{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0001","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-subgroup-qualification；資料={\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"subset\":[0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)","explanation_zh":"Wave6第七層核對=([0], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-subgroup-qualification","seed":202611131001,"parameters":{"group_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"identity":0,"subset":[0],"domain":"Complete_finite_group_normality","operation":"normal-subgroup-qualification","semantic_template":"qf-mod44-normal-subgroup-qualification/{\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"subset\":[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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0002","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-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\":[0,1,2,3,4,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)","explanation_zh":"Wave6第七層核對=([0, 1, 2, 3, 4, 5], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-subgroup-qualification","seed":202611131002,"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":[0,1,2,3,4,5],"domain":"Complete_finite_group_normality","operation":"normal-subgroup-qualification","semantic_template":"qf-mod44-normal-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\":[0,1,2,3,4,5]}"}},"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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0003","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-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, 是, subgroup_obligations_precede_normality)","explanation_zh":"Wave6第七層核對=([0, 2, 4, 6], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-subgroup-qualification","seed":202611131003,"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_normality","operation":"normal-subgroup-qualification","semantic_template":"qf-mod44-normal-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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0004","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-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\":[0,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 5], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)","explanation_zh":"Wave6第七層核對=([0, 5], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-subgroup-qualification","seed":202611131004,"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":[0,5],"domain":"Complete_finite_group_normality","operation":"normal-subgroup-qualification","semantic_template":"qf-mod44-normal-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\":[0,5]}"}},"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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0005","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-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\":[0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1], 是, 否, 否, 否, 否, subgroup_obligations_precede_normality)","explanation_zh":"Wave6第七層核對=([0, 1], 是, 否, 否, 否, 否, subgroup_obligations_precede_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-subgroup-qualification","seed":202611131005,"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":[0,1],"domain":"Complete_finite_group_normality","operation":"normal-subgroup-qualification","semantic_template":"qf-mod44-normal-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\":[0,1]}"}},"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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0006","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-subgroup-qualification；資料={\"group_table\":[[3,0,1,2],[0,1,2,3],[1,2,3,0],[2,3,0,1]],\"identity\":1,\"subset\":[1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)","explanation_zh":"Wave6第七層核對=([1], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-subgroup-qualification","seed":202611131006,"parameters":{"group_table":[[3,0,1,2],[0,1,2,3],[1,2,3,0],[2,3,0,1]],"identity":1,"subset":[1],"domain":"Complete_finite_group_normality","operation":"normal-subgroup-qualification","semantic_template":"qf-mod44-normal-subgroup-qualification/{\"group_table\":[[3,0,1,2],[0,1,2,3],[1,2,3,0],[2,3,0,1]],\"identity\":1,\"subset\":[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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0007","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-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\":[0,1,2,3,4,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)","explanation_zh":"Wave6第七層核對=([0, 1, 2, 3, 4, 5], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-subgroup-qualification","seed":202611131007,"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":[0,1,2,3,4,5],"domain":"Complete_finite_group_normality","operation":"normal-subgroup-qualification","semantic_template":"qf-mod44-normal-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\":[0,1,2,3,4,5]}"}},"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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0008","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-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, 是, subgroup_obligations_precede_normality)","explanation_zh":"Wave6第七層核對=([1, 3, 5, 7], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-subgroup-qualification","seed":202611131008,"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_normality","operation":"normal-subgroup-qualification","semantic_template":"qf-mod44-normal-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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0009","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-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\":[1,6]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 6], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)","explanation_zh":"Wave6第七層核對=([1, 6], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-subgroup-qualification","seed":202611131009,"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":[1,6],"domain":"Complete_finite_group_normality","operation":"normal-subgroup-qualification","semantic_template":"qf-mod44-normal-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\":[1,6]}"}},"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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0010","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-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\":[1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 2], 是, 否, 否, 否, 否, subgroup_obligations_precede_normality)","explanation_zh":"Wave6第七層核對=([1, 2], 是, 否, 否, 否, 否, subgroup_obligations_precede_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-subgroup-qualification","seed":202611131010,"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":[1,2],"domain":"Complete_finite_group_normality","operation":"normal-subgroup-qualification","semantic_template":"qf-mod44-normal-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\":[1,2]}"}},"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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0011","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-subgroup-qualification；資料={\"group_table\":[[2,3,0,1],[3,0,1,2],[0,1,2,3],[1,2,3,0]],\"identity\":2,\"subset\":[2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)","explanation_zh":"Wave6第七層核對=([2], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-subgroup-qualification","seed":202611131011,"parameters":{"group_table":[[2,3,0,1],[3,0,1,2],[0,1,2,3],[1,2,3,0]],"identity":2,"subset":[2],"domain":"Complete_finite_group_normality","operation":"normal-subgroup-qualification","semantic_template":"qf-mod44-normal-subgroup-qualification/{\"group_table\":[[2,3,0,1],[3,0,1,2],[0,1,2,3],[1,2,3,0]],\"identity\":2,\"subset\":[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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0012","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-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\":[0,1,2,3,4,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)","explanation_zh":"Wave6第七層核對=([0, 1, 2, 3, 4, 5], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-subgroup-qualification","seed":202611131012,"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":[0,1,2,3,4,5],"domain":"Complete_finite_group_normality","operation":"normal-subgroup-qualification","semantic_template":"qf-mod44-normal-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\":[0,1,2,3,4,5]}"}},"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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0013","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-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, 是, subgroup_obligations_precede_normality)","explanation_zh":"Wave6第七層核對=([0, 2, 4, 6], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-subgroup-qualification","seed":202611131013,"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_normality","operation":"normal-subgroup-qualification","semantic_template":"qf-mod44-normal-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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0014","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-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\":[2,7]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2, 7], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)","explanation_zh":"Wave6第七層核對=([2, 7], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-subgroup-qualification","seed":202611131014,"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":[2,7],"domain":"Complete_finite_group_normality","operation":"normal-subgroup-qualification","semantic_template":"qf-mod44-normal-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\":[2,7]}"}},"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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0015","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-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\":[2,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2, 3], 是, 否, 否, 否, 否, subgroup_obligations_precede_normality)","explanation_zh":"Wave6第七層核對=([2, 3], 是, 否, 否, 否, 否, subgroup_obligations_precede_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-subgroup-qualification","seed":202611131015,"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":[2,3],"domain":"Complete_finite_group_normality","operation":"normal-subgroup-qualification","semantic_template":"qf-mod44-normal-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\":[2,3]}"}},"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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0016","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-subgroup-qualification；資料={\"group_table\":[[1,2,3,0],[2,3,0,1],[3,0,1,2],[0,1,2,3]],\"identity\":3,\"subset\":[3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)","explanation_zh":"Wave6第七層核對=([3], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-subgroup-qualification","seed":202611131016,"parameters":{"group_table":[[1,2,3,0],[2,3,0,1],[3,0,1,2],[0,1,2,3]],"identity":3,"subset":[3],"domain":"Complete_finite_group_normality","operation":"normal-subgroup-qualification","semantic_template":"qf-mod44-normal-subgroup-qualification/{\"group_table\":[[1,2,3,0],[2,3,0,1],[3,0,1,2],[0,1,2,3]],\"identity\":3,\"subset\":[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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0017","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-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\":[0,1,2,3,4,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)","explanation_zh":"Wave6第七層核對=([0, 