{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0001","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-subgroup-qualification；資料={\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"subset\":[0,2,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 4], 是, 是, 是, 是, complete_Cayley_table_subgroup_certificate)","explanation_zh":"Wave6第五層核對=([0, 2, 4], 是, 是, 是, 是, complete_Cayley_table_subgroup_certificate)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-subgroup-qualification","seed":202611111001,"parameters":{"group_table":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],"identity":0,"subset":[0,2,4],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-subgroup-qualification","semantic_template":"qf-mod42-lagrange-subgroup-qualification/{\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"subset\":[0,2,4]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0002","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-subgroup-qualification；資料={\"group_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],\"identity\":0,\"subset\":[0,2,4,6]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 4, 6], 是, 是, 是, 是, complete_Cayley_table_subgroup_certificate)","explanation_zh":"Wave6第五層核對=([0, 2, 4, 6], 是, 是, 是, 是, complete_Cayley_table_subgroup_certificate)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-subgroup-qualification","seed":202611111002,"parameters":{"group_table":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],"identity":0,"subset":[0,2,4,6],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-subgroup-qualification","semantic_template":"qf-mod42-lagrange-subgroup-qualification/{\"group_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],\"identity\":0,\"subset\":[0,2,4,6]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0003","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-subgroup-qualification；資料={\"group_table\":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8]],\"identity\":0,\"subset\":[0,2,4,6,8]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8], 是, 是, 是, 是, complete_Cayley_table_subgroup_certificate)","explanation_zh":"Wave6第五層核對=([0, 2, 4, 6, 8], 是, 是, 是, 是, complete_Cayley_table_subgroup_certificate)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-subgroup-qualification","seed":202611111003,"parameters":{"group_table":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8]],"identity":0,"subset":[0,2,4,6,8],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-subgroup-qualification","semantic_template":"qf-mod42-lagrange-subgroup-qualification/{\"group_table\":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8]],\"identity\":0,\"subset\":[0,2,4,6,8]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0004","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-subgroup-qualification；資料={\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10]],\"identity\":0,\"subset\":[0,2,4,6,8,10]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10], 是, 是, 是, 是, complete_Cayley_table_subgroup_certificate)","explanation_zh":"Wave6第五層核對=([0, 2, 4, 6, 8, 10], 是, 是, 是, 是, complete_Cayley_table_subgroup_certificate)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-subgroup-qualification","seed":202611111004,"parameters":{"group_table":[[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10]],"identity":0,"subset":[0,2,4,6,8,10],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-subgroup-qualification","semantic_template":"qf-mod42-lagrange-subgroup-qualification/{\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10]],\"identity\":0,\"subset\":[0,2,4,6,8,10]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0005","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-subgroup-qualification；資料={\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12]],\"identity\":0,\"subset\":[0,2,4,6,8,10,12]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10, 12], 是, 是, 是, 是, complete_Cayley_table_subgroup_certificate)","explanation_zh":"Wave6第五層核對=([0, 2, 4, 6, 8, 10, 12], 是, 是, 是, 是, complete_Cayley_table_subgroup_certificate)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-subgroup-qualification","seed":202611111005,"parameters":{"group_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12]],"identity":0,"subset":[0,2,4,6,8,10,12],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-subgroup-qualification","semantic_template":"qf-mod42-lagrange-subgroup-qualification/{\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12]],\"identity\":0,\"subset\":[0,2,4,6,8,10,12]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0006","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-subgroup-qualification；資料={\"group_table\":[[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3]],\"identity\":1,\"subset\":[1,3,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 3, 5], 是, 是, 是, 是, complete_Cayley_table_subgroup_certificate)","explanation_zh":"Wave6第五層核對=([1, 3, 5], 是, 是, 是, 是, complete_Cayley_table_subgroup_certificate)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-subgroup-qualification","seed":202611111006,"parameters":{"group_table":[[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3]],"identity":1,"subset":[1,3,5],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-subgroup-qualification","semantic_template":"qf-mod42-lagrange-subgroup-qualification/{\"group_table\":[[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3]],\"identity\":1,\"subset\":[1,3,5]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0007","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-subgroup-qualification；資料={\"group_table\":[[7,0,1,2,3,4,5,6],[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5]],\"identity\":1,\"subset\":[1,3,5,7]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 3, 5, 7], 是, 是, 是, 是, complete_Cayley_table_subgroup_certificate)","explanation_zh":"Wave6第五層核對=([1, 3, 5, 7], 是, 是, 是, 是, complete_Cayley_table_subgroup_certificate)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-subgroup-qualification","seed":202611111007,"parameters":{"group_table":[[7,0,1,2,3,4,5,6],[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5]],"identity":1,"subset":[1,3,5,7],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-subgroup-qualification","semantic_template":"qf-mod42-lagrange-subgroup-qualification/{\"group_table\":[[7,0,1,2,3,4,5,6],[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5]],\"identity\":1,\"subset\":[1,3,5,7]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0008","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-subgroup-qualification；資料={\"group_table\":[[9,0,1,2,3,4,5,6,7,8],[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7]],\"identity\":1,\"subset\":[1,3,5,7,9]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 3, 5, 7, 9], 是, 是, 是, 是, complete_Cayley_table_subgroup_certificate)","explanation_zh":"Wave6第五層核對=([1, 3, 5, 7, 9], 是, 是, 是, 是, complete_Cayley_table_subgroup_certificate)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-subgroup-qualification","seed":202611111008,"parameters":{"group_table":[[9,0,1,2,3,4,5,6,7,8],[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7]],"identity":1,"subset":[1,3,5,7,9],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-subgroup-qualification","semantic_template":"qf-mod42-lagrange-subgroup-qualification/{\"group_table\":[[9,0,1,2,3,4,5,6,7,8],[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7]],\"identity\":1,\"subset\":[1,3,5,7,9]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0009","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-subgroup-qualification；資料={\"group_table\":[[11,0,1,2,3,4,5,6,7,8,9,10],[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9]],\"identity\":1,\"subset\":[1,3,5,7,9,11]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 3, 5, 7, 9, 11], 是, 是, 是, 是, complete_Cayley_table_subgroup_certificate)","explanation_zh":"Wave6第五層核對=([1, 3, 5, 7, 9, 11], 是, 是, 是, 是, complete_Cayley_table_subgroup_certificate)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-subgroup-qualification","seed":202611111009,"parameters":{"group_table":[[11,0,1,2,3,4,5,6,7,8,9,10],[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9]],"identity":1,"subset":[1,3,5,7,9,11],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-subgroup-qualification","semantic_template":"qf-mod42-lagrange-subgroup-qualification/{\"group_table\":[[11,0,1,2,3,4,5,6,7,8,9,10],[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9]],\"identity\":1,\"subset\":[1,3,5,7,9,11]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0010","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-subgroup-qualification；資料={\"group_table\":[[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11]],\"identity\":1,\"subset\":[1,3,5,7,9,11,13]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 3, 5, 7, 9, 11, 13], 是, 是, 是, 是, complete_Cayley_table_subgroup_certificate)","explanation_zh":"Wave6第五層核對=([1, 3, 5, 7, 9, 11, 13], 是, 是, 是, 是, complete_Cayley_table_subgroup_certificate)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-subgroup-qualification","seed":202611111010,"parameters":{"group_table":[[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11]],"identity":1,"subset":[1,3,5,7,9,11,13],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-subgroup-qualification","semantic_template":"qf-mod42-lagrange-subgroup-qualification/{\"group_table\":[[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11]],\"identity\":1,\"subset\":[1,3,5,7,9,11,13]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0011","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-subgroup-qualification；資料={\"group_table\":[[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2]],\"identity\":2,\"subset\":[0,2,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 4], 是, 是, 是, 是, complete_Cayley_table_subgroup_certificate)","explanation_zh":"Wave6第五層核對=([0, 2, 4], 是, 是, 是, 是, complete_Cayley_table_subgroup_certificate)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-subgroup-qualification","seed":202611111011,"parameters":{"group_table":[[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2]],"identity":2,"subset":[0,2,4],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-subgroup-qualification","semantic_template":"qf-mod42-lagrange-subgroup-qualification/{\"group_table\":[[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2]],\"identity\":2,\"subset\":[0,2,4]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0012","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-subgroup-qualification；資料={\"group_table\":[[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6],[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4]],\"identity\":2,\"subset\":[0,2,4,6]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 4, 6], 是, 是, 是, 是, complete_Cayley_table_subgroup_certificate)","explanation_zh":"Wave6第五層核對=([0, 2, 4, 6], 是, 是, 是, 是, complete_Cayley_table_subgroup_certificate)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-subgroup-qualification","seed":202611111012,"parameters":{"group_table":[[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6],[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4]],"identity":2,"subset":[0,2,4,6],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-subgroup-qualification","semantic_template":"qf-mod42-lagrange-subgroup-qualification/{\"group_table\":[[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6],[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4]],\"identity\":2,\"subset\":[0,2,4,6]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0013","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-subgroup-qualification；資料={\"group_table\":[[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8],[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6]],\"identity\":2,\"subset\":[0,2,4,6,8]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8], 是, 是, 是, 是, complete_Cayley_table_subgroup_certificate)","explanation_zh":"Wave6第五層核對=([0, 2, 4, 6, 8], 是, 是, 是, 是, complete_Cayley_table_subgroup_certificate)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-subgroup-qualification","seed":202611111013,"parameters":{"group_table":[[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8],[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6]],"identity":2,"subset":[0,2,4,6,8],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-subgroup-qualification","semantic_template":"qf-mod42-lagrange-subgroup-qualification/{\"group_table\":[[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8],[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6]],\"identity\":2,\"subset\":[0,2,4,6,8]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0014","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-subgroup-qualification；資料={\"group_table\":[[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10],[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8]],\"identity\":2,\"subset\":[0,2,4,6,8,10]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10], 