, 6, 6, Yes], [13, 12, 5, 5, Yes], [13, 13, 4, 4, Yes], [13, 14, 3, 3, Yes], [14, 0, 1, 1, Yes], [14, 1, 0, 0, Yes], [14, 2, 14, 14, Yes], [14, 3, 13, 13, Yes], [14, 4, 12, 12, Yes], [14, 5, 11, 11, Yes], [14, 6, 10, 10, Yes], [14, 7, 9, 9, Yes], [14, 8, 8, 8, Yes], [14, 9, 7, 7, Yes], [14, 10, 6, 6, Yes], [14, 11, 5, 5, Yes], [14, 12, 4, 4, Yes], [14, 13, 3, 3, Yes], [14, 14, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)", "explanation": "List [x,y,f(x*y),f(x)*f(y),equal] for every source pair. Report the complete homomorphism flag and first failed pair or null. A failed total map blocks normal-kernel, fiber and injectivity theorems; do not silently certify its identity preimage as a normal subgroup. The exact result, retaining the requested order, is (15, 15, [0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 14, 14, Yes], [0, 2, 13, 13, Yes], [0, 3, 12, 12, Yes], [0, 4, 11, 11, Yes], [0, 5, 10, 10, Yes], [0, 6, 9, 9, Yes], [0, 7, 8, 8, Yes], [0, 8, 7, 7, Yes], [0, 9, 6, 6, Yes], [0, 10, 5, 5, Yes], [0, 11, 4, 4, Yes], [0, 12, 3, 3, Yes], [0, 13, 2, 2, Yes], [0, 14, 1, 1, Yes], [1, 0, 14, 14, Yes], [1, 1, 13, 13, Yes], [1, 2, 12, 12, Yes], [1, 3, 11, 11, Yes], [1, 4, 10, 10, Yes], [1, 5, 9, 9, Yes], [1, 6, 8, 8, Yes], [1, 7, 7, 7, Yes], [1, 8, 6, 6, Yes], [1, 9, 5, 5, Yes], [1, 10, 4, 4, Yes], [1, 11, 3, 3, Yes], [1, 12, 2, 2, Yes], [1, 13, 1, 1, Yes], [1, 14, 0, 0, Yes], [2, 0, 13, 13, Yes], [2, 1, 12, 12, Yes], [2, 2, 11, 11, Yes], [2, 3, 10, 10, Yes], [2, 4, 9, 9, Yes], [2, 5, 8, 8, Yes], [2, 6, 7, 7, Yes], [2, 7, 6, 6, Yes], [2, 8, 5, 5, Yes], [2, 9, 4, 4, Yes], [2, 10, 3, 3, Yes], [2, 11, 2, 2, Yes], [2, 12, 1, 1, Yes], [2, 13, 0, 0, Yes], [2, 14, 14, 14, Yes], [3, 0, 12, 12, Yes], [3, 1, 11, 11, Yes], [3, 2, 10, 10, Yes], [3, 3, 9, 9, Yes], [3, 4, 8, 8, Yes], [3, 5, 7, 7, Yes], [3, 6, 6, 6, Yes], [3, 7, 5, 5, Yes], [3, 8, 4, 4, Yes], [3, 9, 3, 3, Yes], [3, 10, 2, 2, Yes], [3, 11, 1, 1, Yes], [3, 12, 0, 0, Yes], [3, 13, 14, 14, Yes], [3, 14, 13, 13, Yes], [4, 0, 11, 11, Yes], [4, 1, 10, 10, Yes], [4, 2, 9, 9, Yes], [4, 3, 8, 8, Yes], [4, 4, 7, 7, Yes], [4, 5, 6, 6, Yes], [4, 6, 5, 5, Yes], [4, 7, 4, 4, Yes], [4, 8, 3, 3, Yes], [4, 9, 2, 2, Yes], [4, 10, 1, 1, Yes], [4, 11, 0, 0, Yes], [4, 12, 14, 14, Yes], [4, 13, 13, 13, Yes], [4, 14, 12, 12, Yes], [5, 0, 10, 10, Yes], [5, 1, 9, 9, Yes], [5, 2, 8, 8, Yes], [5, 3, 7, 7, Yes], [5, 4, 6, 6, Yes], [5, 5, 5, 5, Yes], [5, 6, 4, 4, Yes], [5, 7, 3, 3, Yes], [5, 8, 2, 2, Yes], [5, 9, 1, 1, Yes], [5, 10, 0, 0, Yes], [5, 11, 14, 14, Yes], [5, 12, 13, 13, Yes], [5, 13, 12, 12, Yes], [5, 14, 11, 11, Yes], [6, 0, 9, 9, Yes], [6, 1, 8, 8, Yes], [6, 2, 7, 7, Yes], [6, 3, 6, 6, Yes], [6, 4, 5, 5, Yes], [6, 5, 4, 4, Yes], [6, 6, 3, 3, Yes], [6, 7, 2, 2, Yes], [6, 8, 1, 1, Yes], [6, 9, 0, 0, Yes], [6, 10, 14, 14, Yes], [6, 11, 13, 13, Yes], [6, 12, 12, 12, Yes], [6, 13, 11, 11, Yes], [6, 14, 10, 10, Yes], [7, 0, 8, 8, Yes], [7, 1, 7, 7, Yes], [7, 2, 6, 6, Yes], [7, 3, 5, 5, Yes], [7, 4, 4, 4, Yes], [7, 5, 3, 3, Yes], [7, 6, 2, 2, Yes], [7, 7, 1, 1, Yes], [7, 8, 0, 0, Yes], [7, 9, 14, 14, Yes], [7, 10, 13, 13, Yes], [7, 11, 12, 12, Yes], [7, 12, 11, 11, Yes], [7, 13, 10, 10, Yes], [7, 14, 9, 9, Yes], [8, 0, 7, 7, Yes], [8, 1, 6, 6, Yes], [8, 2, 5, 5, Yes], [8, 3, 4, 4, Yes], [8, 4, 3, 3, Yes], [8, 5, 2, 2, Yes], [8, 6, 1, 1, Yes], [8, 7, 0, 0, Yes], [8, 8, 14, 14, Yes], [8, 9, 13, 13, Yes], [8, 10, 12, 12, Yes], [8, 11, 11, 11, Yes], [8, 12, 10, 10, Yes], [8, 13, 9, 9, Yes], [8, 14, 8, 8, Yes], [9, 0, 6, 