1, 2, 3, 4, 5], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-subgroup-qualification","seed":202611131017,"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":[0,1,2,3,4,5],"domain":"Complete_finite_group_normality","operation":"normal-subgroup-qualification","semantic_template":"qf-mod44-normal-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\":[0,1,2,3,4,5]}"}},"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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0018","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-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, 是, subgroup_obligations_precede_normality)","explanation_zh":"Wave6第七層核對=([1, 3, 5, 7], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-subgroup-qualification","seed":202611131018,"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_normality","operation":"normal-subgroup-qualification","semantic_template":"qf-mod44-normal-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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0019","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-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\":[3,8]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 8], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)","explanation_zh":"Wave6第七層核對=([3, 8], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-subgroup-qualification","seed":202611131019,"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":[3,8],"domain":"Complete_finite_group_normality","operation":"normal-subgroup-qualification","semantic_template":"qf-mod44-normal-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\":[3,8]}"}},"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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0020","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-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\":[3,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 4], 是, 否, 否, 否, 否, subgroup_obligations_precede_normality)","explanation_zh":"Wave6第七層核對=([3, 4], 是, 否, 否, 否, 否, subgroup_obligations_precede_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-subgroup-qualification","seed":202611131020,"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":[3,4],"domain":"Complete_finite_group_normality","operation":"normal-subgroup-qualification","semantic_template":"qf-mod44-normal-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\":[3,4]}"}},"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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0021","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-subgroup-qualification；資料={\"group_table\":[[0,1,2,3],[1,0,3,2],[2,3,1,0],[3,2,0,1]],\"identity\":0,\"subset\":[0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)","explanation_zh":"Wave6第七層核對=([0], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-subgroup-qualification","seed":202611131021,"parameters":{"group_table":[[0,1,2,3],[1,0,3,2],[2,3,1,0],[3,2,0,1]],"identity":0,"subset":[0],"domain":"Complete_finite_group_normality","operation":"normal-subgroup-qualification","semantic_template":"qf-mod44-normal-subgroup-qualification/{\"group_table\":[[0,1,2,3],[1,0,3,2],[2,3,1,0],[3,2,0,1]],\"identity\":0,\"subset\":[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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0022","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-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\":[0,1,2,3,4,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)","explanation_zh":"Wave6第七層核對=([0, 1, 2, 3, 4, 5], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-subgroup-qualification","seed":202611131022,"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":[0,1,2,3,4,5],"domain":"Complete_finite_group_normality","operation":"normal-subgroup-qualification","semantic_template":"qf-mod44-normal-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\":[0,1,2,3,4,5]}"}},"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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0023","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-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, 是, subgroup_obligations_precede_normality)","explanation_zh":"Wave6第七層核對=([0, 2, 4, 6], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-subgroup-qualification","seed":202611131023,"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_normality","operation":"normal-subgroup-qualification","semantic_template":"qf-mod44-normal-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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0024","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-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\":[4,9]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([4, 9], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)","explanation_zh":"Wave6第七層核對=([4, 9], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-subgroup-qualification","seed":202611131024,"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":[4,9],"domain":"Complete_finite_group_normality","operation":"normal-subgroup-qualification","semantic_template":"qf-mod44-normal-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\":[4,9]}"}},"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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0025","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-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\":[4,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([4, 5], 是, 否, 否, 否, 否, subgroup_obligations_precede_normality)","explanation_zh":"Wave6第七層核對=([4, 5], 是, 否, 否, 否, 否, subgroup_obligations_precede_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-subgroup-qualification","seed":202611131025,"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":[4,5],"domain":"Complete_finite_group_normality","operation":"normal-subgroup-qualification","semantic_template":"qf-mod44-normal-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\":[4,5]}"}},"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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0026","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-conjugate-set-audit；資料={\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"subset\":[0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], [[0, [0], 是], [1, [0], 是], [2, [0], 是], [3, [0], 是]], 是, every_group_element_conjugates_H_to_itself)","explanation_zh":"Wave6第七層核對=([0], [[0, [0], 是], [1, [0], 是], [2, [0], 是], [3, [0], 是]], 是, every_group_element_conjugates_H_to_itself)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-conjugate-set-audit","seed":202611131026,"parameters":{"group_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"identity":0,"subset":[0],"domain":"Complete_finite_group_normality","operation":"normal-conjugate-set-audit","semantic_template":"qf-mod44-normal-conjugate-set-audit/{\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"subset\":[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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0027","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-conjugate-set-audit；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\":[0,1,2,3,4,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5], [[0, [0, 1, 2, 3, 4, 5], 是], [1, [0, 1, 2, 3, 4, 5], 是], [2, [0, 1, 2, 3, 4, 5], 是], [3, [0, 1, 2, 3, 4, 5], 是], [4, [0, 1, 2, 3, 4, 5], 是], [5, [0, 1, 2, 3, 4, 5], 是]], 是, every_group_element_conjugates_H_to_itself)","explanation_zh":"Wave6第七層核對=([0, 1, 2, 3, 4, 5], [[0, [0, 1, 2, 3, 4, 5], 是], [1, [0, 1, 2, 3, 4, 5], 是], [2, [0, 1, 2, 3, 4, 5], 是], [3, [0, 1, 2, 3, 4, 5], 是], [4, [0, 1, 2, 3, 4, 5], 是], [5, [0, 1, 2, 3, 4, 5], 是]], 是, every_group_element_conjugates_H_to_itself)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-conjugate-set-audit","seed":202611131027,"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":[0,1,2,3,4,5],"domain":"Complete_finite_group_normality","operation":"normal-conjugate-set-audit","semantic_template":"qf-mod44-normal-conjugate-set-audit/{\"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\":[0,1,2,3,4,5]}"}},"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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0028","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-conjugate-set-audit；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]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], [[0, [0], 是], [1, [0], 是], [2, [0], 是], [3, [0], 是], [4, [0], 是], [5, [0], 是], [6, [0], 是], [7, [0], 是]], 是, every_group_element_conjugates_H_to_itself)","explanation_zh":"Wave6第七層核對=([0], [[0, [0], 是], [1, [0], 是], [2, [0], 是], [3, [0], 是], [4, [0], 是], [5, [0], 是], [6, [0], 是], [7, [0], 是]], 是, every_group_element_conjugates_H_to_itself)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-conjugate-set-audit","seed":202611131028,"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],"domain":"Complete_finite_group_normality","operation":"normal-conjugate-set-audit","semantic_template":"qf-mod44-normal-conjugate-set-audit/{\"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]}"}},"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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0029","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-conjugate-set-audit；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\":[0,1,2,3,4,5,6,7,8,9]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [[0, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [1, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [2, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [3, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [4, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [5, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [6, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [7, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [8, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [9, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是]], 