{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0094", "pilot_number": "R0050-25744", "number": 25744, "family_id": "qf-mod46-iso1-image-codomain-boundary", "title": { "zh-Hant": "第 25744 題：有限群第一同構定理", "en": "Question 25744: First Isomorphism Reaches the Image, Not Automatically the Whole Codomain" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0094", "batch_id": "batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-first-isomorphism-theorem-finite-groups" ], "title_zh": "有限群第一同構定理", "stem_zh": "完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-image-codomain-boundary；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]],\"source_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38]],\"target_identity\":0,\"mapping\":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 40, 20, 否, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, 否, finite_instance_certificate_not_a_general_theorem_proof)", "explanation_zh": "Wave6第九層核對=([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 40, 20, 否, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, 否, finite_instance_certificate_not_a_general_theorem_proof)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。" }, "difficulty": { "band": "undergraduate_core", "level": 3, "skills": [ "有限群第一同構定理" ] }, "variant": { "family_id": "qf-mod46-iso1-image-codomain-boundary", "seed": 202611151094, "parameters": { "source_table": [ [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0 ], [ 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1 ], [ 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2 ], [ 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3 ], [ 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4 ], [ 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5 ], [ 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6 ], [ 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7 ], [ 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8 ], [ 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 ], [ 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ], [ 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 ], [ 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ], [ 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 ], [ 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 ], [ 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 ], [ 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 ], [ 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 ], [ 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 ] ], "source_identity": 0, "target_table": [ [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0 ], [ 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1 ], [ 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2 ], [ 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3 ], [ 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4 ], [ 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5 ], [ 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6 ], [ 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7 ], [ 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8 ], [ 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 ], [ 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ], [ 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 ], [ 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ], [ 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 ], [ 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 ], [ 16, 17, 18, 19, 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} }, "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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r01-first-isomorphism-theorem-finite-groups-0094", "batch_id": "batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "21e69b4196bd00839fbde6ee9eb286347da6f3fa8e97e94a9d9332b7b377d29a", "family_id": "qf-mod46-iso1-image-codomain-boundary", "seed": 202611151094, "semantic_template": 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}, "fields": { "title": "First Isomorphism Reaches the Image, Not Automatically the Whole Codomain", "stem": "Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]], identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 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2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38]], identity=0, and the aligned total mapping=[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary).", "answer": { "kind": "expression", "value": "([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 40, 20, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)", "explanation": "Compute the image and compare its size with the target group. The quotient is isomorphic to that image. Only when f is surjective does it identify the image with the codomain; finite-instance evidence does not replace the general abstract theorem proof. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 40, 20, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)." }, "skills": [ "First Isomorphism Reaches the Image, Not Automatically the Whole Codomain" ], "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": "完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-image-codomain-boundary；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]],\"source_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38]],\"target_identity\":0,\"mapping\":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38]}。