6, Yes], [9, 1, 5, 5, Yes], [9, 2, 4, 4, Yes], [9, 3, 3, 3, Yes], [9, 4, 2, 2, Yes], [9, 5, 1, 1, Yes], [9, 6, 0, 0, Yes], [9, 7, 14, 14, Yes], [9, 8, 13, 13, Yes], [9, 9, 12, 12, Yes], [9, 10, 11, 11, Yes], [9, 11, 10, 10, Yes], [9, 12, 9, 9, Yes], [9, 13, 8, 8, Yes], [9, 14, 7, 7, Yes], [10, 0, 5, 5, Yes], [10, 1, 4, 4, Yes], [10, 2, 3, 3, Yes], [10, 3, 2, 2, Yes], [10, 4, 1, 1, Yes], [10, 5, 0, 0, Yes], [10, 6, 14, 14, Yes], [10, 7, 13, 13, Yes], [10, 8, 12, 12, Yes], [10, 9, 11, 11, Yes], [10, 10, 10, 10, Yes], [10, 11, 9, 9, Yes], [10, 12, 8, 8, Yes], [10, 13, 7, 7, Yes], [10, 14, 6, 6, Yes], [11, 0, 4, 4, Yes], [11, 1, 3, 3, Yes], [11, 2, 2, 2, Yes], [11, 3, 1, 1, Yes], [11, 4, 0, 0, Yes], [11, 5, 14, 14, Yes], [11, 6, 13, 13, Yes], [11, 7, 12, 12, Yes], [11, 8, 11, 11, Yes], [11, 9, 10, 10, Yes], [11, 10, 9, 9, Yes], [11, 11, 8, 8, Yes], [11, 12, 7, 7, Yes], [11, 13, 6, 6, Yes], [11, 14, 5, 5, Yes], [12, 0, 3, 3, Yes], [12, 1, 2, 2, Yes], [12, 2, 1, 1, Yes], [12, 3, 0, 0, Yes], [12, 4, 14, 14, Yes], [12, 5, 13, 13, Yes], [12, 6, 12, 12, Yes], [12, 7, 11, 11, Yes], [12, 8, 10, 10, Yes], [12, 9, 9, 9, Yes], [12, 10, 8, 8, Yes], [12, 11, 7, 7, Yes], [12, 12, 6, 6, Yes], [12, 13, 5, 5, Yes], [12, 14, 4, 4, Yes], [13, 0, 2, 2, Yes], [13, 1, 1, 1, Yes], [13, 2, 0, 0, Yes], [13, 3, 14, 14, Yes], [13, 4, 13, 13, Yes], [13, 5, 12, 12, Yes], [13, 6, 11, 11, Yes], [13, 7, 10, 10, Yes], [13, 8, 9, 9, Yes], [13, 9, 8, 8, Yes], [13, 10, 7, 7, Yes], [13, 11, 6, 6, Yes], [13, 12, 5, 5, Yes], [13, 13, 4, 4, Yes], [13, 14, 3, 3, Yes], [14, 0, 1, 1, Yes], [14, 1, 0, 0, Yes], [14, 2, 14, 14, Yes], [14, 3, 13, 13, Yes], [14, 4, 12, 12, Yes], [14, 5, 11, 11, Yes], [14, 6, 10, 10, Yes], [14, 7, 9, 9, Yes], [14, 8, 8, 8, Yes], [14, 9, 7, 7, Yes], [14, 10, 6, 6, Yes], [14, 11, 5, 5, Yes], [14, 12, 4, 4, Yes], [14, 13, 3, 3, Yes], [14, 14, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)." }, "skills": [ "Qualify Both Groups and Every Product Pair before Applying Kernel Theorems" ], "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review." } }, "givens": [ { "id": "G001", "text": { "zh-Hant": "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\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,0,1,2,3,4,5,6,7,8,9,10,11,12,13]],\"source_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,0,1,2,3,4,5,6,7,8,9,10,11,12,13]],\"target_identity\":0,\"mapping\":[0,14,13,12,11,10,9,8,7,6,5,4,3,2,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]], target identity=0, and total mapping=[0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (source_order, target_order, mapping, product_law_rows, homomorphism, first_failed_pair_or_null, certificate)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [kernel-homomorphism-qualification]", "after": "derived = [[source_order, target_order, mapping, product_law_rows, homomorphism, first_failed_pair_or_null, certificate]]", "relation": "implies", "rule_id": "source_ordered_fields_without_JS", "conditions": [], "explanation": { "zh-Hant": "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。", "en": "List the full source field order without JavaScript,keeping original data,Boolean flags,mathematical objects and nonclaims distinct." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "source roles = [Kernel ORIGINAL SOURCE group, 64, [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]], 0]", "after": "derived = [15, [[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]], 是]", "relation": "implies", "rule_id": "Group_complete_Cayley_signature_actual_identity", "conditions": [], "explanation": { "zh-Hant": "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。", "en": "Validate the complete closed n-by-n table and actual identity label,not an assumed0.Mod44 allows32 and mod45 allows64;do not mix carriers or tables." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "source roles = [Kernel ORIGINAL SOURCE group, 15, FOR EVERY x,y,z in0..n-1]", "after": "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]], 3375, 是]", "relation": "implies", "rule_id": "Group_ALL_associativity_triples_compact_complete_pair_matrix", "conditions": [], "explanation": { "zh-Hant": "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。", "en": "Entry(x,y) checks (xy)z=x(yz) for every z,covering all n^3 triples,not samples.The complete table is already displayed,so do not print millions of redundant lookups." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "source roles = [Kernel ORIGINAL SOURCE group, 0]", "after": "derived = [[0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0], [1, 14, 0, 0], [2, 13, 0, 0], [3, 12, 0, 0], [4, 11, 0, 0], [5, 10, 0, 0], [6, 9, 0, 0], [7, 8, 0, 0], [8, 7, 0, 0], [9, 6, 0, 0], [10, 5, 0, 0], [11, 4, 0, 0], [12, 3, 0, 0], [13, 2, 0, 0], [14, 1, 0, 0]], 是]", "relation": "implies", "rule_id": "Group_ALL_two_sided_inverses", "conditions": [], "explanation": { "zh-Hant": "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。", "en": "Every left and right inverse must return the same actual identity;associativity gives uniqueness.This completes the group premises." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "source roles = [Kernel ORIGINAL TARGET group, 64, [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]], 0]", "after": "derived = [15, [[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]], 是]", "relation": "implies", "rule_id": "Group_complete_Cayley_signature_actual_identity", "conditions": [], "explanation": { "zh-Hant": "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。", "en": "Validate the complete closed n-by-n table and actual identity label,not an assumed0.Mod44 allows32 and mod45 allows64;do not mix carriers or tables." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "source roles = [Kernel ORIGINAL TARGET group, 15, FOR EVERY x,y,z in0..n-1]", "after": "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]], 3375, 是]", "relation": "implies", "rule_id": "Group_ALL_associativity_triples_compact_complete_pair_matrix", "conditions": [], "explanation": { "zh-Hant": "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。", "en": "Entry(x,y) checks (xy)z=x(yz) for every z,covering all n^3 triples,not samples.The complete table is already displayed,so do not print millions of redundant lookups." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "source roles = [Kernel ORIGINAL TARGET group, 0]", "after": "derived = [[0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0], [1, 14, 0, 0], [2, 13, 0, 0], [3, 12, 0, 0], [4, 11, 0, 0], [5, 10, 0, 0], [6, 9, 0, 0], [7, 8, 0, 0], [8, 7, 0, 0], [9, 6, 0, 0], [10, 5, 0, 0], [11, 4, 0, 0], [12, 3, 0, 0], [13, 2, 0, 0], [14, 1, 0, 0]], 是]", "relation": "implies", "rule_id": "Group_ALL_two_sided_inverses", "conditions": [], "explanation": { "zh-Hant": "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。", "en": "Every left and right inverse must return the same actual identity;associativity gives uniqueness.This completes the group premises." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "source roles = [[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]], [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]], [0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]]", "after": "derived = [[[0, 0, 0, 0, 是], [0, 1, 14, 14, 是], [0, 2, 13, 13, 是], [0, 3, 12, 12, 是], [0, 4, 11, 11, 是], [0, 5, 10, 10, 是], [0, 6, 9, 9, 是], [0, 7, 8, 8, 是], [0, 8, 7, 7, 是], [0, 9, 6, 6, 是], [0, 10, 5, 5, 是], [0, 11, 4, 4, 是], [0, 12, 3, 3, 是], [0, 13, 2, 2, 是], [0, 14, 1, 1, 是], [1, 0, 14, 14, 是], [1, 1, 13, 13, 是], [1, 2, 12, 12, 是], [1, 3, 11, 11, 是], [1, 4, 10, 10, 是], [1, 5, 9, 9, 是], [1, 6, 8, 8, 是], [1, 7, 7, 7, 是], [1, 8, 6, 6, 是], [1, 9, 5, 5, 是], [1, 10, 4, 4, 是], [1, 11, 3, 3, 是], [1, 12, 2, 2, 是], [1, 13, 1, 1, 是], [1, 14, 0, 0, 是], [2, 0, 13, 13, 是], [2, 1, 12, 12, 是], [2, 2, 11, 11, 是], [2, 3, 10, 10, 是], [2, 4, 9, 9, 是], [2, 5, 8, 8, 是], [2, 6, 7, 7, 是], [2, 7, 6, 6, 是], [2, 8, 5, 5, 是], [2, 9, 4, 4, 是], [2, 10, 3, 3, 是], [2, 11, 2, 2, 是], [2, 12, 1, 1, 是], [2, 13, 0, 0, 是], [2, 14, 14, 14, 是], [3, 0, 12, 12, 是], [3, 1, 11, 11, 是], [3, 2, 10, 10, 是], [3, 3, 9, 9, 是], [3, 4, 8, 8, 是], [3, 5, 7, 7, 是], [3, 6, 6, 6, 是], [3, 7, 5, 5, 是], [3, 8, 4, 4, 是], [3, 9, 3, 3, 是], [3, 10, 2, 2, 是], [3, 11, 1, 1, 是], [3, 12, 0, 0, 是], [3, 13, 14, 14, 是], [3, 14, 13, 13, 是], [4, 0, 11, 11, 是], [4, 1, 10, 10, 是], [4, 2, 9, 9, 是], [4, 3, 8, 8, 是], [4, 4, 7, 7, 是], [4, 5, 6, 6, 是], [4, 6, 5, 5, 是], [4, 7, 4, 4, 是], [4, 8, 3, 3, 是], [4, 9, 2, 2, 是], [4, 10, 1, 1, 是], [4, 11, 0, 0, 是], [4, 12, 14, 14, 是], [4, 13, 13, 13, 是], [4, 14, 12, 12, 是], [5, 0, 10, 10, 是], [5, 1, 9, 9, 是], [5, 2, 8, 8, 是], [5, 3, 7, 7, 是], [5, 4, 6, 6, 是], [5, 5, 5, 5, 是], [5, 6, 4, 4, 是], [5, 7, 3, 3, 是], [5, 8, 2, 2, 是], [5, 9, 1, 1, 是], [5, 10, 0, 0, 是], [5, 11, 14, 14, 是], [5, 12, 13, 13, 是], [5, 13, 12, 12, 是], [5, 14, 11, 11, 是], [6, 0, 9, 9, 是], [6, 1, 8, 8, 是], [6, 2, 7, 7, 是], [6, 3, 6, 6, 是], [6, 4, 5, 5, 是], [6, 5, 4, 4, 是], [6, 6, 3, 3, 是], [6, 7, 2, 2, 是], [6, 8, 1, 1, 是], [6, 9, 0, 0, 是], [6, 10, 14, 14, 是], [6, 11, 13, 13, 是], [6, 12, 12, 12, 是], [6, 13, 11, 11, 是], [6, 14, 10, 10, 是], [7, 0, 8, 8, 是], [7, 1, 7, 7, 是], [7, 2, 6, 6, 是], [7, 3, 5, 5, 是], [7, 4, 4, 4, 是], [7, 5, 3, 3, 是], [7, 6, 2, 2, 是], [7, 7, 1, 1, 是], [7, 8, 0, 0, 是], [7, 9, 14, 14, 是], [7, 10, 13, 13, 是], [7, 11, 12, 12, 是], [7, 12, 11, 11, 是], [7, 13, 10, 10, 是], [7, 14, 9, 9, 是], [8, 0, 7, 7, 是], [8, 1, 6, 6, 是], [8, 2, 5, 5, 是], [8, 3, 4, 4, 是], [8, 4, 3, 3, 是], [8, 5, 2, 2, 是], [8, 6, 1, 1, 是], [8, 7, 0, 0, 是], [8, 8, 14, 14, 是], [8, 9, 13, 13, 是], [8, 10, 12, 12, 是], [8, 11, 11, 11, 是], [8, 12, 10, 10, 是], [8, 13, 9, 9, 是], [8, 14, 8, 8, 是], [9, 0, 6, 6, 是], [9, 1, 5, 5, 是], [9, 2, 4, 4, 是], [9, 3, 3, 3, 是], [9, 4, 2, 2, 是], [9, 5, 1, 1, 是], [9, 6, 0, 0, 是], [9, 7, 14, 14, 是], [9, 8, 13, 13, 是], [9, 9, 12, 12, 是], [9, 10, 11, 11, 是], [9, 11, 10, 10, 是], [9, 12, 9, 9, 是], [9, 13, 8, 8, 是], [9, 14, 7, 7, 是], [10, 0, 5, 5, 是], [10, 1, 4, 4, 是], [10, 2, 3, 3, 是], [10, 3, 2, 2, 是], [10, 4, 1, 1, 是], [10, 5, 0, 0, 是], [10, 6, 14, 14, 是], [10, 7, 13, 13, 是], [10, 8, 12, 12, 是], [10, 9, 11, 11, 是], [10, 10, 10, 10, 是], [10, 11, 9, 9, 是], [10, 12, 8, 8, 是], [10, 13, 7, 7, 是], [10, 14, 6, 6, 是], [11, 0, 4, 4, 是], [11, 1, 3, 3, 是], [11, 2, 2, 2, 是], [11, 3, 1, 1, 是], [11, 4, 0, 0, 是], [11, 5, 14, 14, 是], [11, 6, 13, 13, 是], [11, 7, 12, 12, 是], [11, 8, 11, 11, 是], [11, 9, 10, 10, 是], [11, 10, 9, 9, 是], [11, 11, 8, 8, 是], [11, 12, 7, 7, 是], [11, 13, 6, 6, 是], [11, 14, 5, 5, 是], [12, 0, 3, 3, 是], [12, 1, 2, 