是, every_group_element_conjugates_H_to_itself)","explanation_zh":"Wave6第七層核對=([0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [[0, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [1, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [2, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [3, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [4, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [5, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [6, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [7, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [8, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [9, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是]], 是, every_group_element_conjugates_H_to_itself)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-conjugate-set-audit","seed":202611131029,"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":[0,1,2,3,4,5,6,7,8,9],"domain":"Complete_finite_group_normality","operation":"normal-conjugate-set-audit","semantic_template":"qf-mod44-normal-conjugate-set-audit/{\"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\":[0,1,2,3,4,5,6,7,8,9]}"}},"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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0030","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-conjugate-set-audit；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]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], [[0, [0], 是], [1, [0], 是], [2, [0], 是], [3, [0], 是], [4, [0], 是], [5, [0], 是], [6, [0], 是], [7, [0], 是], [8, [0], 是], [9, [0], 是], [10, [0], 是], [11, [0], 是]], 是, every_group_element_conjugates_H_to_itself)","explanation_zh":"Wave6第七層核對=([0], [[0, [0], 是], [1, [0], 是], [2, [0], 是], [3, [0], 是], [4, [0], 是], [5, [0], 是], [6, [0], 是], [7, [0], 是], [8, [0], 是], [9, [0], 是], [10, [0], 是], [11, [0], 是]], 是, every_group_element_conjugates_H_to_itself)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-conjugate-set-audit","seed":202611131030,"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],"domain":"Complete_finite_group_normality","operation":"normal-conjugate-set-audit","semantic_template":"qf-mod44-normal-conjugate-set-audit/{\"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]}"}},"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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0031","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-conjugate-set-audit；資料={\"group_table\":[[3,0,1,2],[0,1,2,3],[1,2,3,0],[2,3,0,1]],\"identity\":1,\"subset\":[0,1,2,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3], [[0, [0, 1, 2, 3], 是], [1, [0, 1, 2, 3], 是], [2, [0, 1, 2, 3], 是], [3, [0, 1, 2, 3], 是]], 是, every_group_element_conjugates_H_to_itself)","explanation_zh":"Wave6第七層核對=([0, 1, 2, 3], [[0, [0, 1, 2, 3], 是], [1, [0, 1, 2, 3], 是], [2, [0, 1, 2, 3], 是], [3, [0, 1, 2, 3], 是]], 是, every_group_element_conjugates_H_to_itself)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-conjugate-set-audit","seed":202611131031,"parameters":{"group_table":[[3,0,1,2],[0,1,2,3],[1,2,3,0],[2,3,0,1]],"identity":1,"subset":[0,1,2,3],"domain":"Complete_finite_group_normality","operation":"normal-conjugate-set-audit","semantic_template":"qf-mod44-normal-conjugate-set-audit/{\"group_table\":[[3,0,1,2],[0,1,2,3],[1,2,3,0],[2,3,0,1]],\"identity\":1,\"subset\":[0,1,2,3]}"}},"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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0032","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-conjugate-set-audit；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]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1], [[0, [1], 是], [1, [1], 是], [2, [1], 是], [3, [1], 是], [4, [1], 是], [5, [1], 是]], 是, every_group_element_conjugates_H_to_itself)","explanation_zh":"Wave6第七層核對=([1], [[0, [1], 是], [1, [1], 是], [2, [1], 是], [3, [1], 是], [4, [1], 是], [5, [1], 是]], 是, every_group_element_conjugates_H_to_itself)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-conjugate-set-audit","seed":202611131032,"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],"domain":"Complete_finite_group_normality","operation":"normal-conjugate-set-audit","semantic_template":"qf-mod44-normal-conjugate-set-audit/{\"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]}"}},"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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0033","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-conjugate-set-audit；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\":[0,1,2,3,4,5,6,7]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7], [[0, [0, 1, 2, 3, 4, 5, 6, 7], 是], [1, [0, 1, 2, 3, 4, 5, 6, 7], 是], [2, [0, 1, 2, 3, 4, 5, 6, 7], 是], [3, [0, 1, 2, 3, 4, 5, 6, 7], 是], [4, [0, 1, 2, 3, 4, 5, 6, 7], 是], [5, [0, 1, 2, 3, 4, 5, 6, 7], 是], [6, [0, 1, 2, 3, 4, 5, 6, 7], 是], [7, [0, 1, 2, 3, 4, 5, 6, 7], 是]], 是, every_group_element_conjugates_H_to_itself)","explanation_zh":"Wave6第七層核對=([0, 1, 2, 3, 4, 5, 6, 7], [[0, [0, 1, 2, 3, 4, 5, 6, 7], 是], [1, [0, 1, 2, 3, 4, 5, 6, 7], 是], [2, [0, 1, 2, 3, 4, 5, 6, 7], 是], [3, [0, 1, 2, 3, 4, 5, 6, 7], 是], [4, [0, 1, 2, 3, 4, 5, 6, 7], 是], [5, [0, 1, 2, 3, 4, 5, 6, 7], 是], [6, [0, 1, 2, 3, 4, 5, 6, 7], 是], [7, [0, 1, 2, 3, 4, 5, 6, 7], 是]], 是, every_group_element_conjugates_H_to_itself)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-conjugate-set-audit","seed":202611131033,"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":[0,1,2,3,4,5,6,7],"domain":"Complete_finite_group_normality","operation":"normal-conjugate-set-audit","semantic_template":"qf-mod44-normal-conjugate-set-audit/{\"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\":[0,1,2,3,4,5,6,7]}"}},"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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0034","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-conjugate-set-audit；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]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1], [[0, [1], 是], [1, [1], 是], [2, [1], 是], [3, [1], 是], [4, [1], 是], [5, [1], 是], [6, [1], 是], [7, [1], 是], [8, [1], 是], [9, [1], 是]], 是, every_group_element_conjugates_H_to_itself)","explanation_zh":"Wave6第七層核對=([1], [[0, [1], 是], [1, [1], 是], [2, [1], 是], [3, [1], 是], [4, [1], 是], [5, [1], 是], [6, [1], 是], [7, [1], 是], [8, [1], 是], [9, [1], 是]], 是, every_group_element_conjugates_H_to_itself)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-conjugate-set-audit","seed":202611131034,"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],"domain":"Complete_finite_group_normality","operation":"normal-conjugate-set-audit","semantic_template":"qf-mod44-normal-conjugate-set-audit/{\"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]}"}},"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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0035","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-conjugate-set-audit；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\":[0,1,2,3,4,5,6,7,8,9,10,11]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [[0, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [1, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [2, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [3, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [4, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [5, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [6, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [7, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [8, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [9, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [10, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [11, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是]], 是, every_group_element_conjugates_H_to_itself)","explanation_zh":"Wave6第七層核對=([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [[0, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [1, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [2, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [3, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [4, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [5, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [6, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [7, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [8, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [9, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [10, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [11, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是]], 是, every_group_element_conjugates_H_to_itself)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-conjugate-set-audit","seed":202611131035,"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":[0,1,2,3,4,5,6,7,8,9,10,11],"domain":"Complete_finite_group_normality","operation":"normal-conjugate-set-audit","semantic_template":"qf-mod44-normal-conjugate-set-audit/{\"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\":[0,1,10,11,2,3,4,5,6,7,8,9]}"}},"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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0036","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-conjugate-set-audit；資料={\"group_table\":[[2,3,0,1],[3,0,1,2],[0,1,2,3],[1,2,3,0]],\"identity\":2,\"subset\":[2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2], [[0, [2], 是], [1, [2], 是], [2, [2], 是], [3, [2], 是]], 是, every_group_element_conjugates_H_to_itself)","explanation_zh":"Wave6第七層核對=([2], [[0, [2], 是], [1, [2], 是], [2, [2], 是], [3, [2], 是]], 是, every_group_element_conjugates_H_to_itself)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-conjugate-set-audit","seed":202611131036,"parameters":{"group_table":[[2,3,0,1],[3,0,1,2],[0,1,2,3],[1,2,3,0]],"identity":2,"subset":[2],"domain":"Complete_finite_group_normality","operation":"normal-conjugate-set-audit","semantic_template":"qf-mod44-normal-conjugate-set-audit/{\"group_table\":[[2,3,0,1],[3,0,1,2],[0,1,2,3],[1,2,3,0]],\"identity\":2,\"subset\":[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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0037","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-conjugate-set-audit；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\":[0,1,2,3,4,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5], [[0, [0, 1, 2, 3, 4, 5], 是], [1, [0, 1, 2, 3, 4, 5], 是], [2, [0, 1, 2, 3, 4, 5], 是], [3, [0, 1, 2, 3, 4, 5], 是], [4, [0, 1, 2, 3, 4, 5], 是], [5, [0, 1, 2, 3, 4, 5], 是]], 是, every_group_element_conjugates_H_to_itself)","explanation_zh":"Wave6第七層核對=([0, 1, 2, 3, 4, 5], [[0, [0, 1, 2, 3, 4, 5], 是], [1, [0, 1, 2, 3, 4, 5], 是], [2, [0, 1, 2, 3, 4, 5], 是], [3, [0, 1, 2, 3, 4, 5], 是], [4, [0, 1, 2, 3, 4, 5], 是], [5, [0, 1, 2, 3, 4, 5], 是]], 是, every_group_element_conjugates_H_to_itself)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-conjugate-set-audit","seed":202611131037,"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":[0,1,2,3,4,5],"domain":"Complete_finite_group_normality","operation":"normal-conjugate-set-audit","semantic_template":"qf-mod44-normal-conjugate-set-audit/{\"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\":[0,1,2,3,4,5]}"}},"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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0038","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-conjugate-set-audit；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\":[2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2], [[0, [2], 