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, 15, 16, 17, 18, 19], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]], identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38]], identity=0, and the aligned total mapping=[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [iso1-image-codomain-boundary]", "after": "derived = [[image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary]]", "relation": "implies", "rule_id": "source_ordered_fields_without_JS", "conditions": [], "explanation": { "zh-Hant": "直接列完整來源欄序，不需JavaScript。LP weak為13欄、zero與strong為15欄，前提與角色依原operation／mode，不補欄。", "en": "List the full source field order without JavaScript.LP weak uses13 fields,zero/strong15;premises and roles follow the original operation/mode without padding." }, "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 = [Iso1 ORIGINAL SOURCE, [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]], 0]", "after": "derived = [20, [[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]], 是]", "relation": "implies", "rule_id": "Group_complete_table_actual_identity_bound64", "conditions": [], "explanation": { "zh-Hant": "完整closed Cayley table與實際identity先資格核對，群階可到64，不假設identity0。", "en": "Qualify the complete closed Cayley table and actual identity first,allowing order64 without assuming identity0." }, "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 = [Iso1 ORIGINAL SOURCE, 20, 0]", "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, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是]], 8000, [0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]]", "relation": "implies", "rule_id": "Group_ALL_triples_compact_pair_matrix_and_ALL_inverses", "conditions": [], "explanation": { "zh-Hant": "每pair格逐全部第三元素核結合律，完整n³而非抽樣；inverse同時左右回實際identity。", "en": "Each pair-matrix entry checks every third element for associativity,covering all n³ triples,not samples.Inverses return the actual identity from both sides." }, "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 = [Iso1 ORIGINAL TARGET, [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38]], 0]", "after": "derived = [40, [[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], [22, 22, 22, 22], [23, 23, 23, 23], [24, 24, 24, 24], [25, 25, 25, 25], [26, 26, 26, 26], [27, 27, 27, 27], [28, 28, 28, 28], [29, 29, 29, 29], [30, 30, 30, 30], [31, 31, 31, 31], [32, 32, 32, 32], [33, 33, 33, 33], [34, 34, 34, 34], [35, 35, 35, 35], [36, 36, 36, 36], [37, 37, 37, 37], [38, 38, 38, 38], [39, 39, 39, 39]], 是]", "relation": "implies", "rule_id": "Group_complete_table_actual_identity_bound64", "conditions": [], "explanation": { "zh-Hant": "完整closed Cayley table與實際identity先資格核對，群階可到64，不假設identity0。", "en": "Qualify the complete closed Cayley table and actual identity first,allowing order64 without assuming identity0." }, "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 = [Iso1 ORIGINAL TARGET, 40, 0]", "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, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 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是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是]], 64000, [0, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]]", "relation": "implies", "rule_id": "Group_ALL_triples_compact_pair_matrix_and_ALL_inverses", "conditions": [], "explanation": { "zh-Hant": "每pair格逐全部第三元素核結合律，完整n³而非抽樣；inverse同時左右回實際identity。", "en": "Each pair-matrix entry checks every third element for associativity,covering all n³ triples,not samples.Inverses return the actual identity from both sides." }, "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 = [Iso1 ACTUAL kernel, [0], 0]", "after": "derived = [是, [[0, 0, 0, 是]], 是, [[0, 0, 是]], 是, 是, [[0, [0], 是], [1, [0], 是], [2, [0], 是], [3, [0], 是], [4, [0], 是], [5, [0], 是], [6, [0], 是], [7, [0], 是], [8, [0], 是], [9, [0], 是], [10, [0], 是], [11, [0], 是], [12, [0], 是], [13, [0], 是], [14, [0], 是], [15, [0], 是], [16, [0], 是], [17, [0], 是], [18, [0], 是], [19, [0], 是]], 是]", "relation": "implies", "rule_id": "Subgroup_actual_identity_product_inverse_ALL_conjugates", "conditions": [], "explanation": { "zh-Hant": "含identity、全部product與inverse closure各核，三者合取才成subgroup；再逐全部g驗normal。即使某map失敗，這些實際property仍可分別True，不由定理名稱填旗標。", "en": "Check identity,all products and inverse closure separately before subgroup qualification,then every g for normality.These are actual properties that may individually hold even when a map fails;do not fill flags from a theorem name." }, "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": 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34, 28, 是], [34, 36, 30, 是], [34, 38, 32, 是], [36, 0, 36, 是], [36, 2, 38, 是], [36, 4, 0, 是], [36, 6, 2, 是], [36, 8, 4, 是], [36, 10, 6, 是], [36, 12, 8, 是], [36, 14, 10, 是], [36, 16, 12, 是], [36, 18, 14, 是], [36, 20, 16, 是], [36, 22, 18, 是], [36, 24, 20, 是], [36, 26, 22, 是], [36, 28, 24, 是], [36, 30, 26, 是], [36, 32, 28, 是], [36, 34, 30, 是], [36, 36, 32, 是], [36, 38, 34, 是], [38, 0, 38, 是], [38, 2, 0, 是], [38, 4, 2, 是], [38, 6, 4, 是], [38, 8, 6, 是], [38, 10, 8, 是], [38, 12, 10, 是], [38, 14, 12, 是], [38, 16, 14, 是], [38, 18, 16, 是], [38, 20, 18, 