2, 是], [12, 2, 1, 1, 是], [12, 3, 0, 0, 是], [12, 4, 14, 14, 是], [12, 5, 13, 13, 是], [12, 6, 12, 12, 是], [12, 7, 11, 11, 是], [12, 8, 10, 10, 是], [12, 9, 9, 9, 是], [12, 10, 8, 8, 是], [12, 11, 7, 7, 是], [12, 12, 6, 6, 是], [12, 13, 5, 5, 是], [12, 14, 4, 4, 是], [13, 0, 2, 2, 是], [13, 1, 1, 1, 是], [13, 2, 0, 0, 是], [13, 3, 14, 14, 是], [13, 4, 13, 13, 是], [13, 5, 12, 12, 是], [13, 6, 11, 11, 是], [13, 7, 10, 10, 是], [13, 8, 9, 9, 是], [13, 9, 8, 8, 是], [13, 10, 7, 7, 是], [13, 11, 6, 6, 是], [13, 12, 5, 5, 是], [13, 13, 4, 4, 是], [13, 14, 3, 3, 是], [14, 0, 1, 1, 是], [14, 1, 0, 0, 是], [14, 2, 14, 14, 是], [14, 3, 13, 13, 是], [14, 4, 12, 12, 是], [14, 5, 11, 11, 是], [14, 6, 10, 10, 是], [14, 7, 9, 9, 是], [14, 8, 8, 8, 是], [14, 9, 7, 7, 是], [14, 10, 6, 6, 是], [14, 11, 5, 5, 是], [14, 12, 4, 4, 是], [14, 13, 3, 3, 是], [14, 14, 2, 2, 是]], 是, null]", "relation": "implies", "rule_id": "Kernel_complete_TOTAL_map_ALL_product_pairs_before_theorems", "conditions": [], "explanation": { "zh-Hant": "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。", "en": "For these same source and target groups,check f(xy)=f(x)f(y) for every pair and retain the first failed row/null.A total map need not be a homomorphism.Failure blocks kernel theorems;an internal fallback is not the true identity preimage." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null } ], "math_model": { "result": [ 15, 15, [ 0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1 ], [ [ 0, 0, 0, 0, true ], [ 0, 1, 14, 14, true ], [ 0, 2, 13, 13, true ], [ 0, 3, 12, 12, true ], [ 0, 4, 11, 11, true ], [ 0, 5, 10, 10, true ], [ 0, 6, 9, 9, true ], [ 0, 7, 8, 8, true ], [ 0, 8, 7, 7, true ], [ 0, 9, 6, 6, true ], [ 0, 10, 5, 5, true ], [ 0, 11, 4, 4, true ], [ 0, 12, 3, 3, true ], [ 0, 13, 2, 2, true ], [ 0, 14, 1, 1, true ], [ 1, 0, 14, 14, true ], [ 1, 1, 13, 13, true ], [ 1, 2, 12, 12, true ], [ 1, 3, 11, 11, true ], [ 1, 4, 10, 10, true ], [ 1, 5, 9, 9, true ], [ 1, 6, 8, 8, true ], [ 1, 7, 7, 7, true ], [ 1, 8, 6, 6, true ], [ 1, 9, 5, 5, true ], [ 1, 10, 4, 4, true ], [ 1, 11, 3, 3, true ], [ 1, 12, 2, 2, true ], [ 1, 13, 1, 1, true ], [ 1, 14, 0, 0, true ], [ 2, 0, 13, 13, true ], [ 2, 1, 12, 12, true ], [ 2, 2, 11, 11, true ], [ 2, 3, 10, 10, true ], [ 2, 4, 9, 9, true ], [ 2, 5, 8, 8, true ], [ 2, 6, 7, 7, true ], [ 2, 7, 6, 6, true ], [ 2, 8, 5, 5, true ], [ 2, 9, 4, 4, true ], [ 2, 10, 3, 3, true ], [ 2, 11, 2, 2, true ], [ 2, 12, 1, 1, true ], [ 2, 13, 0, 0, true ], [ 2, 14, 14, 14, true ], [ 3, 0, 12, 12, true ], [ 3, 1, 11, 11, true ], [ 3, 2, 10, 10, true ], [ 3, 3, 9, 9, true ], [ 3, 4, 8, 8, true ], [ 3, 5, 7, 7, true ], [ 3, 6, 6, 6, true ], [ 3, 7, 5, 5, true ], [ 3, 8, 4, 4, true ], [ 3, 9, 3, 3, true ], [ 3, 10, 2, 2, true ], [ 3, 11, 1, 1, true ], [ 3, 12, 0, 0, true ], [ 3, 13, 14, 14, true ], [ 3, 14, 13, 13, true ], [ 4, 0, 11, 11, true ], [ 4, 1, 10, 10, true ], [ 4, 2, 9, 9, true ], [ 4, 3, 8, 8, true ], [ 4, 4, 7, 7, true ], [ 4, 5, 6, 6, true ], [ 4, 6, 5, 5, true ], [ 4, 7, 4, 4, true ], [ 4, 8, 3, 3, true ], [ 4, 9, 2, 2, true ], [ 4, 10, 1, 1, true ], [ 4, 11, 0, 0, true ], [ 4, 12, 14, 14, true ], [ 4, 13, 13, 13, true ], [ 4, 14, 12, 12, true ], [ 5, 0, 10, 10, true ], [ 5, 1, 9, 9, true ], [ 5, 2, 8, 8, true ], [ 5, 3, 7, 7, true ], [ 5, 4, 6, 6, true ], [ 5, 5, 5, 5, true ], [ 5, 6, 4, 4, true ], [ 5, 7, 3, 3, true ], [ 5, 8, 2, 2, true ], [ 5, 9, 1, 1, true ], [ 5, 10, 0, 0, true ], [ 5, 11, 14, 14, true ], [ 5, 12, 13, 13, true ], [ 5, 13, 12, 12, true ], [ 5, 14, 11, 11, true ], [ 6, 0, 