{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261015-advance-20261114-r03-finite-group-conjugacy-class-0099", "pilot_number": "R0048-24899", "number": 24899, "family_id": "qf-mod45-conjugacy-singleton-nonconjugate-boundary", "title": { "zh-Hant": "第 24899 題：有限群共軛類", "en": "Question 24899: A Singleton Class Commutes with Every Group Element, while an Outside Element Is Nonconjugate" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261015-advance-20261114-r03-finite-group-conjugacy-class-0099", "batch_id": "batch-20261015-advance-20261114-r03-finite-group-conjugacy-class-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-finite-group-conjugacy-class" ], "title_zh": "有限群共軛類", "stem_zh": "由完整Cayley table導出全部conjugation action與orbits。操作=conjugacy-singleton-nonconjugate-boundary；資料={\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37],[13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38],[14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44],[20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46],[22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47],[23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48],[24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49],[25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50],[26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7],[33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8],[34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9],[35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10],[36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11],[37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12],[38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13],[39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14],[40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15],[41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16],[42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17],[43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18],[44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19],[45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20],[46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21],[47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22],[48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23],[49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24],[50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25],[51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0,\"element\":0,\"other\":26}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "(0, [0], 是, 26, 否, 是, singleton_class_iff_commutes_with_every_g_and_other_outside_class_is_nonconjugate_without_class_size_formula)", "explanation_zh": "Wave6第八層核對=(0, [0], 是, 26, 否, 是, singleton_class_iff_commutes_with_every_g_and_other_outside_class_is_nonconjugate_without_class_size_formula)；依完整明示資料獨立重算；共軛類是element orbit並形成完整partition；singleton以forall g commute核對；共軛類不是coset或conjugate subgroup；不使用centralizer、class-size公式或類方程。" }, "difficulty": { "band": "undergraduate_core", "level": 3, "skills": [ "有限群共軛類" ] }, "variant": { "family_id": "qf-mod45-conjugacy-singleton-nonconjugate-boundary", "seed": 202611143099, "parameters": { "group_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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51 ], [ 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, 0, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26 ], [ 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27 ], [ 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28 ], [ 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29 ], [ 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30 ], [ 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 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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、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-r03-finite-group-conjugacy-class-0099", "batch_id": "batch-20261015-advance-20261114-r03-finite-group-conjugacy-class-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "53e1f6aea8da5cdd632ecdbb6e6574068209f0be0d8bff0473d2833dbd9a1d23", "family_id": "qf-mod45-conjugacy-singleton-nonconjugate-boundary", "seed": 202611143099, "semantic_template": "qf-mod45-conjugacy-singleton-nonconjugate-boundary/{\"element\":0,\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37],[13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38],[14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44],[20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46],[22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47],[23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48],[24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49],[25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50],[26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7],[33,32,31,30,29,28,27,26,51,50,4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}, "fields": { "title": "A Singleton Class Commutes with Every Group Element, while an Outside Element Is Nonconjugate", "stem": "Use the complete finite group 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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51], [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, 0, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46], [22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48], [24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49], [25, 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, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50], [26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 0, 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], [27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14], [40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15], [41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16], [42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17], [43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18], [44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19], [45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20], [46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21], [47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22], [48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23], [49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24], [50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25], [51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 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, 0]] with identity=0. First certify all group axioms and inverses, then form the full left conjugation action g*x=g*x*g^(-1). Every acting g and acted-on element is included, not a sample. Element=0, other=26. Give (element, singleton_class, commutes_with_all, other, conjugate_to_other, nonconjugate_certificate, boundary).", "answer": { "kind": "expression", "value": "(0, [0], Yes, 26, No, Yes, singleton_class_iff_commutes_with_every_g_and_other_outside_class_is_nonconjugate_without_class_size_formula)", "explanation": "Require a singleton class and an actual other element outside it. Check all-g commutation, then report No conjugacy to that other and its complete nonconjugacy certificate. This does not call every class a singleton or use a class-size formula. The exact result, retaining the requested order, is (0, [0], Yes, 26, No, Yes, singleton_class_iff_commutes_with_every_g_and_other_outside_class_is_nonconjugate_without_class_size_formula)." }, "skills": [ "A Singleton Class Commutes with Every Group Element, while an Outside Element Is Nonconjugate" ], "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": "由完整Cayley table導出全部conjugation action與orbits。