是, 是, 是, 是, complete_Cayley_table_subgroup_certificate)","explanation_zh":"Wave6第五層核對=([0, 2, 4, 6, 8, 10], 是, 是, 是, 是, complete_Cayley_table_subgroup_certificate)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-subgroup-qualification","seed":202611111014,"parameters":{"group_table":[[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10],[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8]],"identity":2,"subset":[0,2,4,6,8,10],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-subgroup-qualification","semantic_template":"qf-mod42-lagrange-subgroup-qualification/{\"group_table\":[[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10],[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8]],\"identity\":2,\"subset\":[0,2,4,6,8,10]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0015","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-subgroup-qualification；資料={\"group_table\":[[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10]],\"identity\":2,\"subset\":[0,2,4,6,8,10,12]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10, 12], 是, 是, 是, 是, complete_Cayley_table_subgroup_certificate)","explanation_zh":"Wave6第五層核對=([0, 2, 4, 6, 8, 10, 12], 是, 是, 是, 是, complete_Cayley_table_subgroup_certificate)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-subgroup-qualification","seed":202611111015,"parameters":{"group_table":[[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10]],"identity":2,"subset":[0,2,4,6,8,10,12],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-subgroup-qualification","semantic_template":"qf-mod42-lagrange-subgroup-qualification/{\"group_table\":[[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10]],\"identity\":2,\"subset\":[0,2,4,6,8,10,12]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0016","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-subgroup-qualification；資料={\"group_table\":[[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1]],\"identity\":3,\"subset\":[1,3,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 3, 5], 是, 是, 是, 是, complete_Cayley_table_subgroup_certificate)","explanation_zh":"Wave6第五層核對=([1, 3, 5], 是, 是, 是, 是, complete_Cayley_table_subgroup_certificate)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-subgroup-qualification","seed":202611111016,"parameters":{"group_table":[[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1]],"identity":3,"subset":[1,3,5],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-subgroup-qualification","semantic_template":"qf-mod42-lagrange-subgroup-qualification/{\"group_table\":[[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1]],\"identity\":3,\"subset\":[1,3,5]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0017","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-subgroup-qualification；資料={\"group_table\":[[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6],[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3]],\"identity\":3,\"subset\":[1,3,5,7]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 3, 5, 7], 是, 是, 是, 是, complete_Cayley_table_subgroup_certificate)","explanation_zh":"Wave6第五層核對=([1, 3, 5, 7], 是, 是, 是, 是, complete_Cayley_table_subgroup_certificate)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-subgroup-qualification","seed":202611111017,"parameters":{"group_table":[[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6],[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3]],"identity":3,"subset":[1,3,5,7],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-subgroup-qualification","semantic_template":"qf-mod42-lagrange-subgroup-qualification/{\"group_table\":[[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6],[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3]],\"identity\":3,\"subset\":[1,3,5,7]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0018","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-subgroup-qualification；資料={\"group_table\":[[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8],[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5]],\"identity\":3,\"subset\":[1,3,5,7,9]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 3, 5, 7, 9], 是, 是, 是, 是, complete_Cayley_table_subgroup_certificate)","explanation_zh":"Wave6第五層核對=([1, 3, 5, 7, 9], 是, 是, 是, 是, complete_Cayley_table_subgroup_certificate)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-subgroup-qualification","seed":202611111018,"parameters":{"group_table":[[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8],[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5]],"identity":3,"subset":[1,3,5,7,9],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-subgroup-qualification","semantic_template":"qf-mod42-lagrange-subgroup-qualification/{\"group_table\":[[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8],[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5]],\"identity\":3,\"subset\":[1,3,5,7,9]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0019","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-subgroup-qualification；資料={\"group_table\":[[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10],[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7]],\"identity\":3,\"subset\":[1,3,5,7,9,11]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 3, 5, 7, 9, 11], 是, 是, 是, 是, complete_Cayley_table_subgroup_certificate)","explanation_zh":"Wave6第五層核對=([1, 3, 5, 7, 9, 11], 是, 是, 是, 是, complete_Cayley_table_subgroup_certificate)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-subgroup-qualification","seed":202611111019,"parameters":{"group_table":[[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10],[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7]],"identity":3,"subset":[1,3,5,7,9,11],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-subgroup-qualification","semantic_template":"qf-mod42-lagrange-subgroup-qualification/{\"group_table\":[[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10],[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7]],\"identity\":3,\"subset\":[1,3,5,7,9,11]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0020","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-subgroup-qualification；資料={\"group_table\":[[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9]],\"identity\":3,\"subset\":[1,3,5,7,9,11,13]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 3, 5, 7, 9, 11, 13], 是, 是, 是, 是, complete_Cayley_table_subgroup_certificate)","explanation_zh":"Wave6第五層核對=([1, 3, 5, 7, 9, 11, 13], 是, 是, 是, 是, complete_Cayley_table_subgroup_certificate)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-subgroup-qualification","seed":202611111020,"parameters":{"group_table":[[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9]],"identity":3,"subset":[1,3,5,7,9,11,13],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-subgroup-qualification","semantic_template":"qf-mod42-lagrange-subgroup-qualification/{\"group_table\":[[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9]],\"identity\":3,\"subset\":[1,3,5,7,9,11,13]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0021","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-subgroup-qualification；資料={\"group_table\":[[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0]],\"identity\":4,\"subset\":[0,2,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 4], 是, 是, 是, 是, complete_Cayley_table_subgroup_certificate)","explanation_zh":"Wave6第五層核對=([0, 2, 4], 是, 是, 是, 是, complete_Cayley_table_subgroup_certificate)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-subgroup-qualification","seed":202611111021,"parameters":{"group_table":[[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0]],"identity":4,"subset":[0,2,4],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-subgroup-qualification","semantic_template":"qf-mod42-lagrange-subgroup-qualification/{\"group_table\":[[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0]],\"identity\":4,\"subset\":[0,2,4]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0022","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-subgroup-qualification；資料={\"group_table\":[[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6],[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2]],\"identity\":4,\"subset\":[0,2,4,6]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 4, 6], 是, 是, 是, 是, complete_Cayley_table_subgroup_certificate)","explanation_zh":"Wave6第五層核對=([0, 2, 4, 6], 是, 是, 是, 是, complete_Cayley_table_subgroup_certificate)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-subgroup-qualification","seed":202611111022,"parameters":{"group_table":[[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6],[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2]],"identity":4,"subset":[0,2,4,6],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-subgroup-qualification","semantic_template":"qf-mod42-lagrange-subgroup-qualification/{\"group_table\":[[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6],[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2]],\"identity\":4,\"subset\":[0,2,4,6]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0023","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-subgroup-qualification；資料={\"group_table\":[[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8],[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4]],\"identity\":4,\"subset\":[0,2,4,6,8]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8], 是, 是, 是, 是, complete_Cayley_table_subgroup_certificate)","explanation_zh":"Wave6第五層核對=([0, 2, 4, 6, 8], 是, 是, 是, 是, complete_Cayley_table_subgroup_certificate)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-subgroup-qualification","seed":202611111023,"parameters":{"group_table":[[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8],[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4]],"identity":4,"subset":[0,2,4,6,8],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-subgroup-qualification","semantic_template":"qf-mod42-lagrange-subgroup-qualification/{\"group_table\":[[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8],[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4]],\"identity\":4,\"subset\":[0,2,4,6,8]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0024","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-subgroup-qualification；資料={\"group_table\":[[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10],[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6]],\"identity\":4,\"subset\":[0,2,4,6,8,10]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10], 是, 是, 是, 是, complete_Cayley_table_subgroup_certificate)","explanation_zh":"Wave6第五層核對=([0, 2, 4, 6, 8, 10], 是, 是, 是, 是, complete_Cayley_table_subgroup_certificate)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-subgroup-qualification","seed":202611111024,"parameters":{"group_table":[[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10],[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6]],"identity":4,"subset":[0,2,4,6,8,10],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-subgroup-qualification","semantic_template":"qf-mod42-lagrange-subgroup-qualification/{\"group_table\":[[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10],[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6]],\"identity\":4,\"subset\":[0,2,4,6,8,10]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0025","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-subgroup-qualification；資料={\"group_table\":[[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8]],\"identity\":4,\"subset\":[0,2,4,6,8,10,12]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10, 12], 是, 是, 是, 是, complete_Cayley_table_subgroup_certificate)","explanation_zh":"Wave6第五層核對=([0, 2, 4, 6, 8, 10, 12], 是, 是, 是, 是, complete_Cayley_table_subgroup_certificate)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-subgroup-qualification","seed":202611111025,"parameters":{"group_table":[[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8]],"identity":4,"subset":[0,2,4,6,8,10,12],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-subgroup-qualification","semantic_template":"qf-mod42-lagrange-subgroup-qualification/{\"group_table\":[[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8]],\"identity\":4,\"subset\":[0,2,4,6,8,10,12]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0026","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-left-coset-partition；資料={\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"subset\":[0,2,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, [0, 2, 4]], [1, [1, 3, 5]]], 是, [0, 1, 2, 3, 4, 5], 是, left_cosets_equal_or_disjoint_and_cover_G)","explanation_zh":"Wave6第五層核對=([[0, [0, 2, 4]], [1, [1, 3, 5]]], 是, [0, 1, 2, 3, 4, 5], 