是], [38, 22, 20, 是], [38, 24, 22, 是], [38, 26, 24, 是], [38, 28, 26, 是], [38, 30, 28, 是], [38, 32, 30, 是], [38, 34, 32, 是], [38, 36, 34, 是], [38, 38, 36, 是]], 是, [[0, 0, 是], [2, 38, 是], [4, 36, 是], [6, 34, 是], [8, 32, 是], [10, 30, 是], [12, 28, 是], [14, 26, 是], [16, 24, 是], [18, 22, 是], [20, 20, 是], [22, 18, 是], [24, 16, 是], [26, 14, 是], [28, 12, 是], [30, 10, 是], [32, 8, 是], [34, 6, 是], [36, 4, 是], [38, 2, 是]], 是, 是, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [34, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [35, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [36, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [37, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [38, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [39, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是]], 是]", "relation": "implies", "rule_id": "Subgroup_actual_identity_product_inverse_ALL_conjugates", "conditions": [], "explanation": { "zh-Hant": "含identity、全部product與inverse closure各核，三者合取才成subgroup；再逐全部g驗normal。即使某map失敗，這些實際property仍可分別True，不由定理名稱填旗標。", "en": "Check identity,all products and inverse closure separately before subgroup qualification,then every g for normality.These are actual properties that may individually hold even when a map fails;do not fill flags from a theorem name." }, "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, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 0, 0]", "after": "derived = [[[0, 0, 0, 0, 是], [0, 1, 2, 2, 是], [0, 2, 4, 4, 是], [0, 3, 6, 6, 是], [0, 4, 8, 8, 是], [0, 5, 10, 10, 是], [0, 6, 12, 12, 是], [0, 7, 14, 14, 是], [0, 8, 16, 16, 是], [0, 9, 18, 18, 是], [0, 10, 20, 20, 是], [0, 11, 22, 22, 是], [0, 12, 24, 24, 是], [0, 13, 26, 26, 是], [0, 14, 28, 28, 是], [0, 15, 30, 30, 是], [0, 16, 32, 32, 是], [0, 17, 34, 34, 是], [0, 18, 36, 36, 是], [0, 19, 38, 38, 是], [1, 0, 2, 2, 是], [1, 1, 4, 4, 是], [1, 2, 6, 6, 是], [1, 3, 8, 8, 是], [1, 4, 10, 10, 是], [1, 5, 12, 12, 是], [1, 6, 14, 14, 是], [1, 7, 16, 16, 是], [1, 8, 18, 18, 是], [1, 9, 20, 20, 是], [1, 10, 22, 22, 是], [1, 11, 24, 24, 是], [1, 12, 26, 26, 是], [1, 13, 28, 28, 是], [1, 14, 30, 30, 是], [1, 15, 32, 32, 是], [1, 16, 34, 34, 是], [1, 17, 36, 36, 是], [1, 18, 38, 38, 是], [1, 19, 0, 0, 是], [2, 0, 4, 4, 是], [2, 1, 6, 6, 是], [2, 2, 8, 8, 是], [2, 3, 10, 10, 是], [2, 4, 12, 12, 是], [2, 5, 14, 14, 是], [2, 6, 16, 16, 是], [2, 7, 18, 18, 是], [2, 8, 20, 20, 是], [2, 9, 22, 22, 是], [2, 10, 24, 24, 是], [2, 11, 26, 26, 是], [2, 12, 28, 28, 是], [2, 13, 30, 30, 是], [2, 14, 32, 32, 是], [2, 15, 34, 34, 是], [2, 16, 36, 36, 是], [2, 17, 38, 38, 是], [2, 18, 0, 0, 是], [2, 19, 2, 2, 是], [3, 0, 6, 6, 是], [3, 1, 8, 8, 是], [3, 2, 10, 10, 是], [3, 3, 12, 12, 是], [3, 4, 14, 14, 是], [3, 5, 16, 16, 是], [3, 6, 18, 18, 是], [3, 7, 20, 20, 是], [3, 8, 22, 22, 是], [3, 9, 24, 24, 是], [3, 10, 26, 26, 是], [3, 11, 28, 28, 是], [3, 12, 30, 30, 是], [3, 13, 32, 32, 是], [3, 14, 34, 34, 是], [3, 15, 36, 36, 是], [3, 16, 38, 38, 是], [3, 17, 0, 0, 是], [3, 18, 2, 2, 是], [3, 19, 4, 4, 是], [4, 0, 8, 8, 是], [4, 1, 10, 10, 是], [4, 2, 12, 12, 是], [4, 3, 14, 14, 是], [4, 4, 16, 16, 是], [4, 5, 18, 18, 是], [4, 6, 20, 20, 是], [4, 7, 22, 22, 是], [4, 8, 24, 24, 是], [4, 9, 26, 26, 是], [4, 10, 28, 28, 是], [4, 11, 30, 30, 是], [4, 12, 32, 32, 是], [4, 13, 34, 34, 是], [4, 14, 36, 36, 是], [4, 15, 38, 38, 是], [4, 16, 0, 0, 是], [4, 17, 2, 2, 是], [4, 18, 4, 4, 是], [4, 19, 6, 6, 是], [5, 0, 10, 10, 是], [5, 1, 12, 12, 是], [5, 2, 14, 14, 是], [5, 3, 16, 16, 是], [5, 4, 18, 18, 是], [5, 5, 20, 20, 是], [5, 6, 22, 22, 是], [5, 7, 24, 24, 是], [5, 8, 26, 26, 是], [5, 9, 28, 28, 是], [5, 10, 30, 30, 是], [5, 11, 32, 32, 是], [5, 12, 34, 34, 是], [5, 13, 36, 36, 是], [5, 14, 38, 38, 是], [5, 15, 0, 0, 是], [5, 16, 2, 2, 是], [5, 17, 4, 4, 是], [5, 18, 6, 6, 是], [5, 19, 8, 8, 是], [6, 0, 12, 12, 是], [6, 1, 14, 14, 是], [6, 2, 16, 16, 是], [6, 3, 18, 18, 是], [6, 4, 20, 20, 是], [6, 5, 22, 22, 是], [6, 6, 24, 24, 是], [6, 7, 26, 26, 是], [6, 8, 28, 28, 是], [6, 9, 30, 30, 是], [6, 10, 32, 32, 是], [6, 11, 34, 34, 是], [6, 12, 36, 36, 是], [6, 13, 38, 38, 是], [6, 14, 0, 0, 是], [6, 15, 2, 2, 是], [6, 16, 4, 4, 是], [6, 17, 6, 6, 是], [6, 18, 8, 8, 是], [6, 19, 10, 10, 是], [7, 0, 14, 14, 是], [7, 1, 16, 16, 是], [7, 2, 18, 18, 是], [7, 3, 20, 20, 是], [7, 4, 22, 22, 是], [7, 5, 24, 24, 是], [7, 6, 26, 26, 是], [7, 7, 28, 28, 是], [7, 8, 30, 30, 是], [7, 9, 32, 32, 是], [7, 10, 34, 34, 是], [7, 11, 36, 36, 是], [7, 12, 38, 38, 是], [7, 13, 0, 0, 是], [7, 14, 2, 2, 是], [7, 15, 4, 4, 是], [7, 16, 6, 6, 是], [7, 17, 8, 8, 是], [7, 18, 10, 10, 是], [7, 19, 12, 12, 是], [8, 0, 16, 16, 是], [8, 1, 18, 18, 是], [8, 2, 20, 20, 是], [8, 3, 22, 22, 是], [8, 4, 24, 24, 是], [8, 5, 26, 26, 是], [8, 6, 28, 28, 是], [8, 7, 30, 30, 是], [8, 8, 32, 32, 是], [8, 9, 34, 34, 是], [8, 10, 36, 36, 是], [8, 11, 38, 38, 是], [8, 12, 0, 0, 是], [8, 13, 2, 2, 是], [8, 14, 4, 4, 是], [8, 15, 6, 6, 是], [8, 16, 8, 8, 是], [8, 17, 10, 10, 是], [8, 18, 12, 12, 是], [8, 19, 14, 14, 是], [9, 0, 18, 18, 是], [9, 1, 20, 20, 是], [9, 2, 22, 22, 是], [9, 3, 24, 24, 是], [9, 4, 26, 26, 是], [9, 5, 28, 28, 是], [9, 6, 30, 30, 是], [9, 7, 32, 32, 是], [9, 8, 34, 34, 是], [9, 9, 36, 36, 是], [9, 10, 38, 38, 是], [9, 11, 0, 0, 是], [9, 12, 2, 2, 是], [9, 13, 4, 4, 是], [9, 14, 6, 6, 是], [9, 15, 8, 8, 是], [9, 16, 10, 10, 是], [9, 17, 12, 12, 是], [9, 18, 14, 14, 是], [9, 19, 16, 16, 是], [10, 0, 20, 20, 是], [10, 1, 22, 