操作=conjugacy-singleton-nonconjugate-boundary；資料={\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37],[13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38],[14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44],[20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46],[22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47],[23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48],[24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49],[25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50],[26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7],[33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8],[34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9],[35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10],[36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11],[37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12],[38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13],[39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14],[40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15],[41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16],[42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17],[43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18],[44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19],[45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20],[46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21],[47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22],[48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23],[49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24],[50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25],[51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0,\"element\":0,\"other\":26}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use the complete finite group 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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51], [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, 0, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46], [22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48], [24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49], [25, 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, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50], [26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 0, 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], [27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14], [40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15], [41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16], [42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17], [43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18], [44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19], [45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20], [46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21], [47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22], [48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23], [49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24], [50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25], [51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 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, 0]] with identity=0. First certify all group axioms and inverses, then form the full left conjugation action g*x=g*x*g^(-1). Every acting g and acted-on element is included, not a sample. Element=0, other=26. Give (element, singleton_class, commutes_with_all, other, conjugate_to_other, nonconjugate_certificate, boundary)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [conjugacy-singleton-nonconjugate-boundary]", "after": "derived = [[element, singleton_class, commutes_with_all, other, conjugate_to_other, nonconjugate_certificate, boundary]]", "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 = [Conjugacy ORIGINAL group, 64, [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51], [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, 0, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46], [22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48], [24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49], [25, 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, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50], [26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 0, 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], [27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 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9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24], [50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25], [51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 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, 0]], 0]", "after": "derived = [52, [[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], [40, 40, 40, 40], [41, 41, 41, 41], [42, 42, 42, 42], [43, 43, 43, 43], [44, 44, 44, 44], [45, 45, 45, 45], [46, 46, 46, 46], [47, 47, 47, 47], [48, 48, 48, 48], [49, 49, 49, 49], [50, 50, 50, 50], [51, 51, 51, 51]], 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": 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"arithmetic": null }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "source roles = [Conjugacy ORIGINAL group, 0]", "after": "derived = [[0, 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, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51], [[0, 0, 0, 0], [1, 25, 0, 0], [2, 24, 0, 0], [3, 23, 0, 0], [4, 22, 0, 0], [5, 21, 0, 0], [6, 20, 0, 0], [7, 19, 0, 0], [8, 18, 0, 0], [9, 17, 0, 0], [10, 16, 0, 0], [11, 15, 0, 0], [12, 14, 0, 0], [13, 13, 0, 0], [14, 12, 0, 0], [15, 11, 0, 0], [16, 10, 0, 0], [17, 9, 0, 0], [18, 8, 0, 0], [19, 7, 0, 0], [20, 6, 0, 0], [21, 5, 0, 0], [22, 4, 0, 0], [23, 3, 0, 0], [24, 2, 0, 0], [25, 1, 0, 0], [26, 26, 0, 0], [27, 27, 0, 0], [28, 28, 0, 0], [29, 29, 0, 0], [30, 30, 0, 0], [31, 31, 0, 0], [32, 32, 0, 0], [33, 33, 0, 0], [34, 34, 0, 0], [35, 35, 0, 0], [36, 36, 0, 0], [37, 37, 0, 0], [38, 38, 0, 0], [39, 39, 0, 0], [40, 40, 0, 0], [41, 41, 0, 0], [42, 42, 0, 0], [43, 43, 0, 0], [44, 44, 0, 0], [45, 45, 0, 0], [46, 46, 0, 0], [47, 47, 0, 0], [48, 48, 0, 0], [49, 49, 0, 0], [50, 50, 0, 0], [51, 51, 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 = [0, 0, [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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]]", "after": "derived = [[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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51], [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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]]", "relation": "implies", "rule_id": "Conjugacy_EVERY_g_EVERY_element_exact_action_row", "conditions": [], "explanation": { "zh-Hant": "對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。", "en": "For every original element x,compute gx then multiply by g inverse to form the complete acting row.This is an element action,not a conjugated subgroup." }, "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 = [1, 25, [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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]]", "after": "derived = [[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, 0, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26], [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, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27]]", "relation": "implies", "rule_id": "Conjugacy_EVERY_g_EVERY_element_exact_action_row", "conditions": [], "explanation": { "zh-Hant": "對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。", "en": "For every original element x,compute gx then multiply by g inverse to form the complete acting row.This is an element action,not a conjugated subgroup." }, "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 = [2, 24, [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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]]", "after": "derived = [[2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27], [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, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29]]", "relation": "implies", "rule_id": "Conjugacy_EVERY_g_EVERY_element_exact_action_row", "conditions": [], "explanation": { "zh-Hant": "對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。", "en": "For every original element x,compute gx then multiply by g inverse to form the complete acting row.This is an element action,not a conjugated subgroup." }, "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 = [3, 