9, 9, true ], [ 6, 1, 8, 8, true ], [ 6, 2, 7, 7, true ], [ 6, 3, 6, 6, true ], [ 6, 4, 5, 5, true ], [ 6, 5, 4, 4, true ], [ 6, 6, 3, 3, true ], [ 6, 7, 2, 2, true ], [ 6, 8, 1, 1, true ], [ 6, 9, 0, 0, true ], [ 6, 10, 14, 14, true ], [ 6, 11, 13, 13, true ], [ 6, 12, 12, 12, true ], [ 6, 13, 11, 11, true ], [ 6, 14, 10, 10, true ], [ 7, 0, 8, 8, true ], [ 7, 1, 7, 7, true ], [ 7, 2, 6, 6, true ], [ 7, 3, 5, 5, true ], [ 7, 4, 4, 4, true ], [ 7, 5, 3, 3, true ], [ 7, 6, 2, 2, true ], [ 7, 7, 1, 1, true ], [ 7, 8, 0, 0, true ], [ 7, 9, 14, 14, true ], [ 7, 10, 13, 13, true ], [ 7, 11, 12, 12, true ], [ 7, 12, 11, 11, true ], [ 7, 13, 10, 10, true ], [ 7, 14, 9, 9, true ], [ 8, 0, 7, 7, true ], [ 8, 1, 6, 6, true ], [ 8, 2, 5, 5, true ], [ 8, 3, 4, 4, true ], [ 8, 4, 3, 3, true ], [ 8, 5, 2, 2, true ], [ 8, 6, 1, 1, true ], [ 8, 7, 0, 0, true ], [ 8, 8, 14, 14, true ], [ 8, 9, 13, 13, true ], [ 8, 10, 12, 12, true ], [ 8, 11, 11, 11, true ], [ 8, 12, 10, 10, true ], [ 8, 13, 9, 9, true ], [ 8, 14, 8, 8, true ], [ 9, 0, 6, 6, true ], [ 9, 1, 5, 5, true ], [ 9, 2, 4, 4, true ], [ 9, 3, 3, 3, true ], [ 9, 4, 2, 2, true ], [ 9, 5, 1, 1, true ], [ 9, 6, 0, 0, true ], [ 9, 7, 14, 14, true ], [ 9, 8, 13, 13, true ], [ 9, 9, 12, 12, true ], [ 9, 10, 11, 11, true ], [ 9, 11, 10, 10, true ], [ 9, 12, 9, 9, true ], [ 9, 13, 8, 8, true ], [ 9, 14, 7, 7, true ], [ 10, 0, 5, 5, true ], [ 10, 1, 4, 4, true ], [ 10, 2, 3, 3, true ], [ 10, 3, 2, 2, true ], [ 10, 4, 1, 1, true ], [ 10, 5, 0, 0, true ], [ 10, 6, 14, 14, true ], [ 10, 7, 13, 13, true ], [ 10, 8, 12, 12, true ], [ 10, 9, 11, 11, true ], [ 10, 10, 10, 10, true ], [ 10, 11, 9, 9, true ], [ 10, 12, 8, 8, true ], [ 10, 13, 7, 7, true ], [ 10, 14, 6, 6, true ], [ 11, 0, 4, 4, true ], [ 11, 1, 3, 3, true ], [ 11, 2, 2, 2, true ], [ 11, 3, 1, 1, true ], [ 11, 4, 0, 0, true ], [ 11, 5, 14, 14, true ], [ 11, 6, 13, 13, true ], [ 11, 7, 12, 12, true ], [ 11, 8, 11, 11, true ], [ 11, 9, 10, 10, true ], [ 11, 10, 9, 9, true ], [ 11, 11, 8, 8, true ], [ 11, 12, 7, 7, true ], [ 11, 13, 6, 6, true ], [ 11, 14, 5, 5, true ], [ 12, 0, 3, 3, true ], [ 12, 1, 2, 2, true ], [ 12, 2, 1, 1, true ], [ 12, 3, 0, 0, true ], [ 12, 4, 14, 14, true ], [ 12, 5, 13, 13, true ], [ 12, 6, 12, 12, true ], [ 12, 7, 11, 11, true ], [ 12, 8, 10, 10, true ], [ 12, 9, 9, 9, true ], [ 12, 10, 8, 8, true ], [ 12, 11, 7, 7, true ], [ 12, 12, 6, 6, true ], [ 12, 13, 5, 5, true ], [ 12, 14, 4, 4, true ], [ 13, 0, 2, 2, true ], [ 13, 1, 1, 1, true ], [ 13, 2, 0, 0, true ], [ 13, 3, 14, 14, true ], [ 13, 4, 13, 13, true ], [ 13, 5, 12, 12, true ], [ 13, 6, 11, 11, true ], [ 13, 7, 10, 10, true ], [ 13, 8, 9, 9, true ], [ 13, 9, 8, 8, true ], [ 13, 10, 7, 7, true ], [ 13, 11, 6, 6, true ], [ 13, 12, 5, 5, true ], [ 13, 13, 4, 4, true ], [ 13, 14, 3, 3, true ], [ 14, 0, 1, 1, true ], [ 14, 1, 0, 0, true ], [ 14, 2, 14, 14, true ], [ 14, 3, 13, 13, true ], [ 14, 4, 12, 12, true ], [ 14, 5, 11, 11, true ], [ 14, 6, 10, 10, true ], [ 14, 7, 9, 9, true ], [ 14, 8, 8, 8, true ], [ 14, 9, 7, 7, true ], [ 14, 10, 6, 6, true ], [ 14, 11, 5, 5, true ], [ 14, 12, 4, 4, true ], [ 14, 13, 3, 3, true ], [ 14, 14, 2, 2, true ] ], true, null, "complete_total_map_satisfies_every_product_pair" ] }, "conclusion": { "canonical_value": "(15, 15, [0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, 是], [0, 1, 14, 14, 是], [0, 2, 13, 13, 是], [0, 3, 12, 12, 是], [0, 4, 11, 11, 是], [0, 5, 10, 10, 是], [0, 6, 9, 9, 是], [0, 7, 8, 8, 是], [0, 8, 7, 7, 是], [0, 9, 6, 6, 是], [0, 10, 5, 5, 