22, 是], [10, 2, 24, 24, 是], [10, 3, 26, 26, 是], [10, 4, 28, 28, 是], [10, 5, 30, 30, 是], [10, 6, 32, 32, 是], [10, 7, 34, 34, 是], [10, 8, 36, 36, 是], [10, 9, 38, 38, 是], [10, 10, 0, 0, 是], [10, 11, 2, 2, 是], [10, 12, 4, 4, 是], [10, 13, 6, 6, 是], [10, 14, 8, 8, 是], [10, 15, 10, 10, 是], [10, 16, 12, 12, 是], [10, 17, 14, 14, 是], [10, 18, 16, 16, 是], [10, 19, 18, 18, 是], [11, 0, 22, 22, 是], [11, 1, 24, 24, 是], [11, 2, 26, 26, 是], [11, 3, 28, 28, 是], [11, 4, 30, 30, 是], [11, 5, 32, 32, 是], [11, 6, 34, 34, 是], [11, 7, 36, 36, 是], [11, 8, 38, 38, 是], [11, 9, 0, 0, 是], [11, 10, 2, 2, 是], [11, 11, 4, 4, 是], [11, 12, 6, 6, 是], [11, 13, 8, 8, 是], [11, 14, 10, 10, 是], [11, 15, 12, 12, 是], [11, 16, 14, 14, 是], [11, 17, 16, 16, 是], [11, 18, 18, 18, 是], [11, 19, 20, 20, 是], [12, 0, 24, 24, 是], [12, 1, 26, 26, 是], [12, 2, 28, 28, 是], [12, 3, 30, 30, 是], [12, 4, 32, 32, 是], [12, 5, 34, 34, 是], [12, 6, 36, 36, 是], [12, 7, 38, 38, 是], [12, 8, 0, 0, 是], [12, 9, 2, 2, 是], [12, 10, 4, 4, 是], [12, 11, 6, 6, 是], [12, 12, 8, 8, 是], [12, 13, 10, 10, 是], [12, 14, 12, 12, 是], [12, 15, 14, 14, 是], [12, 16, 16, 16, 是], [12, 17, 18, 18, 是], [12, 18, 20, 20, 是], [12, 19, 22, 22, 是], [13, 0, 26, 26, 是], [13, 1, 28, 28, 是], [13, 2, 30, 30, 是], [13, 3, 32, 32, 是], [13, 4, 34, 34, 是], [13, 5, 36, 36, 是], [13, 6, 38, 38, 是], [13, 7, 0, 0, 是], [13, 8, 2, 2, 是], [13, 9, 4, 4, 是], [13, 10, 6, 6, 是], [13, 11, 8, 8, 是], [13, 12, 10, 10, 是], [13, 13, 12, 12, 是], [13, 14, 14, 14, 是], [13, 15, 16, 16, 是], [13, 16, 18, 18, 是], [13, 17, 20, 20, 是], [13, 18, 22, 22, 是], [13, 19, 24, 24, 是], [14, 0, 28, 28, 是], [14, 1, 30, 30, 是], [14, 2, 32, 32, 是], [14, 3, 34, 34, 是], [14, 4, 36, 36, 是], [14, 5, 38, 38, 是], [14, 6, 0, 0, 是], [14, 7, 2, 2, 是], [14, 8, 4, 4, 是], [14, 9, 6, 6, 是], [14, 10, 8, 8, 是], [14, 11, 10, 10, 是], [14, 12, 12, 12, 是], [14, 13, 14, 14, 是], [14, 14, 16, 16, 是], [14, 15, 18, 18, 是], [14, 16, 20, 20, 是], [14, 17, 22, 22, 是], [14, 18, 24, 24, 是], [14, 19, 26, 26, 是], [15, 0, 30, 30, 是], [15, 1, 32, 32, 是], [15, 2, 34, 34, 是], [15, 3, 36, 36, 是], [15, 4, 38, 38, 是], [15, 5, 0, 0, 是], [15, 6, 2, 2, 是], [15, 7, 4, 4, 是], [15, 8, 6, 6, 是], [15, 9, 8, 8, 是], [15, 10, 10, 10, 是], [15, 11, 12, 12, 是], [15, 12, 14, 14, 是], [15, 13, 16, 16, 是], [15, 14, 18, 18, 是], [15, 15, 20, 20, 是], [15, 16, 22, 22, 是], [15, 17, 24, 24, 是], [15, 18, 26, 26, 是], [15, 19, 28, 28, 是], [16, 0, 32, 32, 是], [16, 1, 34, 34, 是], [16, 2, 36, 36, 是], [16, 3, 38, 38, 是], [16, 4, 0, 0, 是], [16, 5, 2, 2, 是], [16, 6, 4, 4, 是], [16, 7, 6, 6, 是], [16, 8, 8, 8, 是], [16, 9, 10, 10, 是], [16, 10, 12, 12, 是], [16, 11, 14, 14, 是], [16, 12, 16, 16, 是], [16, 13, 18, 18, 是], [16, 14, 20, 20, 是], [16, 15, 22, 22, 是], [16, 16, 24, 24, 是], [16, 17, 26, 26, 是], [16, 18, 28, 28, 是], [16, 19, 30, 30, 是], [17, 0, 34, 34, 是], [17, 1, 36, 36, 是], [17, 2, 38, 38, 是], [17, 3, 0, 0, 是], [17, 4, 2, 2, 是], [17, 5, 4, 4, 是], [17, 6, 6, 6, 是], [17, 7, 8, 8, 是], [17, 8, 10, 10, 是], [17, 9, 12, 12, 是], [17, 10, 14, 14, 是], [17, 11, 16, 16, 是], [17, 12, 18, 18, 是], [17, 13, 20, 20, 是], [17, 14, 22, 22, 是], [17, 15, 24, 24, 是], [17, 16, 26, 26, 是], [17, 17, 28, 28, 是], [17, 18, 30, 30, 是], [17, 19, 32, 32, 是], [18, 0, 36, 36, 是], [18, 1, 38, 38, 是], [18, 2, 0, 0, 是], [18, 3, 2, 2, 是], [18, 4, 4, 4, 是], [18, 5, 6, 6, 是], [18, 6, 8, 8, 是], [18, 7, 10, 10, 是], [18, 8, 12, 12, 是], [18, 9, 14, 14, 是], [18, 10, 16, 16, 是], [18, 11, 18, 18, 是], [18, 12, 20, 20, 是], [18, 13, 22, 22, 是], [18, 14, 24, 24, 是], [18, 15, 26, 26, 是], [18, 16, 28, 28, 是], [18, 17, 30, 30, 是], [18, 18, 32, 32, 是], [18, 19, 34, 34, 是], [19, 0, 38, 38, 是], [19, 1, 0, 0, 是], [19, 2, 2, 2, 是], [19, 3, 4, 4, 是], [19, 4, 6, 6, 是], [19, 5, 8, 8, 是], [19, 6, 10, 10, 是], [19, 7, 12, 12, 是], [19, 8, 14, 14, 是], [19, 9, 16, 16, 是], [19, 10, 18, 18, 是], [19, 11, 20, 20, 是], [19, 12, 22, 22, 是], [19, 13, 24, 24, 是], [19, 14, 26, 26, 是], [19, 15, 28, 28, 是], [19, 16, 30, 30, 是], [19, 17, 32, 32, 是], [19, 18, 34, 34, 是], [19, 19, 36, 36, 是]], 是, null, [0], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是, 是]", "relation": "implies", "rule_id": "Iso1_ALL_product_laws_ACTUAL_kernel_image_properties_before_theorem", "conditions": [], "explanation": { "zh-Hant": "全pair驗同態，並取完整target identity preimage與image。qualification允許bad map且property按實際核，可能同態False而kernel normal／image subgroup True；空kernel不補identity fallback。", "en": "Audit every homomorphism pair and the full target-identity preimage/image.Qualification allows a bad map and reports actual properties:homFalse can coexist with normal kernel or subgroup image.An empty kernel is not replaced by an identity fallback." }, "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": "S009", "uses": [ "G001", "S008" ], "before": "source roles = [[0], 是, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38]]", "after": "derived = [[[0], [1], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [2], [3], [4], [5], [6], [7], [8], [9]], [0, 1, 12, 13, 14, 15, 16, 17, 18, 19, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [1, 12, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 13, 14, 15, 16, 17, 18, 19, 2], [2, 3, 0, 1, 12, 13, 14, 15, 16, 17, 18, 19, 4, 5, 6, 7, 8, 9, 10, 11], [3, 4, 1, 12, 13, 14, 15, 16, 17, 18, 19, 2, 5, 6, 7, 8, 9, 10, 11, 0], [4, 5, 12, 13, 14, 15, 16, 17, 18, 