23, [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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]]", "after": "derived = [[3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28], [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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31]]", "relation": "implies", "rule_id": "Conjugacy_EVERY_g_EVERY_element_exact_action_row", "conditions": [], "explanation": { "zh-Hant": "對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。", "en": "For every original element x,compute gx then multiply by g inverse to form the complete acting row.This is an element action,not a conjugated subgroup." }, "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 = [4, 22, [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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]]", "after": "derived = [[4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29], [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, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33]]", "relation": "implies", "rule_id": "Conjugacy_EVERY_g_EVERY_element_exact_action_row", "conditions": [], "explanation": { "zh-Hant": "對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。", "en": "For every original element x,compute gx then multiply by g inverse to form the complete acting row.This is an element action,not a conjugated subgroup." }, "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 = [5, 21, [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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]]", "after": "derived = [[5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30], [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, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35]]", "relation": "implies", "rule_id": "Conjugacy_EVERY_g_EVERY_element_exact_action_row", "conditions": [], "explanation": { "zh-Hant": "對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。", "en": "For every original element x,compute gx then multiply by g inverse to form the complete acting row.This is an element action,not a conjugated subgroup." }, "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 = [6, 20, [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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]]", "after": "derived = [[6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31], [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, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37]]", "relation": "implies", "rule_id": "Conjugacy_EVERY_g_EVERY_element_exact_action_row", "conditions": [], "explanation": { "zh-Hant": "對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。", "en": "For every original element x,compute gx then multiply by g inverse to form the complete acting row.This is an element action,not a conjugated subgroup." }, "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": "S012", "uses": [ "G001", "S011" ], "before": "source roles = [7, 19, [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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]]", "after": "derived = [[7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32], [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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39]]", "relation": "implies", "rule_id": "Conjugacy_EVERY_g_EVERY_element_exact_action_row", "conditions": [], "explanation": { "zh-Hant": "對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。", "en": "For every original element x,compute gx then multiply by g inverse to form the complete acting row.This is an element action,not a conjugated subgroup." }, "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": "S013", "uses": [ "G001", "S012" ], "before": "source roles = [8, 18, [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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]]", "after": "derived = [[8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33], [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, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41]]", "relation": "implies", "rule_id": "Conjugacy_EVERY_g_EVERY_element_exact_action_row", "conditions": [], "explanation": { "zh-Hant": "對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。", "en": "For every original element x,compute gx then multiply by g inverse to form the complete acting row.This is an element action,not a conjugated subgroup." }, "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": "S014", "uses": [ "G001", "S013" ], "before": "source roles = [9, 17, [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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]]", "after": "derived = [[9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34], [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, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43]]", "relation": "implies", "rule_id": "Conjugacy_EVERY_g_EVERY_element_exact_action_row", "conditions": [], "explanation": { "zh-Hant": "對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。", "en": "For every original element x,compute gx then multiply by g inverse to form the complete acting row.This is an element action,not a conjugated subgroup." }, "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": "S015", "uses": [ "G001", "S014" ], "before": "source roles = [10, 16, [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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]]", "after": "derived = [[10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [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, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45]]", "relation": "implies", "rule_id": "Conjugacy_EVERY_g_EVERY_element_exact_action_row", "conditions": [], "explanation": { "zh-Hant": "對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。", "en": "For every original element x,compute gx then multiply by g inverse to form the complete acting row.This is an element action,not a conjugated subgroup." }, "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": "S016", "uses": [ "G001", "S015" ], "before": "source roles = [11, 15, [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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]]", "after": "derived = [[11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [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, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47]]", "relation": "implies", "rule_id": "Conjugacy_EVERY_g_EVERY_element_exact_action_row", "conditions": [], "explanation": { "zh-Hant": "對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。", "en": "For every original element x,compute gx then multiply by g inverse to form the complete acting row.This is an element action,not a conjugated subgroup." }, "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": "S017", "uses": [ "G001", "S016" ], "before": "source roles = [12, 14, [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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]]", "after": "derived = [[12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [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, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49]]", "relation": "implies", "rule_id": "Conjugacy_EVERY_g_EVERY_element_exact_action_row", "conditions": [], "explanation": { "zh-Hant": "對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。", "en": "For every original element x,compute gx then multiply by g inverse to form the complete acting row.This is an element action,not a conjugated subgroup." }, "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": "S018", "uses": [ "G001", "S017" ], "before": "source roles = [13, 13, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]]", "after": "derived = [[13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]]", "relation": "implies", "rule_id": "Conjugacy_EVERY_g_EVERY_element_exact_action_row", "conditions": [], "explanation": { "zh-Hant": "對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。", "en": "For every original element x,compute gx then multiply by g inverse to form the complete acting row.This is an element action,not a conjugated