是, left_cosets_equal_or_disjoint_and_cover_G)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-left-coset-partition","seed":202611111026,"parameters":{"group_table":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],"identity":0,"subset":[0,2,4],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-left-coset-partition","semantic_template":"qf-mod42-lagrange-left-coset-partition/{\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"subset\":[0,2,4]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0027","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-left-coset-partition；資料={\"group_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],\"identity\":0,\"subset\":[0,2,4,6]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, [0, 2, 4, 6]], [1, [1, 3, 5, 7]]], 是, [0, 1, 2, 3, 4, 5, 6, 7], 是, left_cosets_equal_or_disjoint_and_cover_G)","explanation_zh":"Wave6第五層核對=([[0, [0, 2, 4, 6]], [1, [1, 3, 5, 7]]], 是, [0, 1, 2, 3, 4, 5, 6, 7], 是, left_cosets_equal_or_disjoint_and_cover_G)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-left-coset-partition","seed":202611111027,"parameters":{"group_table":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],"identity":0,"subset":[0,2,4,6],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-left-coset-partition","semantic_template":"qf-mod42-lagrange-left-coset-partition/{\"group_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],\"identity\":0,\"subset\":[0,2,4,6]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0028","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-left-coset-partition；資料={\"group_table\":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8]],\"identity\":0,\"subset\":[0,2,4,6,8]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, [0, 2, 4, 6, 8]], [1, [1, 3, 5, 7, 9]]], 是, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是, left_cosets_equal_or_disjoint_and_cover_G)","explanation_zh":"Wave6第五層核對=([[0, [0, 2, 4, 6, 8]], [1, [1, 3, 5, 7, 9]]], 是, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是, left_cosets_equal_or_disjoint_and_cover_G)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-left-coset-partition","seed":202611111028,"parameters":{"group_table":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8]],"identity":0,"subset":[0,2,4,6,8],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-left-coset-partition","semantic_template":"qf-mod42-lagrange-left-coset-partition/{\"group_table\":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8]],\"identity\":0,\"subset\":[0,2,4,6,8]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0029","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-left-coset-partition；資料={\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10]],\"identity\":0,\"subset\":[0,2,4,6,8,10]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, [0, 2, 4, 6, 8, 10]], [1, [1, 3, 5, 7, 9, 11]]], 是, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是, left_cosets_equal_or_disjoint_and_cover_G)","explanation_zh":"Wave6第五層核對=([[0, [0, 2, 4, 6, 8, 10]], [1, [1, 3, 5, 7, 9, 11]]], 是, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是, left_cosets_equal_or_disjoint_and_cover_G)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-left-coset-partition","seed":202611111029,"parameters":{"group_table":[[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10]],"identity":0,"subset":[0,2,4,6,8,10],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-left-coset-partition","semantic_template":"qf-mod42-lagrange-left-coset-partition/{\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10]],\"identity\":0,\"subset\":[0,2,4,6,8,10]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0030","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-left-coset-partition；資料={\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12]],\"identity\":0,\"subset\":[0,2,4,6,8,10,12]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, [0, 2, 4, 6, 8, 10, 12]], [1, [1, 3, 5, 7, 9, 11, 13]]], 是, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], 是, left_cosets_equal_or_disjoint_and_cover_G)","explanation_zh":"Wave6第五層核對=([[0, [0, 2, 4, 6, 8, 10, 12]], [1, [1, 3, 5, 7, 9, 11, 13]]], 是, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], 是, left_cosets_equal_or_disjoint_and_cover_G)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-left-coset-partition","seed":202611111030,"parameters":{"group_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12]],"identity":0,"subset":[0,2,4,6,8,10,12],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-left-coset-partition","semantic_template":"qf-mod42-lagrange-left-coset-partition/{\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12]],\"identity\":0,\"subset\":[0,2,4,6,8,10,12]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0031","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-left-coset-partition；資料={\"group_table\":[[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3]],\"identity\":1,\"subset\":[1,3,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, [0, 2, 4]], [1, [1, 3, 5]]], 是, [0, 1, 2, 3, 4, 5], 是, left_cosets_equal_or_disjoint_and_cover_G)","explanation_zh":"Wave6第五層核對=([[0, [0, 2, 4]], [1, [1, 3, 5]]], 是, [0, 1, 2, 3, 4, 5], 是, left_cosets_equal_or_disjoint_and_cover_G)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-left-coset-partition","seed":202611111031,"parameters":{"group_table":[[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3]],"identity":1,"subset":[1,3,5],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-left-coset-partition","semantic_template":"qf-mod42-lagrange-left-coset-partition/{\"group_table\":[[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3]],\"identity\":1,\"subset\":[1,3,5]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0032","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-left-coset-partition；資料={\"group_table\":[[7,0,1,2,3,4,5,6],[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5]],\"identity\":1,\"subset\":[1,3,5,7]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, [0, 2, 4, 6]], [1, [1, 3, 5, 7]]], 是, [0, 1, 2, 3, 4, 5, 6, 7], 是, left_cosets_equal_or_disjoint_and_cover_G)","explanation_zh":"Wave6第五層核對=([[0, [0, 2, 4, 6]], [1, [1, 3, 5, 7]]], 是, [0, 1, 2, 3, 4, 5, 6, 7], 是, left_cosets_equal_or_disjoint_and_cover_G)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-left-coset-partition","seed":202611111032,"parameters":{"group_table":[[7,0,1,2,3,4,5,6],[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5]],"identity":1,"subset":[1,3,5,7],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-left-coset-partition","semantic_template":"qf-mod42-lagrange-left-coset-partition/{\"group_table\":[[7,0,1,2,3,4,5,6],[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5]],\"identity\":1,\"subset\":[1,3,5,7]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0033","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-left-coset-partition；資料={\"group_table\":[[9,0,1,2,3,4,5,6,7,8],[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7]],\"identity\":1,\"subset\":[1,3,5,7,9]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, [0, 2, 4, 6, 8]], [1, [1, 3, 5, 7, 9]]], 是, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是, left_cosets_equal_or_disjoint_and_cover_G)","explanation_zh":"Wave6第五層核對=([[0, [0, 2, 4, 6, 8]], [1, [1, 3, 5, 7, 9]]], 是, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是, left_cosets_equal_or_disjoint_and_cover_G)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-left-coset-partition","seed":202611111033,"parameters":{"group_table":[[9,0,1,2,3,4,5,6,7,8],[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7]],"identity":1,"subset":[1,3,5,7,9],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-left-coset-partition","semantic_template":"qf-mod42-lagrange-left-coset-partition/{\"group_table\":[[9,0,1,2,3,4,5,6,7,8],[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7]],\"identity\":1,\"subset\":[1,3,5,7,9]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0034","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-left-coset-partition；資料={\"group_table\":[[11,0,1,2,3,4,5,6,7,8,9,10],[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9]],\"identity\":1,\"subset\":[1,3,5,7,9,11]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, [0, 2, 4, 6, 8, 10]], [1, [1, 3, 5, 7, 9, 11]]], 是, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是, left_cosets_equal_or_disjoint_and_cover_G)","explanation_zh":"Wave6第五層核對=([[0, [0, 2, 4, 6, 8, 10]], [1, [1, 3, 5, 7, 9, 11]]], 是, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是, left_cosets_equal_or_disjoint_and_cover_G)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-left-coset-partition","seed":202611111034,"parameters":{"group_table":[[11,0,1,2,3,4,5,6,7,8,9,10],[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9]],"identity":1,"subset":[1,3,5,7,9,11],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-left-coset-partition","semantic_template":"qf-mod42-lagrange-left-coset-partition/{\"group_table\":[[11,0,1,2,3,4,5,6,7,8,9,10],[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9]],\"identity\":1,\"subset\":[1,3,5,7,9,11]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0035","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-left-coset-partition；資料={\"group_table\":[[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11]],\"identity\":1,\"subset\":[1,3,5,7,9,11,13]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, [0, 2, 4, 6, 8, 10, 12]], [1, [1, 3, 5, 7, 9, 11, 13]]], 是, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], 是, left_cosets_equal_or_disjoint_and_cover_G)","explanation_zh":"Wave6第五層核對=([[0, [0, 2, 4, 6, 8, 10, 12]], [1, [1, 3, 5, 7, 9, 11, 13]]], 是, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], 是, left_cosets_equal_or_disjoint_and_cover_G)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-left-coset-partition","seed":202611111035,"parameters":{"group_table":[[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11]],"identity":1,"subset":[1,3,5,7,9,11,13],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-left-coset-partition","semantic_template":"qf-mod42-lagrange-left-coset-partition/{\"group_table\":[[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11]],\"identity\":1,\"subset\":[1,3,5,7,9,11,13]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0036","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-left-coset-partition；資料={\"group_table\":[[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2]],\"identity\":2,\"subset\":[0,2,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, [0, 2, 4]], [1, [1, 3, 5]]], 是, [0, 1, 2, 3, 4, 5], 是, left_cosets_equal_or_disjoint_and_cover_G)","explanation_zh":"Wave6第五層核對=([[0, [0, 2, 4]], [1, [1, 3, 5]]], 是, [0, 1, 2, 3, 4, 5], 是, left_cosets_equal_or_disjoint_and_cover_G)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-left-coset-partition","seed":202611111036,"parameters":{"group_table":[[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2]],"identity":2,"subset":[0,2,4],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-left-coset-partition","semantic_template":"qf-mod42-lagrange-left-coset-partition/{\"group_table\":[[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2]],\"identity\":2,\"subset\":[0,2,4]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0037","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-left-coset-partition；資料={\"group_table\":[[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6],[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4]],\"identity\":2,\"subset\":[0,2,4,6]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, [0, 2, 4, 6]], [1, [1, 3, 5, 7]]], 是, [0, 1, 2, 3, 4, 5, 6, 7], 是, left_cosets_equal_or_disjoint_and_cover_G)","explanation_zh":"Wave6第五層核對=([[0, [0, 2, 4, 6]], [1, [1, 3, 5, 7]]], 是, [0, 1, 2, 3, 4, 5, 6, 7], 是, left_cosets_equal_or_disjoint_and_cover_G)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-left-coset-partition","seed":202611111037,"parameters":{"group_table":[[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6],[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4]],"identity":2,"subset":[0,2,4,6],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-left-coset-partition","semantic_template":"qf-mod42-lagrange-left-coset-partition/{\"group_table\":[[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6],[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4]],\"identity\":2,\"subset\":[0,2,4,6]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0038","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-left-coset-partition；資料={\"group_table\":[[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8],[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6]],\"identity\":2,\"subset\":[0,2,4,6,8]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, [0, 2, 4, 6, 8]], [1, [1, 3, 5, 7, 9]]], 是, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是, left_cosets_equal_or_disjoint_and_cover_G)","explanation_zh":"Wave6第五層核對=([[0, [0, 2, 4, 6, 8]], [1, [1, 3, 5, 7, 9]]], 是, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是, left_cosets_equal_or_disjoint_and_cover_G)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-left-coset-partition","seed":202611111038,"parameters":{"group_table":[[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8],[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6]],"identity":2,"subset":[0,2,4,6,8],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-left-coset-partition","semantic_template":"qf-mod42-lagrange-left-coset-partition/{\"group_table\":[[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8],[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6]],\"identity\":2,\"subset\":[0,2,4,6,8]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0039","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-left-coset-partition；資料={\"group_table\":[[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10],[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8]],\"identity\":2,\"subset\":[0,2,4,6,8,10]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, [0, 2, 4, 6, 8, 10]], [1, [1, 3, 5, 7, 9, 11]]], 是, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是, left_cosets_equal_or_disjoint_and_cover_G)","explanation_zh":"Wave6第五層核對=([[0, [0, 2, 4, 6, 8, 10]], [1, [1, 3, 5, 7, 9, 11]]], 是, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是, left_cosets_equal_or_disjoint_and_cover_G)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-left-coset-partition","seed":202611111039,"parameters":{"group_table":[[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10],[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8]],"identity":2,"subset":[0,2,4,6,8,10],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-left-coset-partition","semantic_template":"qf-mod42-lagrange-left-coset-partition/{\"group_table\":[[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10],[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8]],\"identity\":2,\"subset\":[0,2,4,6,8,10]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0040","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-left-coset-partition；資料={\"group_table\":[[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10]],\"identity\":2,\"subset\":[0,2,4,6,8,10,12]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, [0, 2, 4, 6, 8, 10, 12]], [1, [1, 3, 5, 7, 9, 11, 13]]], 是, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], 是, left_cosets_equal_or_disjoint_and_cover_G)","explanation_zh":"Wave6第五層核對=([[0, [0, 2, 4, 6, 8, 10, 12]], [1, [1, 3, 5, 7, 9, 11, 13]]], 是, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], 是, left_cosets_equal_or_disjoint_and_cover_G)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-left-coset-partition","seed":202611111040,"parameters":{"group_table":[[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10]],"identity":2,"subset":[0,2,4,6,8,10,12],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-left-coset-partition","semantic_template":"qf-mod42-lagrange-left-coset-partition/{\"group_table\":[[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10]],\"identity\":2,\"subset\":[0,2,4,6,8,10,12]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0041","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-left-coset-partition；資料={\"group_table\":[[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1]],\"identity\":3,\"subset\":[1,3,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, [0, 2, 4]], [1, [1, 3, 5]]], 是, [0, 1, 2, 3, 4, 5], 是, left_cosets_equal_or_disjoint_and_cover_G)","explanation_zh":"Wave6第五層核對=([[0, [0, 2, 4]], [1, [1, 3, 5]]], 是, [0, 1, 2, 3, 4, 5], 是, left_cosets_equal_or_disjoint_and_cover_G)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-left-coset-partition","seed":202611111041,"parameters":{"group_table":[[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1]],"identity":3,"subset":[1,3,5],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-left-coset-partition","semantic_template":"qf-mod42-lagrange-left-coset-partition/{\"group_table\":[[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1]],\"identity\":3,\"subset\":[1,3,5]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0042","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-left-coset-partition；資料={\"group_table\":[[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6],[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3]],\"identity\":3,\"subset\":[1,3,5,7]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, [0, 2, 4, 6]], [1, [1, 3, 5, 7]]], 是, [0, 1, 2, 3, 4, 5, 6, 7], 是, left_cosets_equal_or_disjoint_and_cover_G)","explanation_zh":"Wave6第五層核對=([[0, [0, 2, 4, 6]], [1, [1, 3, 5, 7]]], 是, [0, 1, 2, 3, 4, 5, 6, 7], 是, left_cosets_equal_or_disjoint_and_cover_G)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-left-coset-partition","seed":202611111042,"parameters":{"group_table":[[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6],[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3]],"identity":3,"subset":[1,3,5,7],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-left-coset-partition","semantic_template":"qf-mod42-lagrange-left-coset-partition/{\"group_table\":[[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6],[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3]],\"identity\":3,\"subset\":[1,3,5,7]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0043","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-left-coset-partition；資料={\"group_table\":[[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8],[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5]],\"identity\":3,\"subset\":[1,3,5,7,9]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, [0, 2, 4, 6, 8]], [1, [1, 3, 5, 7, 9]]], 是, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是, left_cosets_equal_or_disjoint_and_cover_G)","explanation_zh":"Wave6第五層核對=([[0, [0, 2, 4, 6, 8]], [1, [1, 3, 5, 7, 9]]], 是, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是, left_cosets_equal_or_disjoint_and_cover_G)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-left-coset-partition","seed":202611111043,"parameters":{"group_table":[[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8],[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5]],"identity":3,"subset":[1,3,5,7,9],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-left-coset-partition","semantic_template":"qf-mod42-lagrange-left-coset-partition/{\"group_table\":[[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8],[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5]],\"identity\":3,\"subset\":[1,3,5,7,9]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0044","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-left-coset-partition；資料={\"group_table\":[[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10],[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7]],\"identity\":3,\"subset\":[1,3,5,7,9,11]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, [0, 2, 4, 6, 8, 10]], [1, [1, 3, 5, 7, 9, 11]]], 是, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是, left_cosets_equal_or_disjoint_and_cover_G)","explanation_zh":"Wave6第五層核對=([[0, [0, 2, 4, 6, 8, 10]], [1, [1, 3, 5, 7, 9, 11]]], 是, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是, left_cosets_equal_or_disjoint_and_cover_G)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-left-coset-partition","seed":202611111044,"parameters":{"group_table":[[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10],[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7]],"identity":3,"subset":[1,3,5,7,9,11],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-left-coset-partition","semantic_template":"qf-mod42-lagrange-left-coset-partition/{\"group_table\":[[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10],[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7]],\"identity\":3,\"subset\":[1,3,5,7,9,11]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0045","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-left-coset-partition；資料={\"group_table\":[[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9]],\"identity\":3,\"subset\":[1,3,5,7,9,11,13]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, [0, 2, 4, 6, 8, 10, 12]], [1, [1, 3, 5, 7, 9, 11, 13]]], 是, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], 是, left_cosets_equal_or_disjoint_and_cover_G)","explanation_zh":"Wave6第五層核對=([[0, [0, 2, 4, 6, 8, 10, 12]], [1, [1, 3, 5, 7, 9, 11, 13]]], 是, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], 是, left_cosets_equal_or_disjoint_and_cover_G)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-left-coset-partition","seed":202611111045,"parameters":{"group_table":[[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9]],"identity":3,"subset":[1,3,5,7,9,11,13],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-left-coset-partition","semantic_template":"qf-mod42-lagrange-left-coset-partition/{\"group_table\":[[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9]],\"identity\":3,\"subset\":[1,3,5,7,9,11,13]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0046","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-left-coset-partition；資料={\"group_table\":[[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0]],\"identity\":4,\"subset\":[0,2,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, [0, 2, 4]], [1, [1, 3, 5]]], 是, [0, 1, 2, 3, 4, 5], 是, left_cosets_equal_or_disjoint_and_cover_G)","explanation_zh":"Wave6第五層核對=([[0, [0, 2, 4]], [1, [1, 3, 5]]], 是, [0, 1, 2, 3, 4, 5], 是, left_cosets_equal_or_disjoint_and_cover_G)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-left-coset-partition","seed":202611111046,"parameters":{"group_table":[[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0]],"identity":4,"subset":[0,2,4],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-left-coset-partition","semantic_template":"qf-mod42-lagrange-left-coset-partition/{\"group_table\":[[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0]],\"identity\":4,\"subset\":[0,2,4]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0047","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-left-coset-partition；資料={\"group_table\":[[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6],[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2]],\"identity\":4,\"subset\":[0,2,4,6]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, [0, 2, 4, 6]], [1, [1, 3, 5, 7]]], 是, [0, 1, 2, 3, 4, 5, 6, 7], 是, left_cosets_equal_or_disjoint_and_cover_G)","explanation_zh":"Wave6第五層核對=([[0, [0, 2, 4, 6]], [1, [1, 3, 5, 7]]], 是, [0, 1, 2, 3, 4, 5, 6, 7], 是, left_cosets_equal_or_disjoint_and_cover_G)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-left-coset-partition","seed":202611111047,"parameters":{"group_table":[[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6],[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2]],"identity":4,"subset":[0,2,4,6],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-left-coset-partition","semantic_template":"qf-mod42-lagrange-left-coset-partition/{\"group_table\":[[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6],[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2]],\"identity\":4,\"subset\":[0,2,4,6]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0048","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-left-coset-partition；資料={\"group_table\":[[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8],[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4]],\"identity\":4,\"subset\":[0,2,4,6,8]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, [0, 2, 4, 6, 8]], [1, [1, 3, 5, 7, 9]]], 是, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是, left_cosets_equal_or_disjoint_and_cover_G)","explanation_zh":"Wave6第五層核對=([[0, [0, 2, 4, 6, 8]], [1, [1, 3, 5, 7, 9]]], 是, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是, left_cosets_equal_or_disjoint_and_cover_G)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-left-coset-partition","seed":202611111048,"parameters":{"group_table":[[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8],[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4]],"identity":4,"subset":[0,2,4,6,8],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-left-coset-partition","semantic_template":"qf-mod42-lagrange-left-coset-partition/{\"group_table\":[[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8],[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4]],\"identity\":4,\"subset\":[0,2,4,6,8]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0049","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-left-coset-partition；資料={\"group_table\":[[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10],[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6]],\"identity\":4,\"subset\":[0,2,4,6,8,10]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, [0, 