19, 2, 3, 6, 7, 8, 9, 10, 11, 0, 1], [5, 6, 13, 14, 15, 16, 17, 18, 19, 2, 3, 4, 7, 8, 9, 10, 11, 0, 1, 12], [6, 7, 14, 15, 16, 17, 18, 19, 2, 3, 4, 5, 8, 9, 10, 11, 0, 1, 12, 13], [7, 8, 15, 16, 17, 18, 19, 2, 3, 4, 5, 6, 9, 10, 11, 0, 1, 12, 13, 14], [8, 9, 16, 17, 18, 19, 2, 3, 4, 5, 6, 7, 10, 11, 0, 1, 12, 13, 14, 15], [9, 10, 17, 18, 19, 2, 3, 4, 5, 6, 7, 8, 11, 0, 1, 12, 13, 14, 15, 16], [10, 11, 18, 19, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 12, 13, 14, 15, 16, 17], [11, 0, 19, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 12, 13, 14, 15, 16, 17, 18], [12, 13, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 14, 15, 16, 17, 18, 19, 2, 3], [13, 14, 5, 6, 7, 8, 9, 10, 11, 0, 1, 12, 15, 16, 17, 18, 19, 2, 3, 4], [14, 15, 6, 7, 8, 9, 10, 11, 0, 1, 12, 13, 16, 17, 18, 19, 2, 3, 4, 5], [15, 16, 7, 8, 9, 10, 11, 0, 1, 12, 13, 14, 17, 18, 19, 2, 3, 4, 5, 6], [16, 17, 8, 9, 10, 11, 0, 1, 12, 13, 14, 15, 18, 19, 2, 3, 4, 5, 6, 7], [17, 18, 9, 10, 11, 0, 1, 12, 13, 14, 15, 16, 19, 2, 3, 4, 5, 6, 7, 8], [18, 19, 10, 11, 0, 1, 12, 13, 14, 15, 16, 17, 2, 3, 4, 5, 6, 7, 8, 9], [19, 2, 11, 0, 1, 12, 13, 14, 15, 16, 17, 18, 3, 4, 5, 6, 7, 8, 9, 10]], [[0, 0, [0], 是], [0, 1, [1], 是], [0, 2, [2], 是], [0, 3, [3], 是], [0, 4, [4], 是], [0, 5, [5], 是], [0, 6, [6], 是], [0, 7, [7], 是], [0, 8, [8], 是], [0, 9, [9], 是], [0, 10, [10], 是], [0, 11, [11], 是], [0, 12, [12], 是], [0, 13, [13], 是], [0, 14, [14], 是], [0, 15, [15], 是], [0, 16, [16], 是], [0, 17, [17], 是], [0, 18, [18], 是], [0, 19, [19], 是], [1, 0, [1], 是], [1, 1, [12], 是], [1, 2, [3], 是], [1, 3, [4], 是], [1, 4, [5], 是], [1, 5, [6], 是], [1, 6, [7], 是], [1, 7, [8], 是], [1, 8, [9], 是], [1, 9, [10], 是], [1, 10, [11], 是], [1, 11, [0], 是], [1, 12, [13], 是], [1, 13, [14], 是], [1, 14, [15], 是], [1, 15, [16], 是], [1, 16, [17], 是], [1, 17, [18], 是], [1, 18, [19], 是], [1, 19, [2], 是], [2, 0, [2], 是], [2, 1, [3], 是], [2, 2, [0], 是], [2, 3, [1], 是], [2, 4, [12], 是], [2, 5, [13], 是], [2, 6, [14], 是], [2, 7, [15], 是], [2, 8, [16], 是], [2, 9, [17], 是], [2, 10, [18], 是], [2, 11, [19], 是], [2, 12, [4], 是], [2, 13, [5], 是], [2, 14, [6], 是], [2, 15, [7], 是], [2, 16, [8], 是], [2, 17, [9], 是], [2, 18, [10], 是], [2, 19, [11], 是], [3, 0, [3], 是], [3, 1, [4], 是], [3, 2, [1], 是], [3, 3, [12], 是], [3, 4, [13], 是], [3, 5, [14], 是], [3, 6, [15], 是], [3, 7, [16], 是], [3, 8, [17], 是], [3, 9, [18], 是], [3, 10, [19], 是], [3, 11, [2], 是], [3, 12, [5], 是], [3, 13, [6], 是], [3, 14, [7], 是], [3, 15, [8], 是], [3, 16, [9], 是], [3, 17, [10], 是], [3, 18, [11], 是], [3, 19, [0], 是], [4, 0, [4], 是], [4, 1, [5], 是], [4, 2, [12], 是], [4, 3, [13], 是], [4, 4, [14], 是], [4, 5, [15], 是], [4, 6, [16], 是], [4, 7, [17], 是], [4, 8, [18], 是], [4, 9, [19], 是], [4, 10, [2], 是], [4, 11, [3], 是], [4, 12, [6], 是], [4, 13, [7], 是], [4, 14, [8], 是], [4, 15, [9], 是], [4, 16, [10], 是], [4, 17, [11], 是], [4, 18, [0], 是], [4, 19, [1], 是], [5, 0, [5], 是], [5, 1, [6], 是], [5, 2, [13], 是], [5, 3, [14], 是], [5, 4, [15], 是], [5, 5, [16], 是], [5, 6, [17], 是], [5, 7, [18], 是], [5, 8, [19], 是], [5, 9, [2], 是], [5, 10, [3], 是], [5, 11, [4], 是], [5, 12, [7], 是], [5, 13, [8], 是], [5, 14, [9], 是], [5, 15, [10], 是], [5, 16, [11], 是], [5, 17, [0], 是], [5, 18, [1], 是], [5, 19, [12], 是], [6, 0, [6], 是], [6, 1, [7], 是], [6, 2, [14], 是], [6, 3, [15], 是], [6, 4, [16], 是], [6, 5, [17], 是], [6, 6, [18], 是], [6, 7, [19], 是], [6, 8, [2], 是], [6, 9, [3], 是], [6, 10, [4], 是], [6, 11, [5], 是], [6, 12, [8], 是], [6, 13, [9], 是], [6, 14, [10], 是], [6, 15, [11], 是], [6, 16, [0], 是], [6, 17, [1], 是], [6, 18, [12], 是], [6, 19, [13], 是], [7, 0, [7], 是], [7, 1, [8], 是], [7, 2, [15], 是], [7, 3, [16], 是], [7, 4, [17], 是], [7, 5, [18], 是], [7, 6, [19], 是], [7, 7, [2], 是], [7, 8, [3], 是], [7, 9, [4], 是], [7, 10, [5], 是], [7, 11, [6], 是], [7, 12, [9], 是], [7, 13, [10], 是], [7, 14, [11], 是], [7, 15, [0], 是], [7, 16, [1], 是], [7, 17, [12], 是], [7, 18, [13], 是], [7, 19, [14], 是], [8, 0, [8], 是], [8, 1, [9], 是], [8, 2, [16], 是], [8, 3, [17], 是], [8, 4, [18], 是], [8, 5, [19], 是], [8, 6, [2], 是], [8, 7, [3], 是], [8, 8, [4], 是], [8, 9, [5], 是], [8, 10, [6], 是], [8, 11, [7], 是], [8, 12, [10], 是], [8, 13, [11], 是], [8, 14, [0], 是], [8, 15, [1], 是], [8, 16, [12], 是], [8, 17, [13], 是], [8, 18, [14], 是], [8, 19, [15], 是], [9, 0, [9], 是], [9, 1, [10], 是], [9, 2, [17], 是], [9, 3, [18], 是], [9, 4, [19], 是], [9, 5, [2], 是], [9, 6, [3], 是], [9, 7, [4], 是], [9, 8, [5], 是], [9, 9, [6], 是], [9, 10, [7], 是], [9, 11, [8], 是], [9, 12, [11], 是], [9, 13, [0], 是], [9, 14, [1], 是], [9, 15, [12], 是], [9, 16, [13], 是], [9, 17, [14], 是], [9, 18, [15], 是], [9, 19, [16], 是], [10, 0, [10], 是], [10, 1, [11], 是], [10, 2, [18], 是], [10, 3, [19], 是], [10, 4, [2], 是], [10, 5, [3], 是], [10, 6, [4], 是], [10, 7, [5], 是], [10, 8, [6], 是], [10, 9, [7], 是], [10, 10, [8], 是], [10, 11, [9], 是], [10, 12, [0], 是], [10, 13, [1], 是], [10, 14, [12], 是], [10, 15, [13], 是], [10, 16, [14], 是], [10, 17, [15], 是], [10, 18, [16], 是], [10, 19, [17], 是], [11, 0, [11], 是], [11, 1, [0], 是], [11, 2, [19], 是], [11, 3, [2], 是], [11, 4, [3], 是], [11, 5, [4], 是], [11, 6, [5], 是], [11, 7, [6], 是], [11, 8, [7], 是], [11, 9, [8], 是], [11, 10, [9], 是], [11, 11, [10], 是], [11, 12, [1], 是], [11, 13, [12], 是], [11, 14, [13], 是], [11, 15, [14], 是], [11, 16, [15], 是], [11, 17, [16], 是], [11, 18, [17], 