subgroup." }, "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": "S019", "uses": [ "G001", "S018" ], "before": "source roles = [14, 12, [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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]]", "after": "derived = [[14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 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, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27]]", "relation": "implies", "rule_id": "Conjugacy_EVERY_g_EVERY_element_exact_action_row", "conditions": [], "explanation": { "zh-Hant": "對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。", "en": "For every original element x,compute gx then multiply by g inverse to form the complete acting row.This is an element action,not a conjugated subgroup." }, "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": "S020", "uses": [ "G001", "S019" ], "before": "source roles = [15, 11, [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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]]", "after": "derived = [[15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [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, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29]]", "relation": "implies", "rule_id": "Conjugacy_EVERY_g_EVERY_element_exact_action_row", "conditions": [], "explanation": { "zh-Hant": "對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。", "en": "For every original element x,compute gx then multiply by g inverse to form the complete acting row.This is an element action,not a conjugated subgroup." }, "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": "S021", "uses": [ "G001", "S020" ], "before": "source roles = [16, 10, [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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]]", "after": "derived = [[16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31]]", "relation": "implies", "rule_id": "Conjugacy_EVERY_g_EVERY_element_exact_action_row", "conditions": [], "explanation": { "zh-Hant": "對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。", "en": "For every original element x,compute gx then multiply by g inverse to form the complete acting row.This is an element action,not a conjugated subgroup." }, "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": "S022", "uses": [ "G001", "S021" ], "before": "source roles = [17, 9, [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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]]", "after": "derived = [[17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [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, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33]]", "relation": "implies", "rule_id": "Conjugacy_EVERY_g_EVERY_element_exact_action_row", "conditions": [], "explanation": { "zh-Hant": "對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。", "en": "For every original element x,compute gx then multiply by g inverse to form the complete acting row.This is an element action,not a conjugated subgroup." }, "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": "S023", "uses": [ "G001", "S022" ], "before": "source roles = [18, 8, [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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]]", "after": "derived = [[18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [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, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35]]", "relation": "implies", "rule_id": "Conjugacy_EVERY_g_EVERY_element_exact_action_row", "conditions": [], "explanation": { "zh-Hant": "對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。", "en": "For every original element x,compute gx then multiply by g inverse to form the complete acting row.This is an element action,not a conjugated subgroup." }, "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": "S024", "uses": [ "G001", "S023" ], "before": "source roles = [19, 7, [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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]]", "after": "derived = [[19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [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, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37]]", "relation": "implies", "rule_id": "Conjugacy_EVERY_g_EVERY_element_exact_action_row", "conditions": [], "explanation": { "zh-Hant": "對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。", "en": "For every original element x,compute gx then multiply by g inverse to form the complete acting row.This is an element action,not a conjugated subgroup." }, "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": "S025", "uses": [ "G001", "S024" ], "before": "source roles = [20, 6, [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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]]", "after": "derived = [[20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39]]", "relation": "implies", "rule_id": "Conjugacy_EVERY_g_EVERY_element_exact_action_row", "conditions": [], "explanation": { "zh-Hant": "對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。", "en": "For every original element x,compute gx then multiply by g inverse to form the complete acting row.This is an element action,not a conjugated subgroup." }, "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": "S026", "uses": [ "G001", "S025" ], "before": "source roles = [21, 5, [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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]]", "after": "derived = [[21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46], [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, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41]]", "relation": "implies", "rule_id": "Conjugacy_EVERY_g_EVERY_element_exact_action_row", "conditions": [], "explanation": { "zh-Hant": "對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。", "en": "For every original element x,compute gx then multiply by g inverse to form the complete acting row.This is an element action,not a conjugated subgroup." }, "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": "S027", "uses": [ "G001", "S026" ], "before": "source roles = [22, 4, [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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]]", "after": "derived = [[22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [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, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43]]", "relation": "implies", "rule_id": "Conjugacy_EVERY_g_EVERY_element_exact_action_row", "conditions": [], "explanation": { "zh-Hant": "對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。", "en": "For every original element x,compute gx then multiply by g inverse to form the complete acting row.This is an element action,not a conjugated subgroup." }, "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": "S028", "uses": [ "G001", "S027" ], "before": "source roles = [23, 3, [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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]]", "after": "derived = [[23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48], [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, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45]]", "relation": "implies", "rule_id": "Conjugacy_EVERY_g_EVERY_element_exact_action_row", "conditions": [], "explanation": { "zh-Hant": "對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。", "en": "For every original element x,compute gx then multiply by g inverse to form the complete acting row.This is an element action,not a conjugated subgroup." }, "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": "S029", "uses": [ "G001", "S028" ], "before": "source roles = [24, 2, [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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]]", "after": "derived = [[24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49], [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, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47]]", "relation": "implies", "rule_id": "Conjugacy_EVERY_g_EVERY_element_exact_action_row", "conditions": [], "explanation": { "zh-Hant": "對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。", "en": "For