2, 4, 6, 8, 10]], [1, [1, 3, 5, 7, 9, 11]]], 是, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是, left_cosets_equal_or_disjoint_and_cover_G)","explanation_zh":"Wave6第五層核對=([[0, [0, 2, 4, 6, 8, 10]], [1, [1, 3, 5, 7, 9, 11]]], 是, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是, left_cosets_equal_or_disjoint_and_cover_G)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-left-coset-partition","seed":202611111049,"parameters":{"group_table":[[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10],[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6]],"identity":4,"subset":[0,2,4,6,8,10],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-left-coset-partition","semantic_template":"qf-mod42-lagrange-left-coset-partition/{\"group_table\":[[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10],[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6]],\"identity\":4,\"subset\":[0,2,4,6,8,10]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0050","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-left-coset-partition；資料={\"group_table\":[[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8]],\"identity\":4,\"subset\":[0,2,4,6,8,10,12]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, [0, 2, 4, 6, 8, 10, 12]], [1, [1, 3, 5, 7, 9, 11, 13]]], 是, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], 是, left_cosets_equal_or_disjoint_and_cover_G)","explanation_zh":"Wave6第五層核對=([[0, [0, 2, 4, 6, 8, 10, 12]], [1, [1, 3, 5, 7, 9, 11, 13]]], 是, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], 是, left_cosets_equal_or_disjoint_and_cover_G)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-left-coset-partition","seed":202611111050,"parameters":{"group_table":[[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8]],"identity":4,"subset":[0,2,4,6,8,10,12],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-left-coset-partition","semantic_template":"qf-mod42-lagrange-left-coset-partition/{\"group_table\":[[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8]],\"identity\":4,\"subset\":[0,2,4,6,8,10,12]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0051","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-divisibility-identity；資料={\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"subset\":[0,2,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 3, 2, 6, 0, 是)","explanation_zh":"Wave6第五層核對=(6, 3, 2, 6, 0, 是)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-divisibility-identity","seed":202611111051,"parameters":{"group_table":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],"identity":0,"subset":[0,2,4],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-divisibility-identity","semantic_template":"qf-mod42-lagrange-divisibility-identity/{\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"subset\":[0,2,4]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0052","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-divisibility-identity；資料={\"group_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],\"identity\":0,\"subset\":[0,2,4,6]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, 4, 2, 8, 0, 是)","explanation_zh":"Wave6第五層核對=(8, 4, 2, 8, 0, 是)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-divisibility-identity","seed":202611111052,"parameters":{"group_table":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],"identity":0,"subset":[0,2,4,6],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-divisibility-identity","semantic_template":"qf-mod42-lagrange-divisibility-identity/{\"group_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],\"identity\":0,\"subset\":[0,2,4,6]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0053","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-divisibility-identity；資料={\"group_table\":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8]],\"identity\":0,\"subset\":[0,2,4,6,8]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, 5, 2, 10, 0, 是)","explanation_zh":"Wave6第五層核對=(10, 5, 2, 10, 0, 是)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-divisibility-identity","seed":202611111053,"parameters":{"group_table":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8]],"identity":0,"subset":[0,2,4,6,8],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-divisibility-identity","semantic_template":"qf-mod42-lagrange-divisibility-identity/{\"group_table\":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8]],\"identity\":0,\"subset\":[0,2,4,6,8]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0054","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-divisibility-identity；資料={\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10]],\"identity\":0,\"subset\":[0,2,4,6,8,10]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, 6, 2, 12, 0, 是)","explanation_zh":"Wave6第五層核對=(12, 6, 2, 12, 0, 是)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-divisibility-identity","seed":202611111054,"parameters":{"group_table":[[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10]],"identity":0,"subset":[0,2,4,6,8,10],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-divisibility-identity","semantic_template":"qf-mod42-lagrange-divisibility-identity/{\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10]],\"identity\":0,\"subset\":[0,2,4,6,8,10]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0055","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-divisibility-identity；資料={\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12]],\"identity\":0,\"subset\":[0,2,4,6,8,10,12]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(14, 7, 2, 14, 0, 是)","explanation_zh":"Wave6第五層核對=(14, 7, 2, 14, 0, 是)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-divisibility-identity","seed":202611111055,"parameters":{"group_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12]],"identity":0,"subset":[0,2,4,6,8,10,12],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-divisibility-identity","semantic_template":"qf-mod42-lagrange-divisibility-identity/{\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12]],\"identity\":0,\"subset\":[0,2,4,6,8,10,12]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0056","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-divisibility-identity；資料={\"group_table\":[[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3]],\"identity\":1,\"subset\":[1,3,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 3, 2, 6, 0, 是)","explanation_zh":"Wave6第五層核對=(6, 3, 2, 6, 0, 是)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-divisibility-identity","seed":202611111056,"parameters":{"group_table":[[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3]],"identity":1,"subset":[1,3,5],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-divisibility-identity","semantic_template":"qf-mod42-lagrange-divisibility-identity/{\"group_table\":[[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3]],\"identity\":1,\"subset\":[1,3,5]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0057","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-divisibility-identity；資料={\"group_table\":[[7,0,1,2,3,4,5,6],[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5]],\"identity\":1,\"subset\":[1,3,5,7]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, 4, 2, 8, 0, 是)","explanation_zh":"Wave6第五層核對=(8, 4, 2, 8, 0, 是)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-divisibility-identity","seed":202611111057,"parameters":{"group_table":[[7,0,1,2,3,4,5,6],[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5]],"identity":1,"subset":[1,3,5,7],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-divisibility-identity","semantic_template":"qf-mod42-lagrange-divisibility-identity/{\"group_table\":[[7,0,1,2,3,4,5,6],[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5]],\"identity\":1,\"subset\":[1,3,5,7]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0058","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-divisibility-identity；資料={\"group_table\":[[9,0,1,2,3,4,5,6,7,8],[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7]],\"identity\":1,\"subset\":[1,3,5,7,9]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, 5, 2, 10, 0, 是)","explanation_zh":"Wave6第五層核對=(10, 5, 2, 10, 0, 是)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-divisibility-identity","seed":202611111058,"parameters":{"group_table":[[9,0,1,2,3,4,5,6,7,8],[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7]],"identity":1,"subset":[1,3,5,7,9],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-divisibility-identity","semantic_template":"qf-mod42-lagrange-divisibility-identity/{\"group_table\":[[9,0,1,2,3,4,5,6,7,8],[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7]],\"identity\":1,\"subset\":[1,3,5,7,9]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0059","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-divisibility-identity；資料={\"group_table\":[[11,0,1,2,3,4,5,6,7,8,9,10],[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9]],\"identity\":1,\"subset\":[1,3,5,7,9,11]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, 6, 2, 12, 0, 是)","explanation_zh":"Wave6第五層核對=(12, 6, 2, 12, 0, 是)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-divisibility-identity","seed":202611111059,"parameters":{"group_table":[[11,0,1,2,3,4,5,6,7,8,9,10],[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9]],"identity":1,"subset":[1,3,5,7,9,11],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-divisibility-identity","semantic_template":"qf-mod42-lagrange-divisibility-identity/{\"group_table\":[[11,0,1,2,3,4,5,6,7,8,9,10],[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9]],\"identity\":1,\"subset\":[1,3,5,7,9,11]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0060","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-divisibility-identity；資料={\"group_table\":[[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11]],\"identity\":1,\"subset\":[1,3,5,7,9,11,13]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(14, 7, 2, 14, 0, 是)","explanation_zh":"Wave6第五層核對=(14, 7, 2, 14, 0, 是)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-divisibility-identity","seed":202611111060,"parameters":{"group_table":[[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11]],"identity":1,"subset":[1,3,5,7,9,11,13],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-divisibility-identity","semantic_template":"qf-mod42-lagrange-divisibility-identity/{\"group_table\":[[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11]],\"identity\":1,\"subset\":[1,3,5,7,9,11,13]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0061","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-divisibility-identity；資料={\"group_table\":[[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2]],\"identity\":2,\"subset\":[0,2,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 3, 2, 6, 0, 是)","explanation_zh":"Wave6第五層核對=(6, 3, 2, 6, 0, 是)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-divisibility-identity","seed":202611111061,"parameters":{"group_table":[[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2]],"identity":2,"subset":[0,2,4],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-divisibility-identity","semantic_template":"qf-mod42-lagrange-divisibility-identity/{\"group_table\":[[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2]],\"identity\":2,\"subset\":[0,2,4]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0062","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-divisibility-identity；資料={\"group_table\":[[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6],[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4]],\"identity\":2,\"subset\":[0,2,4,6]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, 4, 2, 8, 0, 是)","explanation_zh":"Wave6第五層核對=(8, 4, 2, 8, 0, 是)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-divisibility-identity","seed":202611111062,"parameters":{"group_table":[[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6],[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4]],"identity":2,"subset":[0,2,4,6],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-divisibility-identity","semantic_template":"qf-mod42-lagrange-divisibility-identity/{\"group_table\":[[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6],[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4]],\"identity\":2,\"subset\":[0,2,4,6]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0063","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-divisibility-identity；資料={\"group_table\":[[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8],[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6]],\"identity\":2,\"subset\":[0,2,4,6,8]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, 5, 2, 10, 0, 是)","explanation_zh":"Wave6第五層核對=(10, 