是], [11, 19, [18], 是], [12, 0, [12], 是], [12, 1, [13], 是], [12, 2, [4], 是], [12, 3, [5], 是], [12, 4, [6], 是], [12, 5, [7], 是], [12, 6, [8], 是], [12, 7, [9], 是], [12, 8, [10], 是], [12, 9, [11], 是], [12, 10, [0], 是], [12, 11, [1], 是], [12, 12, [14], 是], [12, 13, [15], 是], [12, 14, [16], 是], [12, 15, [17], 是], [12, 16, [18], 是], [12, 17, [19], 是], [12, 18, [2], 是], [12, 19, [3], 是], [13, 0, [13], 是], [13, 1, [14], 是], [13, 2, [5], 是], [13, 3, [6], 是], [13, 4, [7], 是], [13, 5, [8], 是], [13, 6, [9], 是], [13, 7, [10], 是], [13, 8, [11], 是], [13, 9, [0], 是], [13, 10, [1], 是], [13, 11, [12], 是], [13, 12, [15], 是], [13, 13, [16], 是], [13, 14, [17], 是], [13, 15, [18], 是], [13, 16, [19], 是], [13, 17, [2], 是], [13, 18, [3], 是], [13, 19, [4], 是], [14, 0, [14], 是], [14, 1, [15], 是], [14, 2, [6], 是], [14, 3, [7], 是], [14, 4, [8], 是], [14, 5, [9], 是], [14, 6, [10], 是], [14, 7, [11], 是], [14, 8, [0], 是], [14, 9, [1], 是], [14, 10, [12], 是], [14, 11, [13], 是], [14, 12, [16], 是], [14, 13, [17], 是], [14, 14, [18], 是], [14, 15, [19], 是], [14, 16, [2], 是], [14, 17, [3], 是], [14, 18, [4], 是], [14, 19, [5], 是], [15, 0, [15], 是], [15, 1, [16], 是], [15, 2, [7], 是], [15, 3, [8], 是], [15, 4, [9], 是], [15, 5, [10], 是], [15, 6, [11], 是], [15, 7, [0], 是], [15, 8, [1], 是], [15, 9, [12], 是], [15, 10, [13], 是], [15, 11, [14], 是], [15, 12, [17], 是], [15, 13, [18], 是], [15, 14, [19], 是], [15, 15, [2], 是], [15, 16, [3], 是], [15, 17, [4], 是], [15, 18, [5], 是], [15, 19, [6], 是], [16, 0, [16], 是], [16, 1, [17], 是], [16, 2, [8], 是], [16, 3, [9], 是], [16, 4, [10], 是], [16, 5, [11], 是], [16, 6, [0], 是], [16, 7, [1], 是], [16, 8, [12], 是], [16, 9, [13], 是], [16, 10, [14], 是], [16, 11, [15], 是], [16, 12, [18], 是], [16, 13, [19], 是], [16, 14, [2], 是], [16, 15, [3], 是], [16, 16, [4], 是], [16, 17, [5], 是], [16, 18, [6], 是], [16, 19, [7], 是], [17, 0, [17], 是], [17, 1, [18], 是], [17, 2, [9], 是], [17, 3, [10], 是], [17, 4, [11], 是], [17, 5, [0], 是], [17, 6, [1], 是], [17, 7, [12], 是], [17, 8, [13], 是], [17, 9, [14], 是], [17, 10, [15], 是], [17, 11, [16], 是], [17, 12, [19], 是], [17, 13, [2], 是], [17, 14, [3], 是], [17, 15, [4], 是], [17, 16, [5], 是], [17, 17, [6], 是], [17, 18, [7], 是], [17, 19, [8], 是], [18, 0, [18], 是], [18, 1, [19], 是], [18, 2, [10], 是], [18, 3, [11], 是], [18, 4, [0], 是], [18, 5, [1], 是], [18, 6, [12], 是], [18, 7, [13], 是], [18, 8, [14], 是], [18, 9, [15], 是], [18, 10, [16], 是], [18, 11, [17], 是], [18, 12, [2], 是], [18, 13, [3], 是], [18, 14, [4], 是], [18, 15, [5], 是], [18, 16, [6], 是], [18, 17, [7], 是], [18, 18, [8], 是], [18, 19, [9], 是], [19, 0, [19], 是], [19, 1, [2], 是], [19, 2, [11], 是], [19, 3, [0], 是], [19, 4, [1], 是], [19, 5, [12], 是], [19, 6, [13], 是], [19, 7, [14], 是], [19, 8, [15], 是], [19, 9, [16], 是], [19, 10, [17], 是], [19, 11, [18], 是], [19, 12, [3], 是], [19, 13, [4], 是], [19, 14, [5], 是], [19, 15, [6], 是], [19, 16, [7], 是], [19, 17, [8], 是], [19, 18, [9], 是], [19, 19, [10], 是]], [[0, [0], [0], 是], [1, [1], [2], 是], [2, [10], [20], 是], [3, [11], [22], 是], [4, [12], [24], 是], [5, [13], [26], 是], [6, [14], [28], 是], [7, [15], [30], 是], [8, [16], [32], 是], [9, [17], [34], 是], [10, [18], [36], 是], [11, [19], [38], 是], [12, [2], [4], 是], [13, [3], [6], 是], [14, [4], [8], 是], [15, [5], [10], 是], [16, [6], [12], 是], [17, [7], [14], 是], [18, [8], [16], 是], [19, [9], [18], 是]], 是]", "relation": "implies", "rule_id": "Iso1_full_LEFT_coset_quotient_ALL_products_and_representative_images", "conditions": [], "explanation": { "zh-Hant": "kernel normal使(ah)(bk)=ab(b^-1hb)k∈abK，逐全部代表元驗product coset相同；同一coset的每個f(g)也須同值才定義bar_f。carrier是coset SETS而非首代表元。", "en": "Kernel normality gives(ah)(bk)=ab(b^-1hb)k in abK;check every representative product agrees on its quotient cell.Every f(g) in a coset must also agree before defining bar_f.The carrier consists of coset SETS,not least representatives." }, "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": "S010", "uses": [ "G001", "S009" ], "before": "source roles = [[[0], [1], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [2], [3], [4], [5], [6], [7], [8], [9]], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], [0, 2, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 4, 6, 8, 10, 12, 14, 16, 18]]", "after": "derived = [[[0, 0, 0, 0, 是], [0, 1, 2, 2, 是], [0, 2, 20, 20, 是], [0, 3, 22, 22, 是], [0, 4, 24, 24, 是], [0, 5, 26, 26, 是], [0, 6, 28, 28, 是], [0, 7, 30, 30, 是], [0, 8, 32, 32, 是], [0, 9, 34, 34, 是], [0, 10, 36, 36, 是], [0, 11, 38, 38, 是], [0, 12, 4, 4, 是], [0, 13, 6, 6, 是], [0, 14, 8, 8, 是], [0, 15, 10, 10, 是], [0, 16, 12, 12, 是], [0, 17, 14, 14, 是], [0, 18, 16, 16, 是], [0, 19, 18, 18, 是], [1, 0, 2, 2, 是], [1, 1, 4, 4, 是], [1, 2, 22, 22, 是], [1, 3, 24, 24, 是], [1, 4, 26, 26, 是], [1, 5, 28, 28, 是], [1, 6, 30, 30, 是], [1, 7, 32, 32, 是], [1, 8, 34, 34, 是], [1, 9, 36, 36, 是], [1, 10, 38, 38, 是], [1, 11, 0, 0, 是], [1, 12, 6, 6, 是], [1, 13, 8, 8, 是], [1, 14, 10, 10, 是], [1, 15, 12, 12, 是], [1, 16, 14, 14, 是], [1, 17, 16, 16, 是], [1, 18, 18, 18, 是], [1, 19, 20, 20, 是], [2, 0, 20, 20, 是], [2, 1, 22, 22, 是], [2, 2, 0, 0, 是], [2, 3, 2, 2, 是], [2, 4, 4, 4, 是], [2, 5, 6, 6, 是], [2, 6, 8, 8, 是], [2, 7, 10, 10, 是], [2, 8, 12, 12, 是], [2, 9, 14, 14, 是], [2, 10, 16, 16, 是], [2, 11, 18, 18, 是], [2, 12, 24, 24, 是], [2, 13, 26, 26, 是], [2, 14, 28, 28, 是], [2, 15, 30, 30, 是], [2, 16, 32, 32, 