every original element x,compute gx then multiply by g inverse to form the complete acting row.This is an element action,not a conjugated subgroup." }, "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": "S030", "uses": [ "G001", "S029" ], "before": "source roles = [25, 1, [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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]]", "after": "derived = [[25, 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, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50], [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, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49]]", "relation": "implies", "rule_id": "Conjugacy_EVERY_g_EVERY_element_exact_action_row", "conditions": [], "explanation": { "zh-Hant": "對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。", "en": "For every original element x,compute gx then multiply by g inverse to form the complete acting row.This is an element action,not a conjugated subgroup." }, "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": "S031", "uses": [ "G001", "S030" ], "before": "source roles = [26, 26, [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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]]", "after": "derived = [[26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 0, 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], [0, 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, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27]]", "relation": "implies", "rule_id": "Conjugacy_EVERY_g_EVERY_element_exact_action_row", "conditions": [], "explanation": { "zh-Hant": "對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。", "en": "For every original element x,compute gx then multiply by g inverse to form the complete acting row.This is an element action,not a conjugated subgroup." }, "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": "S032", "uses": [ "G001", "S031" ], "before": "source roles = [27, 27, [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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]]", "after": "derived = [[27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [0, 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, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29]]", "relation": "implies", "rule_id": "Conjugacy_EVERY_g_EVERY_element_exact_action_row", "conditions": [], "explanation": { "zh-Hant": "對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。", "en": "For every original element x,compute gx then multiply by g inverse to form the complete acting row.This is an element action,not a conjugated subgroup." }, "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": "S033", "uses": [ "G001", "S032" ], "before": "source roles = [28, 28, [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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]]", "after": "derived = [[28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [0, 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, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31]]", "relation": "implies", "rule_id": "Conjugacy_EVERY_g_EVERY_element_exact_action_row", "conditions": [], "explanation": { "zh-Hant": "對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。", "en": "For every original element x,compute gx then multiply by g inverse to form the complete acting row.This is an element action,not a conjugated subgroup." }, "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": "S034", "uses": [ "G001", "S033" ], "before": "source roles = [29, 29, [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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]]", "after": "derived = [[29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [0, 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, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33]]", "relation": "implies", "rule_id": "Conjugacy_EVERY_g_EVERY_element_exact_action_row", "conditions": [], "explanation": { "zh-Hant": "對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。", "en": "For every original element x,compute gx then multiply by g inverse to form the complete acting row.This is an element action,not a conjugated subgroup." }, "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": "S035", "uses": [ "G001", "S034" ], "before": "source roles = [30, 30, [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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]]", "after": "derived = [[30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [0, 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, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35]]", "relation": "implies", "rule_id": "Conjugacy_EVERY_g_EVERY_element_exact_action_row", "conditions": [], "explanation": { "zh-Hant": "對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。", "en": "For every original element x,compute gx then multiply by g inverse to form the complete acting row.This is an element action,not a conjugated subgroup." }, "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": "S036", "uses": [ "G001", "S035" ], "before": "source roles = [31, 31, [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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]]", "after": "derived = [[31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [0, 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, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37]]", "relation": "implies", "rule_id": "Conjugacy_EVERY_g_EVERY_element_exact_action_row", "conditions": [], "explanation": { "zh-Hant": "對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。", "en": "For every original element x,compute gx then multiply by g inverse to form the complete acting row.This is an element action,not a conjugated subgroup." }, "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": "S037", "uses": [ "G001", "S036" ], "before": "source roles = [32, 32, [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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]]", "after": "derived = [[32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [0, 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, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39]]", "relation": "implies", "rule_id": "Conjugacy_EVERY_g_EVERY_element_exact_action_row", "conditions": [], "explanation": { "zh-Hant": "對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。", "en": "For every original element x,compute gx then multiply by g inverse to form the complete acting row.This is an element action,not a conjugated subgroup." }, "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": "S038", "uses": [ "G001", "S037" ], "before": "source roles = [33, 33, [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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]]", "after": "derived = [[33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [0, 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, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41]]", "relation": "implies", "rule_id": "Conjugacy_EVERY_g_EVERY_element_exact_action_row", "conditions": [], "explanation": { "zh-Hant": "對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。", "en": "For every original element x,compute gx then multiply by g inverse to form the complete acting row.This is an element action,not a conjugated subgroup." }, "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": "S039", "uses": [ "G001", "S038" ], "before": "source roles = [34, 34, [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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]]", "after": "derived = [[34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [0, 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, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43]]", "relation": "implies", "rule_id": "Conjugacy_EVERY_g_EVERY_element_exact_action_row", "conditions": [], "explanation": { "zh-Hant": "對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。", "en": "For every original element x,compute gx