是], [0, 11, 4, 4, 是], [0, 12, 3, 3, 是], [0, 13, 2, 2, 是], [0, 14, 1, 1, 是], [1, 0, 14, 14, 是], [1, 1, 13, 13, 是], [1, 2, 12, 12, 是], [1, 3, 11, 11, 是], [1, 4, 10, 10, 是], [1, 5, 9, 9, 是], [1, 6, 8, 8, 是], [1, 7, 7, 7, 是], [1, 8, 6, 6, 是], [1, 9, 5, 5, 是], [1, 10, 4, 4, 是], [1, 11, 3, 3, 是], [1, 12, 2, 2, 是], [1, 13, 1, 1, 是], [1, 14, 0, 0, 是], [2, 0, 13, 13, 是], [2, 1, 12, 12, 是], [2, 2, 11, 11, 是], [2, 3, 10, 10, 是], [2, 4, 9, 9, 是], [2, 5, 8, 8, 是], [2, 6, 7, 7, 是], [2, 7, 6, 6, 是], [2, 8, 5, 5, 是], [2, 9, 4, 4, 是], [2, 10, 3, 3, 是], [2, 11, 2, 2, 是], [2, 12, 1, 1, 是], [2, 13, 0, 0, 是], [2, 14, 14, 14, 是], [3, 0, 12, 12, 是], [3, 1, 11, 11, 是], [3, 2, 10, 10, 是], [3, 3, 9, 9, 是], [3, 4, 8, 8, 是], [3, 5, 7, 7, 是], [3, 6, 6, 6, 是], [3, 7, 5, 5, 是], [3, 8, 4, 4, 是], [3, 9, 3, 3, 是], [3, 10, 2, 2, 是], [3, 11, 1, 1, 是], [3, 12, 0, 0, 是], [3, 13, 14, 14, 是], [3, 14, 13, 13, 是], [4, 0, 11, 11, 是], [4, 1, 10, 10, 是], [4, 2, 9, 9, 是], [4, 3, 8, 8, 是], [4, 4, 7, 7, 是], [4, 5, 6, 6, 是], [4, 6, 5, 5, 是], [4, 7, 4, 4, 是], [4, 8, 3, 3, 是], [4, 9, 2, 2, 是], [4, 10, 1, 1, 是], [4, 11, 0, 0, 是], [4, 12, 14, 14, 是], [4, 13, 13, 13, 是], [4, 14, 12, 12, 是], [5, 0, 10, 10, 是], [5, 1, 9, 9, 是], [5, 2, 8, 8, 是], [5, 3, 7, 7, 是], [5, 4, 6, 6, 是], [5, 5, 5, 5, 是], [5, 6, 4, 4, 是], [5, 7, 3, 3, 是], [5, 8, 2, 2, 是], [5, 9, 1, 1, 是], [5, 10, 0, 0, 是], [5, 11, 14, 14, 是], [5, 12, 13, 13, 是], [5, 13, 12, 12, 是], [5, 14, 11, 11, 是], [6, 0, 9, 9, 是], [6, 1, 8, 8, 是], [6, 2, 7, 7, 是], [6, 3, 6, 6, 是], [6, 4, 5, 5, 是], [6, 5, 4, 4, 是], [6, 6, 3, 3, 是], [6, 7, 2, 2, 是], [6, 8, 1, 1, 是], [6, 9, 0, 0, 是], [6, 10, 14, 14, 是], [6, 11, 13, 13, 是], [6, 12, 12, 12, 是], [6, 13, 11, 11, 是], [6, 14, 10, 10, 是], [7, 0, 8, 8, 是], [7, 1, 7, 7, 是], [7, 2, 6, 6, 是], [7, 3, 5, 5, 是], [7, 4, 4, 4, 是], [7, 5, 3, 3, 是], [7, 6, 2, 2, 是], [7, 7, 1, 1, 是], [7, 8, 0, 0, 是], [7, 9, 14, 14, 是], [7, 10, 13, 13, 是], [7, 11, 12, 12, 是], [7, 12, 11, 11, 是], [7, 13, 10, 10, 是], [7, 14, 9, 9, 是], [8, 0, 7, 7, 是], [8, 1, 6, 6, 是], [8, 2, 5, 5, 是], [8, 3, 4, 4, 是], [8, 4, 3, 3, 是], [8, 5, 2, 2, 是], [8, 6, 1, 1, 是], [8, 7, 0, 0, 是], [8, 8, 14, 14, 是], [8, 9, 13, 13, 是], [8, 10, 12, 12, 是], [8, 11, 11, 11, 是], [8, 12, 10, 10, 是], [8, 13, 9, 9, 是], [8, 14, 8, 8, 是], [9, 0, 6, 6, 是], [9, 1, 5, 5, 是], [9, 2, 4, 4, 是], [9, 3, 3, 3, 是], [9, 4, 2, 2, 是], [9, 5, 1, 1, 是], [9, 6, 0, 0, 是], [9, 7, 14, 14, 是], [9, 8, 13, 13, 是], [9, 9, 12, 12, 是], [9, 10, 11, 11, 是], [9, 11, 10, 10, 是], [9, 12, 9, 9, 是], [9, 13, 8, 8, 是], [9, 14, 7, 7, 是], [10, 0, 5, 5, 是], [10, 1, 4, 4, 是], [10, 2, 3, 3, 是], [10, 3, 2, 2, 是], [10, 4, 1, 1, 是], [10, 5, 0, 0, 是], [10, 6, 14, 14, 是], [10, 7, 13, 13, 是], [10, 8, 12, 12, 是], [10, 9, 11, 11, 是], [10, 10, 10, 10, 是], [10, 11, 9, 9, 是], [10, 12, 8, 8, 是], [10, 13, 7, 7, 是], [10, 14, 6, 6, 是], [11, 0, 4, 4, 是], [11, 1, 3, 3, 是], [11, 2, 2, 2, 是], [11, 3, 1, 1, 是], [11, 4, 0, 0, 是], [11, 5, 14, 14, 是], [11, 6, 13, 13, 是], [11, 7, 12, 12, 是], [11, 8, 11, 11, 是], [11, 9, 10, 10, 是], [11, 10, 9, 9, 是], [11, 11, 8, 8, 是], [11, 12, 7, 7, 是], [11, 13, 6, 6, 是], [11, 14, 5, 5, 是], [12, 0, 3, 3, 是], [12, 1, 2, 2, 是], [12, 2, 1, 1, 是], [12, 3, 0, 0, 是], [12, 4, 14, 14, 是], [12, 5, 13, 13, 是], [12, 6, 12, 12, 是], [12, 7, 11, 11, 是], [12, 8, 10, 10, 是], [12, 9, 9, 9, 是], [12, 10, 8, 8, 是], [12, 11, 7, 7, 是], [12, 12, 6, 6, 是], [12, 13, 5, 5, 是], [12, 14, 4, 4, 是], [13, 0, 2, 2, 是], [13, 1, 1, 1, 是], [13, 2, 0, 0, 是], [13, 3, 14, 14, 是], [13, 4, 13, 13, 是], [13, 5, 12, 12, 是], [13, 6, 11, 11, 是], [13, 7, 10, 10, 是], [13, 8, 9, 