是], [1, [2], 是], [2, [2], 是], [3, [2], 是], [4, [2], 是], [5, [2], 是], [6, [2], 是], [7, [2], 是]], 是, every_group_element_conjugates_H_to_itself)","explanation_zh":"Wave6第七層核對=([2], [[0, [2], 是], [1, [2], 是], [2, [2], 是], [3, [2], 是], [4, [2], 是], [5, [2], 是], [6, [2], 是], [7, [2], 是]], 是, every_group_element_conjugates_H_to_itself)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-conjugate-set-audit","seed":202611131038,"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":[2],"domain":"Complete_finite_group_normality","operation":"normal-conjugate-set-audit","semantic_template":"qf-mod44-normal-conjugate-set-audit/{\"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\":[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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0039","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-conjugate-set-audit；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\":[0,1,2,3,4,5,6,7,8,9]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [[0, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [1, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [2, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [3, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [4, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [5, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [6, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [7, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [8, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [9, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是]], 是, every_group_element_conjugates_H_to_itself)","explanation_zh":"Wave6第七層核對=([0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [[0, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [1, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [2, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [3, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [4, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [5, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [6, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [7, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [8, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [9, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是]], 是, every_group_element_conjugates_H_to_itself)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-conjugate-set-audit","seed":202611131039,"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":[0,1,2,3,4,5,6,7,8,9],"domain":"Complete_finite_group_normality","operation":"normal-conjugate-set-audit","semantic_template":"qf-mod44-normal-conjugate-set-audit/{\"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\":[0,1,2,3,4,5,6,7,8,9]}"}},"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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0040","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-conjugate-set-audit；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\":[2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2], [[0, [2], 是], [1, [2], 是], [2, [2], 是], [3, [2], 是], [4, [2], 是], [5, [2], 是], [6, [2], 是], [7, [2], 是], [8, [2], 是], [9, [2], 是], [10, [2], 是], [11, [2], 是]], 是, every_group_element_conjugates_H_to_itself)","explanation_zh":"Wave6第七層核對=([2], [[0, [2], 是], [1, [2], 是], [2, [2], 是], [3, [2], 是], [4, [2], 是], [5, [2], 是], [6, [2], 是], [7, [2], 是], [8, [2], 是], [9, [2], 是], [10, [2], 是], [11, [2], 是]], 是, every_group_element_conjugates_H_to_itself)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-conjugate-set-audit","seed":202611131040,"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":[2],"domain":"Complete_finite_group_normality","operation":"normal-conjugate-set-audit","semantic_template":"qf-mod44-normal-conjugate-set-audit/{\"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\":[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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0041","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-conjugate-set-audit；資料={\"group_table\":[[1,2,3,0],[2,3,0,1],[3,0,1,2],[0,1,2,3]],\"identity\":3,\"subset\":[0,1,2,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3], [[0, [0, 1, 2, 3], 是], [1, [0, 1, 2, 3], 是], [2, [0, 1, 2, 3], 是], [3, [0, 1, 2, 3], 是]], 是, every_group_element_conjugates_H_to_itself)","explanation_zh":"Wave6第七層核對=([0, 1, 2, 3], [[0, [0, 1, 2, 3], 是], [1, [0, 1, 2, 3], 是], [2, [0, 1, 2, 3], 是], [3, [0, 1, 2, 3], 是]], 是, every_group_element_conjugates_H_to_itself)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-conjugate-set-audit","seed":202611131041,"parameters":{"group_table":[[1,2,3,0],[2,3,0,1],[3,0,1,2],[0,1,2,3]],"identity":3,"subset":[0,1,2,3],"domain":"Complete_finite_group_normality","operation":"normal-conjugate-set-audit","semantic_template":"qf-mod44-normal-conjugate-set-audit/{\"group_table\":[[1,2,3,0],[2,3,0,1],[3,0,1,2],[0,1,2,3]],\"identity\":3,\"subset\":[0,1,2,3]}"}},"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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0042","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-conjugate-set-audit；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\":[3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3], [[0, [3], 是], [1, [3], 是], [2, [3], 是], [3, [3], 是], [4, [3], 是], [5, [3], 是]], 是, every_group_element_conjugates_H_to_itself)","explanation_zh":"Wave6第七層核對=([3], [[0, [3], 是], [1, [3], 是], [2, [3], 是], [3, [3], 是], [4, [3], 是], [5, [3], 是]], 是, every_group_element_conjugates_H_to_itself)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-conjugate-set-audit","seed":202611131042,"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":[3],"domain":"Complete_finite_group_normality","operation":"normal-conjugate-set-audit","semantic_template":"qf-mod44-normal-conjugate-set-audit/{\"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\":[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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0043","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-conjugate-set-audit；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\":[0,1,2,3,4,5,6,7]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7], [[0, [0, 1, 2, 3, 4, 5, 6, 7], 是], [1, [0, 1, 2, 3, 4, 5, 6, 7], 是], [2, [0, 1, 2, 3, 4, 5, 6, 7], 是], [3, [0, 1, 2, 3, 4, 5, 6, 7], 是], [4, [0, 1, 2, 3, 4, 5, 6, 7], 是], [5, [0, 1, 2, 3, 4, 5, 6, 7], 是], [6, [0, 1, 2, 3, 4, 5, 6, 7], 是], [7, [0, 1, 2, 3, 4, 5, 6, 7], 是]], 是, every_group_element_conjugates_H_to_itself)","explanation_zh":"Wave6第七層核對=([0, 1, 2, 3, 4, 5, 6, 7], [[0, [0, 1, 2, 3, 4, 5, 6, 7], 是], [1, [0, 1, 2, 3, 4, 5, 6, 7], 是], [2, [0, 1, 2, 3, 4, 5, 6, 7], 是], [3, [0, 1, 2, 3, 4, 5, 6, 7], 是], [4, [0, 1, 2, 3, 4, 5, 6, 7], 是], [5, [0, 1, 2, 3, 4, 5, 6, 7], 是], [6, [0, 1, 2, 3, 4, 5, 6, 7], 是], [7, [0, 1, 2, 3, 4, 5, 6, 7], 是]], 是, every_group_element_conjugates_H_to_itself)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-conjugate-set-audit","seed":202611131043,"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":[0,1,2,3,4,5,6,7],"domain":"Complete_finite_group_normality","operation":"normal-conjugate-set-audit","semantic_template":"qf-mod44-normal-conjugate-set-audit/{\"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\":[0,1,2,3,4,5,6,7]}"}},"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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0044","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-conjugate-set-audit；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\":[3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3], [[0, [3], 是], [1, [3], 是], [2, [3], 是], [3, [3], 是], [4, [3], 是], [5, [3], 是], [6, [3], 是], [7, [3], 是], [8, [3], 是], [9, [3], 是]], 是, every_group_element_conjugates_H_to_itself)","explanation_zh":"Wave6第七層核對=([3], [[0, [3], 是], [1, [3], 是], [2, [3], 是], [3, [3], 是], [4, [3], 是], [5, [3], 是], [6, [3], 是], [7, [3], 是], [8, [3], 是], [9, [3], 是]], 是, every_group_element_conjugates_H_to_itself)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-conjugate-set-audit","seed":202611131044,"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":[3],"domain":"Complete_finite_group_normality","operation":"normal-conjugate-set-audit","semantic_template":"qf-mod44-normal-conjugate-set-audit/{\"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\":[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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0045","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-conjugate-set-audit；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\":[0,1,2,3,4,5,6,7,8,9,10,11]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [[0, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [1, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [2, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [3, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [4, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [5, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [6, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [7, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [8, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [9, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [10, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [11, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是]], 是, every_group_element_conjugates_H_to_itself)","explanation_zh":"Wave6第七層核對=([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [[0, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [1, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [2, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [3, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [4, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [5, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [6, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [7, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [8, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [9, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [10, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [11, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是]], 是, every_group_element_conjugates_H_to_itself)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-conjugate-set-audit","seed":202611131045,"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":[0,1,2,3,4,5,6,7,8,9,10,11],"domain":"Complete_finite_group_normality","operation":"normal-conjugate-set-audit","semantic_template":"qf-mod44-normal-conjugate-set-audit/{\"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\":[0,1,10,11,2,3,4,5,6,7,8,9]}"}},"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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0046","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-conjugate-set-audit；資料={\"group_table\":[[0,1,2,3],[1,0,3,2],[2,3,1,0],[3,2,0,1]],\"identity\":0,\"subset\":[0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], [[0, [0], 