5, 2, 10, 0, 是)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-divisibility-identity","seed":202611111063,"parameters":{"group_table":[[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8],[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6]],"identity":2,"subset":[0,2,4,6,8],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-divisibility-identity","semantic_template":"qf-mod42-lagrange-divisibility-identity/{\"group_table\":[[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8],[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6]],\"identity\":2,\"subset\":[0,2,4,6,8]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0064","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-divisibility-identity；資料={\"group_table\":[[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10],[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8]],\"identity\":2,\"subset\":[0,2,4,6,8,10]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, 6, 2, 12, 0, 是)","explanation_zh":"Wave6第五層核對=(12, 6, 2, 12, 0, 是)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-divisibility-identity","seed":202611111064,"parameters":{"group_table":[[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10],[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8]],"identity":2,"subset":[0,2,4,6,8,10],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-divisibility-identity","semantic_template":"qf-mod42-lagrange-divisibility-identity/{\"group_table\":[[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10],[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8]],\"identity\":2,\"subset\":[0,2,4,6,8,10]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0065","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-divisibility-identity；資料={\"group_table\":[[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10]],\"identity\":2,\"subset\":[0,2,4,6,8,10,12]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(14, 7, 2, 14, 0, 是)","explanation_zh":"Wave6第五層核對=(14, 7, 2, 14, 0, 是)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-divisibility-identity","seed":202611111065,"parameters":{"group_table":[[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10]],"identity":2,"subset":[0,2,4,6,8,10,12],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-divisibility-identity","semantic_template":"qf-mod42-lagrange-divisibility-identity/{\"group_table\":[[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10]],\"identity\":2,\"subset\":[0,2,4,6,8,10,12]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0066","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-divisibility-identity；資料={\"group_table\":[[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1]],\"identity\":3,\"subset\":[1,3,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 3, 2, 6, 0, 是)","explanation_zh":"Wave6第五層核對=(6, 3, 2, 6, 0, 是)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-divisibility-identity","seed":202611111066,"parameters":{"group_table":[[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1]],"identity":3,"subset":[1,3,5],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-divisibility-identity","semantic_template":"qf-mod42-lagrange-divisibility-identity/{\"group_table\":[[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1]],\"identity\":3,\"subset\":[1,3,5]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0067","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-divisibility-identity；資料={\"group_table\":[[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6],[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3]],\"identity\":3,\"subset\":[1,3,5,7]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, 4, 2, 8, 0, 是)","explanation_zh":"Wave6第五層核對=(8, 4, 2, 8, 0, 是)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-divisibility-identity","seed":202611111067,"parameters":{"group_table":[[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6],[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3]],"identity":3,"subset":[1,3,5,7],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-divisibility-identity","semantic_template":"qf-mod42-lagrange-divisibility-identity/{\"group_table\":[[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6],[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3]],\"identity\":3,\"subset\":[1,3,5,7]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0068","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-divisibility-identity；資料={\"group_table\":[[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8],[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5]],\"identity\":3,\"subset\":[1,3,5,7,9]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, 5, 2, 10, 0, 是)","explanation_zh":"Wave6第五層核對=(10, 5, 2, 10, 0, 是)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-divisibility-identity","seed":202611111068,"parameters":{"group_table":[[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8],[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5]],"identity":3,"subset":[1,3,5,7,9],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-divisibility-identity","semantic_template":"qf-mod42-lagrange-divisibility-identity/{\"group_table\":[[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8],[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5]],\"identity\":3,\"subset\":[1,3,5,7,9]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0069","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-divisibility-identity；資料={\"group_table\":[[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10],[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7]],\"identity\":3,\"subset\":[1,3,5,7,9,11]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, 6, 2, 12, 0, 是)","explanation_zh":"Wave6第五層核對=(12, 6, 2, 12, 0, 是)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-divisibility-identity","seed":202611111069,"parameters":{"group_table":[[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10],[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7]],"identity":3,"subset":[1,3,5,7,9,11],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-divisibility-identity","semantic_template":"qf-mod42-lagrange-divisibility-identity/{\"group_table\":[[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10],[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7]],\"identity\":3,\"subset\":[1,3,5,7,9,11]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0070","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-divisibility-identity；資料={\"group_table\":[[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9]],\"identity\":3,\"subset\":[1,3,5,7,9,11,13]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(14, 7, 2, 14, 0, 是)","explanation_zh":"Wave6第五層核對=(14, 7, 2, 14, 0, 是)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-divisibility-identity","seed":202611111070,"parameters":{"group_table":[[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9]],"identity":3,"subset":[1,3,5,7,9,11,13],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-divisibility-identity","semantic_template":"qf-mod42-lagrange-divisibility-identity/{\"group_table\":[[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9]],\"identity\":3,\"subset\":[1,3,5,7,9,11,13]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0071","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-divisibility-identity；資料={\"group_table\":[[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0]],\"identity\":4,\"subset\":[0,2,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 3, 2, 6, 0, 是)","explanation_zh":"Wave6第五層核對=(6, 3, 2, 6, 0, 是)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-divisibility-identity","seed":202611111071,"parameters":{"group_table":[[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0]],"identity":4,"subset":[0,2,4],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-divisibility-identity","semantic_template":"qf-mod42-lagrange-divisibility-identity/{\"group_table\":[[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0]],\"identity\":4,\"subset\":[0,2,4]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0072","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-divisibility-identity；資料={\"group_table\":[[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6],[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2]],\"identity\":4,\"subset\":[0,2,4,6]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, 4, 2, 8, 0, 是)","explanation_zh":"Wave6第五層核對=(8, 4, 2, 8, 0, 是)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-divisibility-identity","seed":202611111072,"parameters":{"group_table":[[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6],[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2]],"identity":4,"subset":[0,2,4,6],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-divisibility-identity","semantic_template":"qf-mod42-lagrange-divisibility-identity/{\"group_table\":[[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6],[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2]],\"identity\":4,\"subset\":[0,2,4,6]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0073","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-divisibility-identity；資料={\"group_table\":[[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8],[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4]],\"identity\":4,\"subset\":[0,2,4,6,8]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, 5, 2, 10, 0, 是)","explanation_zh":"Wave6第五層核對=(10, 5, 2, 10, 0, 是)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-divisibility-identity","seed":202611111073,"parameters":{"group_table":[[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8],[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4]],"identity":4,"subset":[0,2,4,6,8],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-divisibility-identity","semantic_template":"qf-mod42-lagrange-divisibility-identity/{\"group_table\":[[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8],[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4]],\"identity\":4,\"subset\":[0,2,4,6,8]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0074","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-divisibility-identity；資料={\"group_table\":[[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10],[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6]],\"identity\":4,\"subset\":[0,2,4,6,8,10]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, 6, 2, 12, 0, 是)","explanation_zh":"Wave6第五層核對=(12, 6, 2, 12, 0, 是)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-divisibility-identity","seed":202611111074,"parameters":{"group_table":[[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10],[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6]],"identity":4,"subset":[0,2,4,6,8,10],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-divisibility-identity","semantic_template":"qf-mod42-lagrange-divisibility-identity/{\"group_table\":[[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10],[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6]],\"identity\":4,\"subset\":[0,2,4,6,8,10]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0075","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-divisibility-identity；資料={\"group_table\":[[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8]],\"identity\":4,\"subset\":[0,2,4,6,8,10,12]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(14, 7, 2, 14, 0, 是)","explanation_zh":"Wave6第五層核對=(14, 7, 2, 14, 0, 是)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-divisibility-identity","seed":202611111075,"parameters":{"group_table":[[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8]],"identity":4,"subset":[0,2,4,6,8,10,12],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-divisibility-identity","semantic_template":"qf-mod42-lagrange-divisibility-identity/{\"group_table\":[[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8]],\"identity\":4,\"subset\":[0,2,4,6,8,10,12]}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0076","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-converse-boundary；資料={\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"candidate_subset\":[0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, [0, 1], 2, 是, 否, 否, divisor_sized_subset_need_not_be_a_subgroup_no_existence_converse_claim)","explanation_zh":"Wave6第五層核對=(6, [0, 1], 2, 是, 否, 否, divisor_sized_subset_need_not_be_a_subgroup_no_existence_converse_claim)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-converse-boundary","seed":202611111076,"parameters":{"group_table":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],"identity":0,"candidate_subset":[0,1],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-converse-boundary","semantic_template":"qf-mod42-lagrange-converse-boundary/{\"candidate_subset\":[0,1],\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0077","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-converse-boundary；資料={\"group_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],\"identity\":0,\"candidate_subset\":[0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, [0, 