9, 是], [13, 9, 8, 8, 是], [13, 10, 7, 7, 是], [13, 11, 6, 6, 是], [13, 12, 5, 5, 是], [13, 13, 4, 4, 是], [13, 14, 3, 3, 是], [14, 0, 1, 1, 是], [14, 1, 0, 0, 是], [14, 2, 14, 14, 是], [14, 3, 13, 13, 是], [14, 4, 12, 12, 是], [14, 5, 11, 11, 是], [14, 6, 10, 10, 是], [14, 7, 9, 9, 是], [14, 8, 8, 8, 是], [14, 9, 7, 7, 是], [14, 10, 6, 6, 是], [14, 11, 5, 5, 是], [14, 12, 4, 4, 是], [14, 13, 3, 3, 是], [14, 14, 2, 2, 是]], 是, null, complete_total_map_satisfies_every_product_pair)", "explanation": { "zh-Hant": "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。", "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged." } }, "definition_note": { "zh-Hant": "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。", "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate." }, "source": { "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db", "question_binding": { "batch_id": "batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations", "batch_path": "public/data/questions/batches/batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations/questions.jsonl", "batch_sha256": "3f7d62a92a3b7dc4115370810169c8da49b4e2bf4b448b54efee63c57f32120d", "original_line": 13, "original_line_sha256": "b7958beaf2b8a7b8552ad0103a601d397d29e2f2bcc966748426a441fb5b78a6", "question_compact_sha256": "5eaa9d545a11ecf939a939d7ab7f21bdafa6ea16d4d706c373de5b9aa6bc909c", "english_row_compact_sha256": "dc3b53a19c3491e7d5e28d570fa0d07afd0d7bbfbec231d6221a984a6a385fe7", "english_manifest_path": "public/data/i18n/editions/batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations.en.jsonl" }, "mko_bindings": { "mko-finite-group-homomorphism-kernel": { "sha256": "2d8504a53dc49db696cced0fde35020459fee128373b3d2efc52f01168374d11", "english_sha256": "4c8c7e9116861cd389dc48d302f74f3de3e520a7450771096acb9a4ec434b806", "original_path": "public/data/mko/finite-group-homomorphism-kernel.json", "english_path": "public/data/i18n/editions/mko-finite-group-homomorphism-kernel.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "7f1649723787c6d7cf1abe85636fd3740f2c3cc1784071db7e38069361c81fd4", "original_candidate_sha256": "10ff4259418010af8d1f1d629cd6d2782c40fe7c8b1f0b260276338796950685", "original_candidate_line_sha256": "10ff4259418010af8d1f1d629cd6d2782c40fe7c8b1f0b260276338796950685" }, "historical_source_state": { "review_state": { "derivation": "candidate_authored_and_locally_checked", "source_question_state": "candidate", "independent_AI_or_human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN" }, "scope_note": { "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。", "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged." }, "label": "Frozen producer metadata; current per-question review is separate below." }, "revision": { "id": "review-20261026-009-p49", "review_date": "2026-10-26", "owner_authorized_reader_publication": true, "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.", "review_receipts": { "semantic_review_sha256": "049a9e07e69fead88d93b34f0937b9470a06d71c1a8d097058fc83554529bdd0", "question_first_witness_sha256": "0c71ebb418f3de347d232d159fc25e93036f4ab9166bcfb5ee5b4be86d5c55b9" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 8, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "61b551e9e5941dca495a035c8b4f57a767ca3854dc1267ec4cdb5e1cafa5e9e7", "arithmetic_steps": 0 }, "uses_answer_as_premise": false, "human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN", "Canonical_truth_promotion": false, "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged." } }