是], [1, [0], 是], [2, [0], 是], [3, [0], 是]], 是, every_group_element_conjugates_H_to_itself)","explanation_zh":"Wave6第七層核對=([0], [[0, [0], 是], [1, [0], 是], [2, [0], 是], [3, [0], 是]], 是, every_group_element_conjugates_H_to_itself)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-conjugate-set-audit","seed":202611131046,"parameters":{"group_table":[[0,1,2,3],[1,0,3,2],[2,3,1,0],[3,2,0,1]],"identity":0,"subset":[0],"domain":"Complete_finite_group_normality","operation":"normal-conjugate-set-audit","semantic_template":"qf-mod44-normal-conjugate-set-audit/{\"group_table\":[[0,1,2,3],[1,0,3,2],[2,3,1,0],[3,2,0,1]],\"identity\":0,\"subset\":[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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0047","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-conjugate-set-audit；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\":[0,1,2,3,4,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5], [[0, [0, 1, 2, 3, 4, 5], 是], [1, [0, 1, 2, 3, 4, 5], 是], [2, [0, 1, 2, 3, 4, 5], 是], [3, [0, 1, 2, 3, 4, 5], 是], [4, [0, 1, 2, 3, 4, 5], 是], [5, [0, 1, 2, 3, 4, 5], 是]], 是, every_group_element_conjugates_H_to_itself)","explanation_zh":"Wave6第七層核對=([0, 1, 2, 3, 4, 5], [[0, [0, 1, 2, 3, 4, 5], 是], [1, [0, 1, 2, 3, 4, 5], 是], [2, [0, 1, 2, 3, 4, 5], 是], [3, [0, 1, 2, 3, 4, 5], 是], [4, [0, 1, 2, 3, 4, 5], 是], [5, [0, 1, 2, 3, 4, 5], 是]], 是, every_group_element_conjugates_H_to_itself)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-conjugate-set-audit","seed":202611131047,"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":[0,1,2,3,4,5],"domain":"Complete_finite_group_normality","operation":"normal-conjugate-set-audit","semantic_template":"qf-mod44-normal-conjugate-set-audit/{\"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\":[0,1,2,3,4,5]}"}},"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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0048","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-conjugate-set-audit；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\":[4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([4], [[0, [4], 是], [1, [4], 是], [2, [4], 是], [3, [4], 是], [4, [4], 是], [5, [4], 是], [6, [4], 是], [7, [4], 是]], 是, every_group_element_conjugates_H_to_itself)","explanation_zh":"Wave6第七層核對=([4], [[0, [4], 是], [1, [4], 是], [2, [4], 是], [3, [4], 是], [4, [4], 是], [5, [4], 是], [6, [4], 是], [7, [4], 是]], 是, every_group_element_conjugates_H_to_itself)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-conjugate-set-audit","seed":202611131048,"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":[4],"domain":"Complete_finite_group_normality","operation":"normal-conjugate-set-audit","semantic_template":"qf-mod44-normal-conjugate-set-audit/{\"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\":[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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0049","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-conjugate-set-audit；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\":[0,1,2,3,4,5,6,7,8,9]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [[0, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [1, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [2, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [3, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [4, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [5, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [6, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [7, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [8, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [9, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是]], 是, every_group_element_conjugates_H_to_itself)","explanation_zh":"Wave6第七層核對=([0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [[0, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [1, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [2, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [3, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [4, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [5, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [6, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [7, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [8, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [9, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是]], 是, every_group_element_conjugates_H_to_itself)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-conjugate-set-audit","seed":202611131049,"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":[0,1,2,3,4,5,6,7,8,9],"domain":"Complete_finite_group_normality","operation":"normal-conjugate-set-audit","semantic_template":"qf-mod44-normal-conjugate-set-audit/{\"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\":[0,1,2,3,4,5,6,7,8,9]}"}},"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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0050","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-conjugate-set-audit；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\":[4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([4], [[0, [4], 是], [1, [4], 是], [2, [4], 是], [3, [4], 是], [4, [4], 是], [5, [4], 是], [6, [4], 是], [7, [4], 是], [8, [4], 是], [9, [4], 是], [10, [4], 是], [11, [4], 是]], 是, every_group_element_conjugates_H_to_itself)","explanation_zh":"Wave6第七層核對=([4], [[0, [4], 是], [1, [4], 是], [2, [4], 是], [3, [4], 是], [4, [4], 是], [5, [4], 是], [6, [4], 是], [7, [4], 是], [8, [4], 是], [9, [4], 是], [10, [4], 是], [11, [4], 是]], 是, every_group_element_conjugates_H_to_itself)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-conjugate-set-audit","seed":202611131050,"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":[4],"domain":"Complete_finite_group_normality","operation":"normal-conjugate-set-audit","semantic_template":"qf-mod44-normal-conjugate-set-audit/{\"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\":[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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0051","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-left-right-coset-audit；資料={\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"subset\":[0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], [[0, [0], [0], 是], [1, [1], [1], 是], [2, [2], [2], 是], [3, [3], [3], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)","explanation_zh":"Wave6第七層核對=([0], [[0, [0], [0], 是], [1, [1], [1], 是], [2, [2], [2], 是], [3, [3], [3], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-left-right-coset-audit","seed":202611131051,"parameters":{"group_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"identity":0,"subset":[0],"domain":"Complete_finite_group_normality","operation":"normal-left-right-coset-audit","semantic_template":"qf-mod44-normal-left-right-coset-audit/{\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"subset\":[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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0052","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-left-right-coset-audit；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\":[0,1,2,3,4,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5], [[0, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是], [1, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是], [2, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是], [3, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是], [4, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是], [5, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)","explanation_zh":"Wave6第七層核對=([0, 1, 2, 3, 4, 5], [[0, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是], [1, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是], [2, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是], [3, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是], [4, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是], [5, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-left-right-coset-audit","seed":202611131052,"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":[0,1,2,3,4,5],"domain":"Complete_finite_group_normality","operation":"normal-left-right-coset-audit","semantic_template":"qf-mod44-normal-left-right-coset-audit/{\"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\":[0,1,2,3,4,5]}"}},"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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0053","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-left-right-coset-audit；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]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], [[0, [0], [0], 是], [1, [1], [1], 是], [2, [2], [2], 是], [3, [3], [3], 是], [4, [4], [4], 是], [5, [5], [5], 是], [6, [6], [6], 是], [7, [7], [7], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)","explanation_zh":"Wave6第七層核對=([0], [[0, [0], [0], 是], [1, [1], [1], 是], [2, [2], [2], 是], [3, [3], [3], 是], [4, [4], [4], 是], [5, [5], [5], 是], [6, [6], [6], 是], [7, [7], [7], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-left-right-coset-audit","seed":202611131053,"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],"domain":"Complete_finite_group_normality","operation":"normal-left-right-coset-audit","semantic_template":"qf-mod44-normal-left-right-coset-audit/{\"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]}"}},"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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0054","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-left-right-coset-audit；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\":[0,1,2,3,4,5,6,7,8,9]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [[0, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [1, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [2, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [3, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [4, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [5, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [6, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [7, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [8, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [9, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)","explanation_zh":"Wave6第七層核對=([0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [[0, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [1, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [2, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [3, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [4, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [5, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [6, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [7, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [8, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [9, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-left-right-coset-audit","seed":202611131054,"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":[0,1,2,3,4,5,6,7,8,9],"domain":"Complete_finite_group_normality","operation":"normal-left-right-coset-audit","semantic_template":"qf-mod44-normal-left-right-coset-audit/{\"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\":[0,1,2,3,4,5,6,7,8,9]}"}},"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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0055","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-left-right-coset-audit；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]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], [[0, [0], [0], 