then multiply by g inverse to form the complete acting row.This is an element action,not a conjugated subgroup." }, "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": "S040", "uses": [ "G001", "S039" ], "before": "source roles = [35, 35, [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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]]", "after": "derived = [[35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [0, 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, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45]]", "relation": "implies", "rule_id": "Conjugacy_EVERY_g_EVERY_element_exact_action_row", "conditions": [], "explanation": { "zh-Hant": "對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。", "en": "For every original element x,compute gx then multiply by g inverse to form the complete acting row.This is an element action,not a conjugated subgroup." }, "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": "S041", "uses": [ "G001", "S040" ], "before": "source roles = [36, 36, [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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]]", "after": "derived = [[36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [0, 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, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47]]", "relation": "implies", "rule_id": "Conjugacy_EVERY_g_EVERY_element_exact_action_row", "conditions": [], "explanation": { "zh-Hant": "對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。", "en": "For every original element x,compute gx then multiply by g inverse to form the complete acting row.This is an element action,not a conjugated subgroup." }, "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": "S042", "uses": [ "G001", "S041" ], "before": "source roles = [37, 37, [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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]]", "after": "derived = [[37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12], [0, 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, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49]]", "relation": "implies", "rule_id": "Conjugacy_EVERY_g_EVERY_element_exact_action_row", "conditions": [], "explanation": { "zh-Hant": "對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。", "en": "For every original element x,compute gx then multiply by g inverse to form the complete acting row.This is an element action,not a conjugated subgroup." }, "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": "S043", "uses": [ "G001", "S042" ], "before": "source roles = [38, 38, [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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]]", "after": "derived = [[38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13], [0, 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, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51]]", "relation": "implies", "rule_id": "Conjugacy_EVERY_g_EVERY_element_exact_action_row", "conditions": [], "explanation": { "zh-Hant": "對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。", "en": "For every original element x,compute gx then multiply by g inverse to form the complete acting row.This is an element action,not a conjugated subgroup." }, "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": "S044", "uses": [ "G001", "S043" ], "before": "source roles = [39, 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, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]]", "after": "derived = [[39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14], [0, 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, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27]]", "relation": "implies", "rule_id": "Conjugacy_EVERY_g_EVERY_element_exact_action_row", "conditions": [], "explanation": { "zh-Hant": "對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。", "en": "For every original element x,compute gx then multiply by g inverse to form the complete acting row.This is an element action,not a conjugated subgroup." }, "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": "S045", "uses": [ "G001", "S044" ], "before": "source roles = [40, 40, [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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]]", "after": "derived = [[40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15], [0, 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, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29]]", "relation": "implies", "rule_id": "Conjugacy_EVERY_g_EVERY_element_exact_action_row", "conditions": [], "explanation": { "zh-Hant": "對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。", "en": "For every original element x,compute gx then multiply by g inverse to form the complete acting row.This is an element action,not a conjugated subgroup." }, "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": "S046", "uses": [ "G001", "S045" ], "before": "source roles = [41, 41, [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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]]", "after": "derived = [[41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16], [0, 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, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31]]", "relation": "implies", "rule_id": "Conjugacy_EVERY_g_EVERY_element_exact_action_row", "conditions": [], "explanation": { "zh-Hant": "對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。", "en": "For every original element x,compute gx then multiply by g inverse to form the complete acting row.This is an element action,not a conjugated subgroup." }, "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": "S047", "uses": [ "G001", "S046" ], "before": "source roles = [42, 42, [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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]]", "after": "derived = [[42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17], [0, 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, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33]]", "relation": "implies", "rule_id": "Conjugacy_EVERY_g_EVERY_element_exact_action_row", "conditions": [], "explanation": { "zh-Hant": "對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。", "en": "For every original element x,compute gx then multiply by g inverse to form the complete acting row.This is an element action,not a conjugated subgroup." }, "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": "S048", "uses": [ "G001", "S047" ], "before": "source roles = [43, 43, [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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]]", "after": "derived = [[43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18], [0, 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, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35]]", "relation": "implies", "rule_id": "Conjugacy_EVERY_g_EVERY_element_exact_action_row", "conditions": [], "explanation": { "zh-Hant": "對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。", "en": "For every original element x,compute gx then multiply by g inverse to form the complete acting row.This is an element action,not a conjugated subgroup." }, "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": "S049", "uses": [ "G001", "S048" ], "before": "source roles = [44, 44, [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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]]", "after": "derived = [[44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19], [0, 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, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37]]", "relation": "implies", "rule_id": "Conjugacy_EVERY_g_EVERY_element_exact_action_row", "conditions": [], "explanation": { "zh-Hant": "對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。", "en": "For every original element x,compute gx then multiply by g inverse to form the complete acting row.This is an element action,not a conjugated subgroup." }, "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": "S050", "uses": [ "G001", "S049" ], "before": "source roles = [45, 