1], 2, 是, 否, 否, divisor_sized_subset_need_not_be_a_subgroup_no_existence_converse_claim)","explanation_zh":"Wave6第五層核對=(8, [0, 1], 2, 是, 否, 否, divisor_sized_subset_need_not_be_a_subgroup_no_existence_converse_claim)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-converse-boundary","seed":202611111077,"parameters":{"group_table":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],"identity":0,"candidate_subset":[0,1],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-converse-boundary","semantic_template":"qf-mod42-lagrange-converse-boundary/{\"candidate_subset\":[0,1],\"group_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],\"identity\":0}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0078","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-converse-boundary；資料={\"group_table\":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8]],\"identity\":0,\"candidate_subset\":[0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, [0, 1], 2, 是, 否, 否, divisor_sized_subset_need_not_be_a_subgroup_no_existence_converse_claim)","explanation_zh":"Wave6第五層核對=(10, [0, 1], 2, 是, 否, 否, divisor_sized_subset_need_not_be_a_subgroup_no_existence_converse_claim)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-converse-boundary","seed":202611111078,"parameters":{"group_table":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8]],"identity":0,"candidate_subset":[0,1],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-converse-boundary","semantic_template":"qf-mod42-lagrange-converse-boundary/{\"candidate_subset\":[0,1],\"group_table\":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8]],\"identity\":0}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0079","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-converse-boundary；資料={\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10]],\"identity\":0,\"candidate_subset\":[0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, [0, 1], 2, 是, 否, 否, divisor_sized_subset_need_not_be_a_subgroup_no_existence_converse_claim)","explanation_zh":"Wave6第五層核對=(12, [0, 1], 2, 是, 否, 否, divisor_sized_subset_need_not_be_a_subgroup_no_existence_converse_claim)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-converse-boundary","seed":202611111079,"parameters":{"group_table":[[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10]],"identity":0,"candidate_subset":[0,1],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-converse-boundary","semantic_template":"qf-mod42-lagrange-converse-boundary/{\"candidate_subset\":[0,1],\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10]],\"identity\":0}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0080","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-converse-boundary；資料={\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12]],\"identity\":0,\"candidate_subset\":[0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(14, [0, 1], 2, 是, 否, 否, divisor_sized_subset_need_not_be_a_subgroup_no_existence_converse_claim)","explanation_zh":"Wave6第五層核對=(14, [0, 1], 2, 是, 否, 否, divisor_sized_subset_need_not_be_a_subgroup_no_existence_converse_claim)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-converse-boundary","seed":202611111080,"parameters":{"group_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12]],"identity":0,"candidate_subset":[0,1],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-converse-boundary","semantic_template":"qf-mod42-lagrange-converse-boundary/{\"candidate_subset\":[0,1],\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12]],\"identity\":0}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0081","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-converse-boundary；資料={\"group_table\":[[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3]],\"identity\":1,\"candidate_subset\":[1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, [1, 2], 2, 是, 否, 否, divisor_sized_subset_need_not_be_a_subgroup_no_existence_converse_claim)","explanation_zh":"Wave6第五層核對=(6, [1, 2], 2, 是, 否, 否, divisor_sized_subset_need_not_be_a_subgroup_no_existence_converse_claim)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-converse-boundary","seed":202611111081,"parameters":{"group_table":[[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3]],"identity":1,"candidate_subset":[1,2],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-converse-boundary","semantic_template":"qf-mod42-lagrange-converse-boundary/{\"candidate_subset\":[1,2],\"group_table\":[[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3]],\"identity\":1}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0082","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-converse-boundary；資料={\"group_table\":[[7,0,1,2,3,4,5,6],[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5]],\"identity\":1,\"candidate_subset\":[1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, [1, 2], 2, 是, 否, 否, divisor_sized_subset_need_not_be_a_subgroup_no_existence_converse_claim)","explanation_zh":"Wave6第五層核對=(8, [1, 2], 2, 是, 否, 否, divisor_sized_subset_need_not_be_a_subgroup_no_existence_converse_claim)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-converse-boundary","seed":202611111082,"parameters":{"group_table":[[7,0,1,2,3,4,5,6],[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5]],"identity":1,"candidate_subset":[1,2],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-converse-boundary","semantic_template":"qf-mod42-lagrange-converse-boundary/{\"candidate_subset\":[1,2],\"group_table\":[[7,0,1,2,3,4,5,6],[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5]],\"identity\":1}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0083","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-converse-boundary；資料={\"group_table\":[[9,0,1,2,3,4,5,6,7,8],[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7]],\"identity\":1,\"candidate_subset\":[1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, [1, 2], 2, 是, 否, 否, divisor_sized_subset_need_not_be_a_subgroup_no_existence_converse_claim)","explanation_zh":"Wave6第五層核對=(10, [1, 2], 2, 是, 否, 否, divisor_sized_subset_need_not_be_a_subgroup_no_existence_converse_claim)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-converse-boundary","seed":202611111083,"parameters":{"group_table":[[9,0,1,2,3,4,5,6,7,8],[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7]],"identity":1,"candidate_subset":[1,2],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-converse-boundary","semantic_template":"qf-mod42-lagrange-converse-boundary/{\"candidate_subset\":[1,2],\"group_table\":[[9,0,1,2,3,4,5,6,7,8],[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7]],\"identity\":1}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0084","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-converse-boundary；資料={\"group_table\":[[11,0,1,2,3,4,5,6,7,8,9,10],[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9]],\"identity\":1,\"candidate_subset\":[1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, [1, 2], 2, 是, 否, 否, divisor_sized_subset_need_not_be_a_subgroup_no_existence_converse_claim)","explanation_zh":"Wave6第五層核對=(12, [1, 2], 2, 是, 否, 否, divisor_sized_subset_need_not_be_a_subgroup_no_existence_converse_claim)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-converse-boundary","seed":202611111084,"parameters":{"group_table":[[11,0,1,2,3,4,5,6,7,8,9,10],[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9]],"identity":1,"candidate_subset":[1,2],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-converse-boundary","semantic_template":"qf-mod42-lagrange-converse-boundary/{\"candidate_subset\":[1,2],\"group_table\":[[11,0,1,2,3,4,5,6,7,8,9,10],[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9]],\"identity\":1}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0085","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-converse-boundary；資料={\"group_table\":[[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11]],\"identity\":1,\"candidate_subset\":[1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(14, [1, 2], 2, 是, 否, 否, divisor_sized_subset_need_not_be_a_subgroup_no_existence_converse_claim)","explanation_zh":"Wave6第五層核對=(14, [1, 2], 2, 是, 否, 否, divisor_sized_subset_need_not_be_a_subgroup_no_existence_converse_claim)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-converse-boundary","seed":202611111085,"parameters":{"group_table":[[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11]],"identity":1,"candidate_subset":[1,2],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-converse-boundary","semantic_template":"qf-mod42-lagrange-converse-boundary/{\"candidate_subset\":[1,2],\"group_table\":[[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11]],\"identity\":1}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0086","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-converse-boundary；資料={\"group_table\":[[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2]],\"identity\":2,\"candidate_subset\":[2,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, [2, 3], 2, 是, 否, 否, divisor_sized_subset_need_not_be_a_subgroup_no_existence_converse_claim)","explanation_zh":"Wave6第五層核對=(6, [2, 3], 2, 是, 否, 否, divisor_sized_subset_need_not_be_a_subgroup_no_existence_converse_claim)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-converse-boundary","seed":202611111086,"parameters":{"group_table":[[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2]],"identity":2,"candidate_subset":[2,3],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-converse-boundary","semantic_template":"qf-mod42-lagrange-converse-boundary/{\"candidate_subset\":[2,3],\"group_table\":[[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2]],\"identity\":2}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0087","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-converse-boundary；資料={\"group_table\":[[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6],[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4]],\"identity\":2,\"candidate_subset\":[2,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, [2, 3], 2, 是, 否, 否, divisor_sized_subset_need_not_be_a_subgroup_no_existence_converse_claim)","explanation_zh":"Wave6第五層核對=(8, [2, 3], 2, 是, 否, 否, divisor_sized_subset_need_not_be_a_subgroup_no_existence_converse_claim)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-converse-boundary","seed":202611111087,"parameters":{"group_table":[[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6],[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4]],"identity":2,"candidate_subset":[2,3],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-converse-boundary","semantic_template":"qf-mod42-lagrange-converse-boundary/{\"candidate_subset\":[2,3],\"group_table\":[[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6],[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4]],\"identity\":2}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0088","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-converse-boundary；資料={\"group_table\":[[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8],[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6]],\"identity\":2,\"candidate_subset\":[2,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, [2, 3], 2, 是, 否, 否, divisor_sized_subset_need_not_be_a_subgroup_no_existence_converse_claim)","explanation_zh":"Wave6第五層核對=(10, [2, 3], 2, 是, 否, 否, divisor_sized_subset_need_not_be_a_subgroup_no_existence_converse_claim)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-converse-boundary","seed":202611111088,"parameters":{"group_table":[[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8],[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6]],"identity":2,"candidate_subset":[2,3],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-converse-boundary","semantic_template":"qf-mod42-lagrange-converse-boundary/{\"candidate_subset\":[2,3],\"group_table\":[[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8],[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6]],\"identity\":2}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0089","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-converse-boundary；資料={\"group_table\":[[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10],[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8]],\"identity\":2,\"candidate_subset\":[2,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, [2, 3], 2, 是, 否, 否, divisor_sized_subset_need_not_be_a_subgroup_no_existence_converse_claim)","explanation_zh":"Wave6第五層核對=(12, [2, 3], 2, 是, 否, 