是], [1, [1], [1], 是], [2, [2], [2], 是], [3, [3], [3], 是], [4, [4], [4], 是], [5, [5], [5], 是], [6, [6], [6], 是], [7, [7], [7], 是], [8, [8], [8], 是], [9, [9], [9], 是], [10, [10], [10], 是], [11, [11], [11], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)","explanation_zh":"Wave6第七層核對=([0], [[0, [0], [0], 是], [1, [1], [1], 是], [2, [2], [2], 是], [3, [3], [3], 是], [4, [4], [4], 是], [5, [5], [5], 是], [6, [6], [6], 是], [7, [7], [7], 是], [8, [8], [8], 是], [9, [9], [9], 是], [10, [10], [10], 是], [11, [11], [11], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-left-right-coset-audit","seed":202611131055,"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],"domain":"Complete_finite_group_normality","operation":"normal-left-right-coset-audit","semantic_template":"qf-mod44-normal-left-right-coset-audit/{\"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]}"}},"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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0056","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-left-right-coset-audit；資料={\"group_table\":[[3,0,1,2],[0,1,2,3],[1,2,3,0],[2,3,0,1]],\"identity\":1,\"subset\":[0,1,2,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3], [[0, [0, 1, 2, 3], [0, 1, 2, 3], 是], [1, [0, 1, 2, 3], [0, 1, 2, 3], 是], [2, [0, 1, 2, 3], [0, 1, 2, 3], 是], [3, [0, 1, 2, 3], [0, 1, 2, 3], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)","explanation_zh":"Wave6第七層核對=([0, 1, 2, 3], [[0, [0, 1, 2, 3], [0, 1, 2, 3], 是], [1, [0, 1, 2, 3], [0, 1, 2, 3], 是], [2, [0, 1, 2, 3], [0, 1, 2, 3], 是], [3, [0, 1, 2, 3], [0, 1, 2, 3], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-left-right-coset-audit","seed":202611131056,"parameters":{"group_table":[[3,0,1,2],[0,1,2,3],[1,2,3,0],[2,3,0,1]],"identity":1,"subset":[0,1,2,3],"domain":"Complete_finite_group_normality","operation":"normal-left-right-coset-audit","semantic_template":"qf-mod44-normal-left-right-coset-audit/{\"group_table\":[[3,0,1,2],[0,1,2,3],[1,2,3,0],[2,3,0,1]],\"identity\":1,\"subset\":[0,1,2,3]}"}},"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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0057","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-left-right-coset-audit；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]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1], [[0, [0], [0], 是], [1, [1], [1], 是], [2, [2], [2], 是], [3, [3], [3], 是], [4, [4], [4], 是], [5, [5], [5], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)","explanation_zh":"Wave6第七層核對=([1], [[0, [0], [0], 是], [1, [1], [1], 是], [2, [2], [2], 是], [3, [3], [3], 是], [4, [4], [4], 是], [5, [5], [5], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-left-right-coset-audit","seed":202611131057,"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],"domain":"Complete_finite_group_normality","operation":"normal-left-right-coset-audit","semantic_template":"qf-mod44-normal-left-right-coset-audit/{\"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]}"}},"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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0058","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-left-right-coset-audit；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\":[0,1,2,3,4,5,6,7]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7], [[0, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [1, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [2, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [3, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [4, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [5, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [6, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [7, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)","explanation_zh":"Wave6第七層核對=([0, 1, 2, 3, 4, 5, 6, 7], [[0, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [1, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [2, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [3, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [4, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [5, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [6, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [7, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-left-right-coset-audit","seed":202611131058,"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":[0,1,2,3,4,5,6,7],"domain":"Complete_finite_group_normality","operation":"normal-left-right-coset-audit","semantic_template":"qf-mod44-normal-left-right-coset-audit/{\"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\":[0,1,2,3,4,5,6,7]}"}},"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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0059","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-left-right-coset-audit；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]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1], [[0, [0], [0], 是], [1, [1], [1], 是], [2, [2], [2], 是], [3, [3], [3], 是], [4, [4], [4], 是], [5, [5], [5], 是], [6, [6], [6], 是], [7, [7], [7], 是], [8, [8], [8], 是], [9, [9], [9], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)","explanation_zh":"Wave6第七層核對=([1], [[0, [0], [0], 是], [1, [1], [1], 是], [2, [2], [2], 是], [3, [3], [3], 是], [4, [4], [4], 是], [5, [5], [5], 是], [6, [6], [6], 是], [7, [7], [7], 是], [8, [8], [8], 是], [9, [9], [9], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-left-right-coset-audit","seed":202611131059,"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],"domain":"Complete_finite_group_normality","operation":"normal-left-right-coset-audit","semantic_template":"qf-mod44-normal-left-right-coset-audit/{\"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]}"}},"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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0060","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-left-right-coset-audit；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\":[0,1,2,3,4,5,6,7,8,9,10,11]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [[0, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [1, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [2, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [3, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [4, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [5, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [6, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [7, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [8, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [9, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [10, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [11, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)","explanation_zh":"Wave6第七層核對=([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [[0, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [1, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [2, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [3, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [4, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [5, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [6, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [7, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [8, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [9, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [10, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [11, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-left-right-coset-audit","seed":202611131060,"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":[0,1,2,3,4,5,6,7,8,9,10,11],"domain":"Complete_finite_group_normality","operation":"normal-left-right-coset-audit","semantic_template":"qf-mod44-normal-left-right-coset-audit/{\"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\":[0,1,10,11,2,3,4,5,6,7,8,9]}"}},"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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0061","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-left-right-coset-audit；資料={\"group_table\":[[2,3,0,1],[3,0,1,2],[0,1,2,3],[1,2,3,0]],\"identity\":2,\"subset\":[2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2], [[0, [0], [0], 是], [1, [1], [1], 是], [2, [2], [2], 是], [3, [3], [3], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)","explanation_zh":"Wave6第七層核對=([2], [[0, [0], [0], 是], [1, [1], [1], 是], [2, [2], [2], 是], [3, [3], [3], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-left-right-coset-audit","seed":202611131061,"parameters":{"group_table":[[2,3,0,1],[3,0,1,2],[0,1,2,3],[1,2,3,0]],"identity":2,"subset":[2],"domain":"Complete_finite_group_normality","operation":"normal-left-right-coset-audit","semantic_template":"qf-mod44-normal-left-right-coset-audit/{\"group_table\":[[2,3,0,1],[3,0,1,2],[0,1,2,3],[1,2,3,0]],\"identity\":2,\"subset\":[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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0062","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-left-right-coset-audit；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\":[0,1,2,3,4,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5], [[0, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是], [1, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是], [2, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是], [3, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是], [4, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是], [5, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)","explanation_zh":"Wave6第七層核對=([0, 1, 2, 3, 4, 5], [[0, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是], [1, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是], [2, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是], [3, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是], [4, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是], [5, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-left-right-coset-audit","seed":202611131062,"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":[0,1,2,3,4,5],"domain":"Complete_finite_group_normality","operation":"normal-left-right-coset-audit","semantic_template":"qf-mod44-normal-left-right-coset-audit/{\"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\":[0,1,2,3,4,5]}"}},"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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0063","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-left-right-coset-audit；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\":[2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2], [[0, [0], [0], 