是], [2, 17, 34, 34, 是], [2, 18, 36, 36, 是], [2, 19, 38, 38, 是], [3, 0, 22, 22, 是], [3, 1, 24, 24, 是], [3, 2, 2, 2, 是], [3, 3, 4, 4, 是], [3, 4, 6, 6, 是], [3, 5, 8, 8, 是], [3, 6, 10, 10, 是], [3, 7, 12, 12, 是], [3, 8, 14, 14, 是], [3, 9, 16, 16, 是], [3, 10, 18, 18, 是], [3, 11, 20, 20, 是], [3, 12, 26, 26, 是], [3, 13, 28, 28, 是], [3, 14, 30, 30, 是], [3, 15, 32, 32, 是], [3, 16, 34, 34, 是], [3, 17, 36, 36, 是], [3, 18, 38, 38, 是], [3, 19, 0, 0, 是], [4, 0, 24, 24, 是], [4, 1, 26, 26, 是], [4, 2, 4, 4, 是], [4, 3, 6, 6, 是], [4, 4, 8, 8, 是], [4, 5, 10, 10, 是], [4, 6, 12, 12, 是], [4, 7, 14, 14, 是], [4, 8, 16, 16, 是], [4, 9, 18, 18, 是], [4, 10, 20, 20, 是], [4, 11, 22, 22, 是], [4, 12, 28, 28, 是], [4, 13, 30, 30, 是], [4, 14, 32, 32, 是], [4, 15, 34, 34, 是], [4, 16, 36, 36, 是], [4, 17, 38, 38, 是], [4, 18, 0, 0, 是], [4, 19, 2, 2, 是], [5, 0, 26, 26, 是], [5, 1, 28, 28, 是], [5, 2, 6, 6, 是], [5, 3, 8, 8, 是], [5, 4, 10, 10, 是], [5, 5, 12, 12, 是], [5, 6, 14, 14, 是], [5, 7, 16, 16, 是], [5, 8, 18, 18, 是], [5, 9, 20, 20, 是], [5, 10, 22, 22, 是], [5, 11, 24, 24, 是], [5, 12, 30, 30, 是], [5, 13, 32, 32, 是], [5, 14, 34, 34, 是], [5, 15, 36, 36, 是], [5, 16, 38, 38, 是], [5, 17, 0, 0, 是], [5, 18, 2, 2, 是], [5, 19, 4, 4, 是], [6, 0, 28, 28, 是], [6, 1, 30, 30, 是], [6, 2, 8, 8, 是], [6, 3, 10, 10, 是], [6, 4, 12, 12, 是], [6, 5, 14, 14, 是], [6, 6, 16, 16, 是], [6, 7, 18, 18, 是], [6, 8, 20, 20, 是], [6, 9, 22, 22, 是], [6, 10, 24, 24, 是], [6, 11, 26, 26, 是], [6, 12, 32, 32, 是], [6, 13, 34, 34, 是], [6, 14, 36, 36, 是], [6, 15, 38, 38, 是], [6, 16, 0, 0, 是], [6, 17, 2, 2, 是], [6, 18, 4, 4, 是], [6, 19, 6, 6, 是], [7, 0, 30, 30, 是], [7, 1, 32, 32, 是], [7, 2, 10, 10, 是], [7, 3, 12, 12, 是], [7, 4, 14, 14, 是], [7, 5, 16, 16, 是], [7, 6, 18, 18, 是], [7, 7, 20, 20, 是], [7, 8, 22, 22, 是], [7, 9, 24, 24, 是], [7, 10, 26, 26, 是], [7, 11, 28, 28, 是], [7, 12, 34, 34, 是], [7, 13, 36, 36, 是], [7, 14, 38, 38, 是], [7, 15, 0, 0, 是], [7, 16, 2, 2, 是], [7, 17, 4, 4, 是], [7, 18, 6, 6, 是], [7, 19, 8, 8, 是], [8, 0, 32, 32, 是], [8, 1, 34, 34, 是], [8, 2, 12, 12, 是], [8, 3, 14, 14, 是], [8, 4, 16, 16, 是], [8, 5, 18, 18, 是], [8, 6, 20, 20, 是], [8, 7, 22, 22, 是], [8, 8, 24, 24, 是], [8, 9, 26, 26, 是], [8, 10, 28, 28, 是], [8, 11, 30, 30, 是], [8, 12, 36, 36, 是], [8, 13, 38, 38, 是], [8, 14, 0, 0, 是], [8, 15, 2, 2, 是], [8, 16, 4, 4, 是], [8, 17, 6, 6, 是], [8, 18, 8, 8, 是], [8, 19, 10, 10, 是], [9, 0, 34, 34, 是], [9, 1, 36, 36, 是], [9, 2, 14, 14, 是], [9, 3, 16, 16, 是], [9, 4, 18, 18, 是], [9, 5, 20, 20, 是], [9, 6, 22, 22, 是], [9, 7, 24, 24, 是], [9, 8, 26, 26, 是], [9, 9, 28, 28, 是], [9, 10, 30, 30, 是], [9, 11, 32, 32, 是], [9, 12, 38, 38, 是], [9, 13, 0, 0, 是], [9, 14, 2, 2, 是], [9, 15, 4, 4, 是], [9, 16, 6, 6, 是], [9, 17, 8, 8, 是], [9, 18, 10, 10, 是], [9, 19, 12, 12, 是], [10, 0, 36, 36, 是], [10, 1, 38, 38, 是], [10, 2, 16, 16, 是], [10, 3, 18, 18, 是], [10, 4, 20, 20, 是], [10, 5, 22, 22, 是], [10, 6, 24, 24, 是], [10, 7, 26, 26, 是], [10, 8, 28, 28, 是], [10, 9, 30, 30, 是], [10, 10, 32, 32, 是], [10, 11, 34, 34, 是], [10, 12, 0, 0, 是], [10, 13, 2, 2, 是], [10, 14, 4, 4, 是], [10, 15, 6, 6, 是], [10, 16, 8, 8, 是], [10, 17, 10, 10, 是], [10, 18, 12, 12, 是], [10, 19, 14, 14, 是], [11, 0, 38, 38, 是], [11, 1, 0, 0, 是], [11, 2, 18, 18, 是], [11, 3, 20, 20, 是], [11, 4, 22, 22, 是], [11, 5, 24, 24, 是], [11, 6, 26, 26, 是], [11, 7, 28, 28, 是], [11, 8, 30, 30, 是], [11, 9, 32, 32, 是], [11, 10, 34, 34, 是], [11, 11, 36, 36, 是], [11, 12, 2, 2, 是], [11, 13, 4, 4, 是], [11, 14, 6, 6, 是], [11, 15, 8, 8, 是], [11, 16, 10, 10, 是], [11, 17, 12, 12, 是], [11, 18, 14, 14, 是], [11, 19, 16, 16, 是], [12, 0, 4, 4, 是], [12, 1, 6, 6, 是], [12, 2, 24, 24, 是], [12, 3, 26, 26, 是], [12, 4, 28, 28, 是], [12, 5, 30, 30, 是], [12, 6, 32, 32, 是], [12, 7, 34, 34, 是], [12, 8, 36, 36, 是], [12, 9, 38, 38, 是], [12, 10, 0, 0, 是], [12, 11, 2, 2, 是], [12, 12, 8, 8, 是], [12, 13, 10, 10, 是], [12, 14, 12, 12, 是], [12, 15, 14, 14, 是], [12, 16, 16, 16, 是], [12, 17, 18, 18, 是], [12, 18, 20, 20, 是], [12, 19, 22, 22, 是], [13, 0, 6, 6, 是], [13, 1, 8, 8, 是], [13, 2, 26, 26, 是], [13, 3, 28, 28, 是], [13, 4, 30, 30, 是], [13, 5, 32, 32, 是], [13, 6, 34, 34, 是], [13, 7, 36, 36, 是], [13, 8, 38, 38, 是], [13, 9, 0, 0, 是], [13, 10, 2, 2, 是], [13, 11, 4, 4, 是], [13, 12, 10, 10, 是], [13, 13, 12, 12, 是], [13, 14, 14, 14, 是], [13, 15, 16, 16, 是], [13, 16, 18, 18, 是], [13, 17, 20, 20, 是], [13, 18, 22, 22, 是], [13, 19, 24, 24, 是], [14, 0, 8, 8, 是], [14, 1, 10, 10, 是], [14, 2, 28, 28, 是], [14, 3, 30, 30, 是], [14, 4, 32, 32, 是], [14, 5, 34, 34, 是], [14, 6, 36, 36, 是], [14, 7, 38, 38, 是], [14, 8, 0, 0, 是], [14, 9, 2, 2, 是], [14, 10, 4, 4, 是], [14, 11, 6, 6, 是], [14, 12, 12, 12, 是], [14, 13, 14, 14, 是], [14, 14, 16, 16, 是], [14, 15, 18, 18, 是], [14, 16, 20, 20, 是], [14, 17, 22, 22, 是], [14, 18, 24, 24, 是], [14, 19, 26, 26, 是], [15, 0, 10, 10, 是], [15, 1, 12, 12, 是], [15, 2, 30, 30, 是], [15, 3, 32, 32, 是], [15, 4, 34, 34, 是], [15, 5, 36, 36, 是], [15, 6, 38, 38, 是], [15, 7, 0, 0, 是], [15, 8, 2, 2, 是], [15, 9, 4, 4, 是], [15, 10, 6, 6, 是], [15, 11, 8, 8, 是], [15, 12, 14, 14, 是], [15, 13, 16, 16, 是], [15, 14, 18, 18, 是], [15, 15, 20, 20, 是], [15, 16, 22, 22, 是], [15, 17, 24, 24, 是], [15, 18, 26, 26, 是], [15, 19, 28, 28, 是], [16, 0, 12, 12, 是], [16, 1, 14, 14, 是], [16, 2, 32, 32, 是], [16, 3, 34, 34, 是], [16, 4, 36, 36, 是], [16, 5, 38, 38, 是], [16, 6, 0, 0, 是], [16, 7, 2, 2, 是], [16, 