45, [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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]]", "after": "derived = [[45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20], [0, 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, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39]]", "relation": "implies", "rule_id": "Conjugacy_EVERY_g_EVERY_element_exact_action_row", "conditions": [], "explanation": { "zh-Hant": "對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。", "en": "For every original element x,compute gx then multiply by g inverse to form the complete acting row.This is an element action,not a conjugated subgroup." }, "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": "S051", "uses": [ "G001", "S050" ], "before": "source roles = [46, 46, [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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]]", "after": "derived = [[46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21], [0, 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, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41]]", "relation": "implies", "rule_id": "Conjugacy_EVERY_g_EVERY_element_exact_action_row", "conditions": [], "explanation": { "zh-Hant": "對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。", "en": "For every original element x,compute gx then multiply by g inverse to form the complete acting row.This is an element action,not a conjugated subgroup." }, "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": "S052", "uses": [ "G001", "S051" ], "before": "source roles = [47, 47, [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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]]", "after": "derived = [[47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22], [0, 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, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43]]", "relation": "implies", "rule_id": "Conjugacy_EVERY_g_EVERY_element_exact_action_row", "conditions": [], "explanation": { "zh-Hant": "對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。", "en": "For every original element x,compute gx then multiply by g inverse to form the complete acting row.This is an element action,not a conjugated subgroup." }, "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": "S053", "uses": [ "G001", "S052" ], "before": "source roles = [48, 48, [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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]]", "after": "derived = [[48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23], [0, 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, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45]]", "relation": "implies", "rule_id": "Conjugacy_EVERY_g_EVERY_element_exact_action_row", "conditions": [], "explanation": { "zh-Hant": "對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。", "en": "For every original element x,compute gx then multiply by g inverse to form the complete acting row.This is an element action,not a conjugated subgroup." }, "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": "S054", "uses": [ "G001", "S053" ], "before": "source roles = [49, 49, [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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]]", "after": "derived = [[49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24], [0, 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, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47]]", "relation": "implies", "rule_id": "Conjugacy_EVERY_g_EVERY_element_exact_action_row", "conditions": [], "explanation": { "zh-Hant": "對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。", "en": "For every original element x,compute gx then multiply by g inverse to form the complete acting row.This is an element action,not a conjugated subgroup." }, "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": "S055", "uses": [ "G001", "S054" ], "before": "source roles = [50, 50, [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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]]", "after": "derived = [[50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25], [0, 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, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49]]", "relation": "implies", "rule_id": "Conjugacy_EVERY_g_EVERY_element_exact_action_row", "conditions": [], "explanation": { "zh-Hant": "對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。", "en": "For every original element x,compute gx then multiply by g inverse to form the complete acting row.This is an element action,not a conjugated subgroup." }, "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": "S056", "uses": [ "G001", "S055" ], "before": "source roles = [51, 51, [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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]]", "after": "derived = [[51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 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, 0], [0, 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, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51]]", "relation": "implies", "rule_id": "Conjugacy_EVERY_g_EVERY_element_exact_action_row", "conditions": [], "explanation": { "zh-Hant": "對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。", "en": "For every original element x,compute gx then multiply by g inverse to form the complete acting row.This is an element action,not a conjugated subgroup." }, "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": "S057", "uses": [ "G001", "S056" ], "before": "source roles = [0, 52, [[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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51], [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, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27], [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, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29], [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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31], [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, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33], [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, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [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, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 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, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [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, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [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, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [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, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [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, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49], [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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51], [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, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27], [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, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29], [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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31], [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, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33], [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, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [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, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 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, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [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, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [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, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [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, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [0, 1, 2, 