是], [1, [1], [1], 是], [2, [2], [2], 是], [3, [3], [3], 是], [4, [4], [4], 是], [5, [5], [5], 是], [6, [6], [6], 是], [7, [7], [7], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)","explanation_zh":"Wave6第七層核對=([2], [[0, [0], [0], 是], [1, [1], [1], 是], [2, [2], [2], 是], [3, [3], [3], 是], [4, [4], [4], 是], [5, [5], [5], 是], [6, [6], [6], 是], [7, [7], [7], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-left-right-coset-audit","seed":202611131063,"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":[2],"domain":"Complete_finite_group_normality","operation":"normal-left-right-coset-audit","semantic_template":"qf-mod44-normal-left-right-coset-audit/{\"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\":[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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0064","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-left-right-coset-audit；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\":[0,1,2,3,4,5,6,7,8,9]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [[0, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [1, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [2, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [3, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [4, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [5, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [6, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [7, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [8, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [9, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)","explanation_zh":"Wave6第七層核對=([0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [[0, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [1, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [2, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [3, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [4, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [5, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [6, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [7, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [8, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [9, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-left-right-coset-audit","seed":202611131064,"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":[0,1,2,3,4,5,6,7,8,9],"domain":"Complete_finite_group_normality","operation":"normal-left-right-coset-audit","semantic_template":"qf-mod44-normal-left-right-coset-audit/{\"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\":[0,1,2,3,4,5,6,7,8,9]}"}},"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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0065","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-left-right-coset-audit；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\":[2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2], [[0, [0], [0], 是], [1, [1], [1], 是], [2, [2], [2], 是], [3, [3], [3], 是], [4, [4], [4], 是], [5, [5], [5], 是], [6, [6], [6], 是], [7, [7], [7], 是], [8, [8], [8], 是], [9, [9], [9], 是], [10, [10], [10], 是], [11, [11], [11], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)","explanation_zh":"Wave6第七層核對=([2], [[0, [0], [0], 是], [1, [1], [1], 是], [2, [2], [2], 是], [3, [3], [3], 是], [4, [4], [4], 是], [5, [5], [5], 是], [6, [6], [6], 是], [7, [7], [7], 是], [8, [8], [8], 是], [9, [9], [9], 是], [10, [10], [10], 是], [11, [11], [11], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-left-right-coset-audit","seed":202611131065,"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":[2],"domain":"Complete_finite_group_normality","operation":"normal-left-right-coset-audit","semantic_template":"qf-mod44-normal-left-right-coset-audit/{\"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\":[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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0066","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-left-right-coset-audit；資料={\"group_table\":[[1,2,3,0],[2,3,0,1],[3,0,1,2],[0,1,2,3]],\"identity\":3,\"subset\":[0,1,2,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3], [[0, [0, 1, 2, 3], [0, 1, 2, 3], 是], [1, [0, 1, 2, 3], [0, 1, 2, 3], 是], [2, [0, 1, 2, 3], [0, 1, 2, 3], 是], [3, [0, 1, 2, 3], [0, 1, 2, 3], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)","explanation_zh":"Wave6第七層核對=([0, 1, 2, 3], [[0, [0, 1, 2, 3], [0, 1, 2, 3], 是], [1, [0, 1, 2, 3], [0, 1, 2, 3], 是], [2, [0, 1, 2, 3], [0, 1, 2, 3], 是], [3, [0, 1, 2, 3], [0, 1, 2, 3], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-left-right-coset-audit","seed":202611131066,"parameters":{"group_table":[[1,2,3,0],[2,3,0,1],[3,0,1,2],[0,1,2,3]],"identity":3,"subset":[0,1,2,3],"domain":"Complete_finite_group_normality","operation":"normal-left-right-coset-audit","semantic_template":"qf-mod44-normal-left-right-coset-audit/{\"group_table\":[[1,2,3,0],[2,3,0,1],[3,0,1,2],[0,1,2,3]],\"identity\":3,\"subset\":[0,1,2,3]}"}},"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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0067","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-left-right-coset-audit；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\":[3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3], [[0, [0], [0], 是], [1, [1], [1], 是], [2, [2], [2], 是], [3, [3], [3], 是], [4, [4], [4], 是], [5, [5], [5], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)","explanation_zh":"Wave6第七層核對=([3], [[0, [0], [0], 是], [1, [1], [1], 是], [2, [2], [2], 是], [3, [3], [3], 是], [4, [4], [4], 是], [5, [5], [5], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-left-right-coset-audit","seed":202611131067,"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":[3],"domain":"Complete_finite_group_normality","operation":"normal-left-right-coset-audit","semantic_template":"qf-mod44-normal-left-right-coset-audit/{\"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\":[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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0068","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-left-right-coset-audit；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\":[0,1,2,3,4,5,6,7]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7], [[0, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [1, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [2, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [3, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [4, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [5, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [6, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [7, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)","explanation_zh":"Wave6第七層核對=([0, 1, 2, 3, 4, 5, 6, 7], [[0, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [1, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [2, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [3, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [4, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [5, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [6, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [7, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-left-right-coset-audit","seed":202611131068,"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":[0,1,2,3,4,5,6,7],"domain":"Complete_finite_group_normality","operation":"normal-left-right-coset-audit","semantic_template":"qf-mod44-normal-left-right-coset-audit/{\"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\":[0,1,2,3,4,5,6,7]}"}},"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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0069","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-left-right-coset-audit；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\":[3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3], [[0, [0], [0], 是], [1, [1], [1], 是], [2, [2], [2], 是], [3, [3], [3], 是], [4, [4], [4], 是], [5, [5], [5], 是], [6, [6], [6], 是], [7, [7], [7], 是], [8, [8], [8], 是], [9, [9], [9], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)","explanation_zh":"Wave6第七層核對=([3], [[0, [0], [0], 是], [1, [1], [1], 是], [2, [2], [2], 是], [3, [3], [3], 是], [4, [4], [4], 是], [5, [5], [5], 是], [6, [6], [6], 是], [7, [7], [7], 是], [8, [8], [8], 是], [9, [9], [9], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-left-right-coset-audit","seed":202611131069,"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":[3],"domain":"Complete_finite_group_normality","operation":"normal-left-right-coset-audit","semantic_template":"qf-mod44-normal-left-right-coset-audit/{\"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\":[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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0070","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-left-right-coset-audit；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\":[0,1,2,3,4,5,6,7,8,9,10,11]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [[0, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [1, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [2, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [3, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [4, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [5, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [6, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [7, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [8, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [9, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [10, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [11, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)","explanation_zh":"Wave6第七層核對=([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [[0, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [1, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [2, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [3, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [4, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [5, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [6, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [7, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [8, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [9, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [10, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [11, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-left-right-coset-audit","seed":202611131070,"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":[0,1,2,3,4,5,6,7,8,9,10,11],"domain":"Complete_finite_group_normality","operation":"normal-left-right-coset-audit","semantic_template":"qf-mod44-normal-left-right-coset-audit/{\"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\":[0,1,10,11,2,3,4,5,6,7,8,9]}"}},"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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0071","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-left-right-coset-audit；資料={\"group_table\":[[0,1,2,3],[1,0,3,2],[2,3,1,0],[3,2,0,1]],\"identity\":0,\"subset\":[0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], [[0, [0], [0], 