否, divisor_sized_subset_need_not_be_a_subgroup_no_existence_converse_claim)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-converse-boundary","seed":202611111089,"parameters":{"group_table":[[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10],[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8]],"identity":2,"candidate_subset":[2,3],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-converse-boundary","semantic_template":"qf-mod42-lagrange-converse-boundary/{\"candidate_subset\":[2,3],\"group_table\":[[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10],[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8]],\"identity\":2}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0090","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-converse-boundary；資料={\"group_table\":[[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10]],\"identity\":2,\"candidate_subset\":[2,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(14, [2, 3], 2, 是, 否, 否, divisor_sized_subset_need_not_be_a_subgroup_no_existence_converse_claim)","explanation_zh":"Wave6第五層核對=(14, [2, 3], 2, 是, 否, 否, divisor_sized_subset_need_not_be_a_subgroup_no_existence_converse_claim)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-converse-boundary","seed":202611111090,"parameters":{"group_table":[[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10]],"identity":2,"candidate_subset":[2,3],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-converse-boundary","semantic_template":"qf-mod42-lagrange-converse-boundary/{\"candidate_subset\":[2,3],\"group_table\":[[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10]],\"identity\":2}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0091","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-converse-boundary；資料={\"group_table\":[[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1]],\"identity\":3,\"candidate_subset\":[3,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, [3, 4], 2, 是, 否, 否, divisor_sized_subset_need_not_be_a_subgroup_no_existence_converse_claim)","explanation_zh":"Wave6第五層核對=(6, [3, 4], 2, 是, 否, 否, divisor_sized_subset_need_not_be_a_subgroup_no_existence_converse_claim)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-converse-boundary","seed":202611111091,"parameters":{"group_table":[[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1]],"identity":3,"candidate_subset":[3,4],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-converse-boundary","semantic_template":"qf-mod42-lagrange-converse-boundary/{\"candidate_subset\":[3,4],\"group_table\":[[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1]],\"identity\":3}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0092","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-converse-boundary；資料={\"group_table\":[[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6],[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3]],\"identity\":3,\"candidate_subset\":[3,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, [3, 4], 2, 是, 否, 否, divisor_sized_subset_need_not_be_a_subgroup_no_existence_converse_claim)","explanation_zh":"Wave6第五層核對=(8, [3, 4], 2, 是, 否, 否, divisor_sized_subset_need_not_be_a_subgroup_no_existence_converse_claim)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-converse-boundary","seed":202611111092,"parameters":{"group_table":[[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6],[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3]],"identity":3,"candidate_subset":[3,4],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-converse-boundary","semantic_template":"qf-mod42-lagrange-converse-boundary/{\"candidate_subset\":[3,4],\"group_table\":[[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6],[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3]],\"identity\":3}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0093","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-converse-boundary；資料={\"group_table\":[[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8],[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5]],\"identity\":3,\"candidate_subset\":[3,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, [3, 4], 2, 是, 否, 否, divisor_sized_subset_need_not_be_a_subgroup_no_existence_converse_claim)","explanation_zh":"Wave6第五層核對=(10, [3, 4], 2, 是, 否, 否, divisor_sized_subset_need_not_be_a_subgroup_no_existence_converse_claim)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-converse-boundary","seed":202611111093,"parameters":{"group_table":[[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8],[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5]],"identity":3,"candidate_subset":[3,4],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-converse-boundary","semantic_template":"qf-mod42-lagrange-converse-boundary/{\"candidate_subset\":[3,4],\"group_table\":[[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8],[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5]],\"identity\":3}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0094","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-converse-boundary；資料={\"group_table\":[[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10],[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7]],\"identity\":3,\"candidate_subset\":[3,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, [3, 4], 2, 是, 否, 否, divisor_sized_subset_need_not_be_a_subgroup_no_existence_converse_claim)","explanation_zh":"Wave6第五層核對=(12, [3, 4], 2, 是, 否, 否, divisor_sized_subset_need_not_be_a_subgroup_no_existence_converse_claim)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-converse-boundary","seed":202611111094,"parameters":{"group_table":[[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10],[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7]],"identity":3,"candidate_subset":[3,4],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-converse-boundary","semantic_template":"qf-mod42-lagrange-converse-boundary/{\"candidate_subset\":[3,4],\"group_table\":[[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10],[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7]],\"identity\":3}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0095","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-converse-boundary；資料={\"group_table\":[[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9]],\"identity\":3,\"candidate_subset\":[3,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(14, [3, 4], 2, 是, 否, 否, divisor_sized_subset_need_not_be_a_subgroup_no_existence_converse_claim)","explanation_zh":"Wave6第五層核對=(14, [3, 4], 2, 是, 否, 否, divisor_sized_subset_need_not_be_a_subgroup_no_existence_converse_claim)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-converse-boundary","seed":202611111095,"parameters":{"group_table":[[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9]],"identity":3,"candidate_subset":[3,4],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-converse-boundary","semantic_template":"qf-mod42-lagrange-converse-boundary/{\"candidate_subset\":[3,4],\"group_table\":[[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9]],\"identity\":3}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0096","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-converse-boundary；資料={\"group_table\":[[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0]],\"identity\":4,\"candidate_subset\":[4,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, [4, 5], 2, 是, 否, 否, divisor_sized_subset_need_not_be_a_subgroup_no_existence_converse_claim)","explanation_zh":"Wave6第五層核對=(6, [4, 5], 2, 是, 否, 否, divisor_sized_subset_need_not_be_a_subgroup_no_existence_converse_claim)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-converse-boundary","seed":202611111096,"parameters":{"group_table":[[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0]],"identity":4,"candidate_subset":[4,5],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-converse-boundary","semantic_template":"qf-mod42-lagrange-converse-boundary/{\"candidate_subset\":[4,5],\"group_table\":[[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0]],\"identity\":4}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0097","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-converse-boundary；資料={\"group_table\":[[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6],[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2]],\"identity\":4,\"candidate_subset\":[4,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, [4, 5], 2, 是, 否, 否, divisor_sized_subset_need_not_be_a_subgroup_no_existence_converse_claim)","explanation_zh":"Wave6第五層核對=(8, [4, 5], 2, 是, 否, 否, divisor_sized_subset_need_not_be_a_subgroup_no_existence_converse_claim)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-converse-boundary","seed":202611111097,"parameters":{"group_table":[[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6],[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2]],"identity":4,"candidate_subset":[4,5],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-converse-boundary","semantic_template":"qf-mod42-lagrange-converse-boundary/{\"candidate_subset\":[4,5],\"group_table\":[[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6],[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2]],\"identity\":4}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0098","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-converse-boundary；資料={\"group_table\":[[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8],[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4]],\"identity\":4,\"candidate_subset\":[4,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, [4, 5], 2, 是, 否, 否, divisor_sized_subset_need_not_be_a_subgroup_no_existence_converse_claim)","explanation_zh":"Wave6第五層核對=(10, [4, 5], 2, 是, 否, 否, divisor_sized_subset_need_not_be_a_subgroup_no_existence_converse_claim)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-converse-boundary","seed":202611111098,"parameters":{"group_table":[[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8],[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4]],"identity":4,"candidate_subset":[4,5],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-converse-boundary","semantic_template":"qf-mod42-lagrange-converse-boundary/{\"candidate_subset\":[4,5],\"group_table\":[[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8],[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4]],\"identity\":4}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0099","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-converse-boundary；資料={\"group_table\":[[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10],[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6]],\"identity\":4,\"candidate_subset\":[4,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, [4, 5], 2, 是, 否, 否, divisor_sized_subset_need_not_be_a_subgroup_no_existence_converse_claim)","explanation_zh":"Wave6第五層核對=(12, [4, 5], 2, 是, 否, 否, divisor_sized_subset_need_not_be_a_subgroup_no_existence_converse_claim)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-converse-boundary","seed":202611111099,"parameters":{"group_table":[[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10],[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6]],"identity":4,"candidate_subset":[4,5],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-converse-boundary","semantic_template":"qf-mod42-lagrange-converse-boundary/{\"candidate_subset\":[4,5],\"group_table\":[[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10],[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6]],\"identity\":4}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0100","batch_id":"batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-theorem-finite-groups"],"title_zh":"有限群的Lagrange定理","stem_zh":"完整有限Cayley table、完整subgroup證書與全部left cosets。操作=lagrange-converse-boundary；資料={\"group_table\":[[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8]],\"identity\":4,\"candidate_subset\":[4,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(14, [4, 5], 2, 是, 否, 否, divisor_sized_subset_need_not_be_a_subgroup_no_existence_converse_claim)","explanation_zh":"Wave6第五層核對=(14, [4, 5], 2, 是, 否, 否, divisor_sized_subset_need_not_be_a_subgroup_no_existence_converse_claim)；依完整明示資料獨立重算；left cosets必須相等或不交、覆蓋G且每個由H到gH的平移是bijection；本family只證divisor-sized指定subset不必是subgroup；不據此證明或否定同階subgroup的存在。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Lagrange定理"]},"variant":{"family_id":"qf-mod42-lagrange-converse-boundary","seed":202611111100,"parameters":{"group_table":[[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8]],"identity":4,"candidate_subset":[4,5],"domain":"Complete_finite_group_coset_partition","operation":"lagrange-converse-boundary","semantic_template":"qf-mod42-lagrange-converse-boundary/{\"candidate_subset\":[4,5],\"group_table\":[[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8]],\"identity\":4}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}