3, 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23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35], [0, 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, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37], [0, 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, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39], [0, 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, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41], [0, 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, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43], [0, 25, 24, 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identity separately.Listing an action table alone would not prove its axioms." }, "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": "S058", "uses": [ "G001", "S057" ], "before": "source roles = [[[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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51], [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, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27], [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, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 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49], [0, 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, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51], [0, 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, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27], [0, 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, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29], [0, 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, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31], [0, 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, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33], [0, 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, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35], [0, 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, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37], [0, 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, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39], [0, 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, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41], [0, 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, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43], [0, 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, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45], [0, 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, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47], [0, 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, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49], [0, 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, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51]]]", "after": "derived = [[[0], [1, 25], [10, 16], [11, 15], [12, 14], [13], [2, 24], [26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], [27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51], [3, 23], [4, 22], [5, 21], [6, 20], [7, 19], [8, 18], [9, 17]], 52, 52, 是]", "relation": "implies", "rule_id": "Conjugacy_complete_orbits_equal_or_disjoint_partition", "conditions": [], "explanation": { "zh-Hant": "e給自反、g^-1給反向、群乘積給傳遞；故所有完整orbits相等或不交並覆蓋G。按來源coset-like列表排序不代表它們是cosets；不引centralizer/class-size或類方程。", "en": "Identity gives reflexivity,inverses symmetry and products transitivity,so complete orbits are equal or disjoint and cover G.Source family sorting does not make classes cosets;no centralizer/class-size formula or class equation is introduced." }, "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": "S059", "uses": [ "G001", "S058" ], "before": "source roles = [0]", "after": "derived = [[[0, 0], [1, 0], [2, 0], [3, 0], [4, 0], [5, 0], [6, 0], [7, 0], [8, 0], [9, 0], [10, 0], [11, 0], [12, 0], [13, 0], [14, 0], [15, 0], [16, 0], [17, 0], [18, 0], [19, 0], [20, 0], [21, 0], [22, 0], [23, 0], [24, 0], [25, 0], [26, 0], [27, 0], [28, 0], [29, 0], [30, 0], [31, 0], [32, 0], [33, 0], [34, 0], [35, 0], [36, 0], [37, 0], [38, 0], [39, 0], [40, 0], [41, 0], [42, 0], [43, 0], [44, 0], [45, 0], [46, 0], [47, 0], [48, 0], [49, 0], [50, 0], [51, 0]], [0]]", "relation": "implies", "rule_id": "Conjugacy_actual_element_ALL_images_not_coset", "conditions": [], "explanation": { "zh-Hant": "保每個acting g的像，再去重成排序的完整class。", "en": "Retain the image for every acting g,then deduplicate into the full sorted class." }, "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": "S060", "uses": [ "G001", "S059" ], "before": "source roles = [0, 26, [0]]", "after": "derived = [[[0, 0, 0, 是], [1, 1, 1, 是], [2, 2, 2, 是], [3, 3, 3, 是], [4, 4, 4, 是], [5, 5, 5, 是], [6, 6, 6, 是], [7, 7, 7, 是], [8, 8, 8, 是], [9, 9, 9, 是], [10, 10, 10, 是], [11, 11, 11, 是], [12, 12, 12, 是], [13, 13, 13, 是], [14, 14, 14, 是], [15, 15, 15, 是], [16, 16, 16, 是], [17, 17, 17, 是], [18, 18, 18, 是], [19, 19, 19, 是], [20, 20, 20, 是], [21, 21, 21, 是], [22, 22, 22, 是], [23, 23, 23, 是], [24, 24, 24, 是], [25, 25, 25, 是], [26, 26, 26, 是], [27, 27, 27, 是], [28, 28, 28, 是], [29, 29, 29, 是], [30, 30, 30, 是], [31, 31, 31, 是], [32, 32, 32, 是], [33, 33, 33, 是], [34, 34, 34, 是], [35, 35, 35, 是], [36, 36, 36, 是], [37, 37, 37, 是], [38, 38, 38, 是], [39, 39, 39, 是], [40, 40, 40, 是], [41, 41, 41, 是], [42, 42, 42, 是], [43, 43, 43, 是], [44, 44, 44, 是], [45, 45, 45, 是], [46, 46, 46, 是], [47, 47, 47, 是], [48, 48, 48, 是], [49, 49, 49, 是], [50, 50, 50, 是], [51, 51, 51, 是]], 是, 否, 是]", "relation": "implies", "rule_id": "Conjugacy_singleton_iff_ALL_commute_and_actual_other_outside", "conditions": [], "explanation": { "zh-Hant": "gxg^-1=x右乘g等價gx=xg，故singleton恰當全g commute。other真不在完整orbit才可給nonconjugate證書，不聲稱所有class單點。", "en": "Right multiplying gxg^-1=x by g yields gx=xg,so singleton is equivalent to all-g commutation.Other is actually outside the complete orbit,justifying nonconjugacy without declaring every class singleton." }, "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, [ 0 ], true, 26, false, true, "singleton_class_iff_commutes_with_every_g_and_other_outside_class_is_nonconjugate_without_class_size_formula" ] }, "conclusion": { "canonical_value": "(0, [0], 是, 26, 否, 是, singleton_class_iff_commutes_with_every_g_and_other_outside_class_is_nonconjugate_without_class_size_formula)", "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-r03-finite-group-conjugacy-class-foundations", "batch_path": "public/data/questions/batches/batch-20261015-advance-20261114-r03-finite-group-conjugacy-class-foundations/questions.jsonl", "batch_sha256": "fb55ac17a6e4b309293fc0e8de28b716f1db87d926ccf462fbc46a1cf7e273f2", "original_line": 99, "original_line_sha256": "541dd43e6d8ed57ca10eea54e13995170277b5836171e549ca83a0153c3ffafe", "question_compact_sha256": "53e1f6aea8da5cdd632ecdbb6e6574068209f0be0d8bff0473d2833dbd9a1d23", "english_row_compact_sha256": "a5923eaaf659e28bfec5e2cd2cf90c051466d010a97eec1841af80fb840490c4", "english_manifest_path": "public/data/i18n/editions/batch-20261015-advance-20261114-r03-finite-group-conjugacy-class-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261015-advance-20261114-r03-finite-group-conjugacy-class-foundations.en.jsonl" }, "mko_bindings": { "mko-finite-group-conjugacy-class": { "sha256": "64bbc8cfc6df1fef690cba391584bc0ae2770f3af76115c24634669363f3bb26", "english_sha256": "026cb09f3fdfee6e3b2779e163410961306f437b84d80dc9b8673be1d56d9511", "original_path": "public/data/mko/finite-group-conjugacy-class.json", "english_path": "public/data/i18n/editions/mko-finite-group-conjugacy-class.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "7f1649723787c6d7cf1abe85636fd3740f2c3cc1784071db7e38069361c81fd4", "original_candidate_sha256": "74abdd8d27873746109af6906f901861afea71679b401b74521dcbd698a53b18", "original_candidate_line_sha256": "74abdd8d27873746109af6906f901861afea71679b401b74521dcbd698a53b18" }, "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": 60, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "c8c3755fef77be20b92d8e67e684581e74a5360d274eb9836b4d27b46477f8d3", "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." } }