8, 4, 4, 是], [16, 9, 6, 6, 是], [16, 10, 8, 8, 是], [16, 11, 10, 10, 是], [16, 12, 16, 16, 是], [16, 13, 18, 18, 是], [16, 14, 20, 20, 是], [16, 15, 22, 22, 是], [16, 16, 24, 24, 是], [16, 17, 26, 26, 是], [16, 18, 28, 28, 是], [16, 19, 30, 30, 是], [17, 0, 14, 14, 是], [17, 1, 16, 16, 是], [17, 2, 34, 34, 是], [17, 3, 36, 36, 是], [17, 4, 38, 38, 是], [17, 5, 0, 0, 是], [17, 6, 2, 2, 是], [17, 7, 4, 4, 是], [17, 8, 6, 6, 是], [17, 9, 8, 8, 是], [17, 10, 10, 10, 是], [17, 11, 12, 12, 是], [17, 12, 18, 18, 是], [17, 13, 20, 20, 是], [17, 14, 22, 22, 是], [17, 15, 24, 24, 是], [17, 16, 26, 26, 是], [17, 17, 28, 28, 是], [17, 18, 30, 30, 是], [17, 19, 32, 32, 是], [18, 0, 16, 16, 是], [18, 1, 18, 18, 是], [18, 2, 36, 36, 是], [18, 3, 38, 38, 是], [18, 4, 0, 0, 是], [18, 5, 2, 2, 是], [18, 6, 4, 4, 是], [18, 7, 6, 6, 是], [18, 8, 8, 8, 是], [18, 9, 10, 10, 是], [18, 10, 12, 12, 是], [18, 11, 14, 14, 是], [18, 12, 20, 20, 是], [18, 13, 22, 22, 是], [18, 14, 24, 24, 是], [18, 15, 26, 26, 是], [18, 16, 28, 28, 是], [18, 17, 30, 30, 是], [18, 18, 32, 32, 是], [18, 19, 34, 34, 是], [19, 0, 18, 18, 是], [19, 1, 20, 20, 是], [19, 2, 38, 38, 是], [19, 3, 0, 0, 是], [19, 4, 2, 2, 是], [19, 5, 4, 4, 是], [19, 6, 6, 6, 是], [19, 7, 8, 8, 是], [19, 8, 10, 10, 是], [19, 9, 12, 12, 是], [19, 10, 14, 14, 是], [19, 11, 16, 16, 是], [19, 12, 22, 22, 是], [19, 13, 24, 24, 是], [19, 14, 26, 26, 是], [19, 15, 28, 28, 是], [19, 16, 30, 30, 是], [19, 17, 32, 32, 是], [19, 18, 34, 34, 是], [19, 19, 36, 36, 是]], 是, 是, 是, 是, 否]", "relation": "implies", "rule_id": "Iso1_ALL_induced_product_pairs_injective_and_onto_IMAGE_separate", "conditions": [], "explanation": { "zh-Hant": "bar_f保乘法由f(ab)=f(a)f(b)；若f(a)=f(b)，a^-1b∈K故同coset，給injective。任意image值f(g)由gK命中，給onto IMAGE；不由同cardinality代替這些責任，也不自動onto整個target。", "en": "The identity f(ab)=f(a)f(b) preserves products.If f(a)=f(b),a^-1b is in K and the cosets agree,giving injectivity.Every image value f(g) is hit by gK,giving surjectivity onto the IMAGE.Equal cardinalities do not replace these obligations or give surjectivity to the whole target." }, "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": "S011", "uses": [ "G001", "S010" ], "before": "source roles = [[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 40, 20]", "after": "derived = [否, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, 否]", "relation": "implies", "rule_id": "Iso1_image_VS_codomain_scope_and_finite_general_proof_boundary", "conditions": [], "explanation": { "zh-Hant": "只有實際f滿射，image才等同整個codomain。最後No保一般抽象證明／形式化責任，不否定本題完整有限同構證書或教學理由。", "en": "Only actual surjectivity identifies image with the whole codomain.The final No preserves general abstract/formal proof obligations,not denial of this complete finite isomorphism certificate or its pedagogical argument." }, "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": [ [ 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38 ], 40, 20, false, "G/kernel_is_isomorphic_to_image", "non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain", false, "finite_instance_certificate_not_a_general_theorem_proof" ] }, "conclusion": { "canonical_value": "([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 40, 20, 否, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, 否, finite_instance_certificate_not_a_general_theorem_proof)", "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-20261115-r01-first-isomorphism-theorem-finite-groups-foundations", "batch_path": "public/data/questions/batches/batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations/questions.jsonl", "batch_sha256": "b27c8fc97af12e9d77e259f367fbae66d666796519a9c4014e73f875c727b827", "original_line": 94, "original_line_sha256": "f3cdc9593a13ff866a2973ff82449c42cd02c3ff6fbbc807a9feeba5275a7947", "question_compact_sha256": "21e69b4196bd00839fbde6ee9eb286347da6f3fa8e97e94a9d9332b7b377d29a", "english_row_compact_sha256": "a80aae218cdeee37e9bedb8f64bfa55e53b4f17112e2b3bd4b17eacbb7d4a917", "english_manifest_path": "public/data/i18n/editions/batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations.en.jsonl" }, "mko_bindings": { "mko-first-isomorphism-theorem-finite-groups": { "sha256": "5586ce5ed3d47467ffd6c485dfcb495844143a17783d770ce2572420bfa28065", "english_sha256": "f9e622d65b015b588d1e29eb89655426b79abaf193e2f4763bec7b9e8c6aeadb", "original_path": "public/data/mko/first-isomorphism-theorem-finite-groups.json", "english_path": "public/data/i18n/editions/mko-first-isomorphism-theorem-finite-groups.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "4624c29c8cf22048332daea19f430f922e032c30410aa690ba3f1924d8b9ee45", "original_candidate_sha256": "dbc623be6e52af2f5d6a3a6c4c4d2b44245e79b67f5c3b2e8b857020bbe1897c", "original_candidate_line_sha256": "dbc623be6e52af2f5d6a3a6c4c4d2b44245e79b67f5c3b2e8b857020bbe1897c" }, "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-20261026-010-p51", "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": "74c63a2c12285a3b2073f8d9f102bc2ba5fb7ed890c5b7c3b8afa94f032eda82", "question_first_witness_sha256": "6d56428610f8e86c45fd00a2ad3b27ad9c052956946f0b91d71a68536f79d95f" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 11, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "4cb397bb86dadd6e039b722da90e06b317ea86ebf1897939fd66a2627e31ab93", "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." } }