是], [1, [1], [1], 是], [2, [2], [2], 是], [3, [3], [3], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)","explanation_zh":"Wave6第七層核對=([0], [[0, [0], [0], 是], [1, [1], [1], 是], [2, [2], [2], 是], [3, [3], [3], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-left-right-coset-audit","seed":202611131071,"parameters":{"group_table":[[0,1,2,3],[1,0,3,2],[2,3,1,0],[3,2,0,1]],"identity":0,"subset":[0],"domain":"Complete_finite_group_normality","operation":"normal-left-right-coset-audit","semantic_template":"qf-mod44-normal-left-right-coset-audit/{\"group_table\":[[0,1,2,3],[1,0,3,2],[2,3,1,0],[3,2,0,1]],\"identity\":0,\"subset\":[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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0072","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-left-right-coset-audit；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\":[0,1,2,3,4,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5], [[0, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是], [1, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是], [2, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是], [3, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是], [4, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是], [5, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)","explanation_zh":"Wave6第七層核對=([0, 1, 2, 3, 4, 5], [[0, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是], [1, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是], [2, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是], [3, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是], [4, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是], [5, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-left-right-coset-audit","seed":202611131072,"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":[0,1,2,3,4,5],"domain":"Complete_finite_group_normality","operation":"normal-left-right-coset-audit","semantic_template":"qf-mod44-normal-left-right-coset-audit/{\"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\":[0,1,2,3,4,5]}"}},"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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0073","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-left-right-coset-audit；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\":[4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([4], [[0, [0], [0], 是], [1, [1], [1], 是], [2, [2], [2], 是], [3, [3], [3], 是], [4, [4], [4], 是], [5, [5], [5], 是], [6, [6], [6], 是], [7, [7], [7], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)","explanation_zh":"Wave6第七層核對=([4], [[0, [0], [0], 是], [1, [1], [1], 是], [2, [2], [2], 是], [3, [3], [3], 是], [4, [4], [4], 是], [5, [5], [5], 是], [6, [6], [6], 是], [7, [7], [7], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-left-right-coset-audit","seed":202611131073,"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":[4],"domain":"Complete_finite_group_normality","operation":"normal-left-right-coset-audit","semantic_template":"qf-mod44-normal-left-right-coset-audit/{\"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\":[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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0074","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-left-right-coset-audit；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\":[0,1,2,3,4,5,6,7,8,9]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [[0, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [1, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [2, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [3, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [4, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [5, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [6, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [7, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [8, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [9, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)","explanation_zh":"Wave6第七層核對=([0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [[0, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [1, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [2, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [3, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [4, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [5, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [6, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [7, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [8, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [9, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-left-right-coset-audit","seed":202611131074,"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":[0,1,2,3,4,5,6,7,8,9],"domain":"Complete_finite_group_normality","operation":"normal-left-right-coset-audit","semantic_template":"qf-mod44-normal-left-right-coset-audit/{\"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\":[0,1,2,3,4,5,6,7,8,9]}"}},"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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0075","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-left-right-coset-audit；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\":[4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([4], [[0, [0], [0], 是], [1, [1], [1], 是], [2, [2], [2], 是], [3, [3], [3], 是], [4, [4], [4], 是], [5, [5], [5], 是], [6, [6], [6], 是], [7, [7], [7], 是], [8, [8], [8], 是], [9, [9], [9], 是], [10, [10], [10], 是], [11, [11], [11], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)","explanation_zh":"Wave6第七層核對=([4], [[0, [0], [0], 是], [1, [1], [1], 是], [2, [2], [2], 是], [3, [3], [3], 是], [4, [4], [4], 是], [5, [5], [5], 是], [6, [6], [6], 是], [7, [7], [7], 是], [8, [8], [8], 是], [9, [9], [9], 是], [10, [10], [10], 是], [11, [11], [11], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-left-right-coset-audit","seed":202611131075,"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":[4],"domain":"Complete_finite_group_normality","operation":"normal-left-right-coset-audit","semantic_template":"qf-mod44-normal-left-right-coset-audit/{\"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\":[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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0076","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-nonnormal-witness；資料={\"group_table\":[[0,1,2,3,4,5],[1,0,5,4,3,2],[2,4,0,5,1,3],[3,5,4,0,2,1],[4,2,3,1,5,0],[5,3,1,2,0,4]],\"identity\":0,\"subset\":[0,1],\"witness_g\":4,\"witness_h\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1], 4, 1, 5, 3, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)","explanation_zh":"Wave6第七層核對=([0, 1], 4, 1, 5, 3, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-nonnormal-witness","seed":202611131076,"parameters":{"group_table":[[0,1,2,3,4,5],[1,0,5,4,3,2],[2,4,0,5,1,3],[3,5,4,0,2,1],[4,2,3,1,5,0],[5,3,1,2,0,4]],"identity":0,"subset":[0,1],"witness_g":4,"witness_h":1,"domain":"Complete_finite_group_normality","operation":"normal-nonnormal-witness","semantic_template":"qf-mod44-normal-nonnormal-witness/{\"group_table\":[[0,1,2,3,4,5],[1,0,5,4,3,2],[2,4,0,5,1,3],[3,5,4,0,2,1],[4,2,3,1,5,0],[5,3,1,2,0,4]],\"identity\":0,\"subset\":[0,1],\"witness_g\":4,\"witness_h\":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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0077","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-nonnormal-witness；資料={\"group_table\":[[0,1,2,3,4,5],[1,0,4,5,2,3],[2,5,0,4,3,1],[3,4,5,0,1,2],[4,3,1,2,5,0],[5,2,3,1,0,4]],\"identity\":0,\"subset\":[0,1],\"witness_g\":5,\"witness_h\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1], 5, 1, 4, 3, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)","explanation_zh":"Wave6第七層核對=([0, 1], 5, 1, 4, 3, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-nonnormal-witness","seed":202611131077,"parameters":{"group_table":[[0,1,2,3,4,5],[1,0,4,5,2,3],[2,5,0,4,3,1],[3,4,5,0,1,2],[4,3,1,2,5,0],[5,2,3,1,0,4]],"identity":0,"subset":[0,1],"witness_g":5,"witness_h":1,"domain":"Complete_finite_group_normality","operation":"normal-nonnormal-witness","semantic_template":"qf-mod44-normal-nonnormal-witness/{\"group_table\":[[0,1,2,3,4,5],[1,0,4,5,2,3],[2,5,0,4,3,1],[3,4,5,0,1,2],[4,3,1,2,5,0],[5,2,3,1,0,4]],\"identity\":0,\"subset\":[0,1],\"witness_g\":5,\"witness_h\":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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0078","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-nonnormal-witness；資料={\"group_table\":[[0,1,2,3,4,5],[1,0,5,4,3,2],[2,3,0,1,5,4],[3,2,4,5,1,0],[4,5,3,2,0,1],[5,4,1,0,2,3]],\"identity\":0,\"subset\":[0,1],\"witness_g\":3,\"witness_h\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1], 3, 1, 5, 4, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)","explanation_zh":"Wave6第七層核對=([0, 1], 3, 1, 5, 4, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-nonnormal-witness","seed":202611131078,"parameters":{"group_table":[[0,1,2,3,4,5],[1,0,5,4,3,2],[2,3,0,1,5,4],[3,2,4,5,1,0],[4,5,3,2,0,1],[5,4,1,0,2,3]],"identity":0,"subset":[0,1],"witness_g":3,"witness_h":1,"domain":"Complete_finite_group_normality","operation":"normal-nonnormal-witness","semantic_template":"qf-mod44-normal-nonnormal-witness/{\"group_table\":[[0,1,2,3,4,5],[1,0,5,4,3,2],[2,3,0,1,5,4],[3,2,4,5,1,0],[4,5,3,2,0,1],[5,4,1,0,2,3]],\"identity\":0,\"subset\":[0,1],\"witness_g\":3,\"witness_h\":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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0079","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-nonnormal-witness；資料={\"group_table\":[[0,1,2,3,4,5],[1,0,3,2,5,4],[2,5,0,4,3,1],[3,4,1,5,2,0],[4,3,5,1,0,2],[5,2,4,0,1,3]],\"identity\":0,\"subset\":[0,1],\"witness_g\":5,\"witness_h\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1], 5, 1, 3, 4, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)","explanation_zh":"Wave6第七層核對=([0, 1], 5, 1, 3, 4, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-nonnormal-witness","seed":202611131079,"parameters":{"group_table":[[0,1,2,3,4,5],[1,0,3,2,5,4],[2,5,0,4,3,1],[3,4,1,5,2,0],[4,3,5,1,0,2],[5,2,4,0,1,3]],"identity":0,"subset":[0,1],"witness_g":5,"witness_h":1,"domain":"Complete_finite_group_normality","operation":"normal-nonnormal-witness","semantic_template":"qf-mod44-normal-nonnormal-witness/{\"group_table\":[[0,1,2,3,4,5],[1,0,3,2,5,4],[2,5,0,4,3,1],[3,4,1,5,2,0],[4,3,5,1,0,2],[5,2,4,0,1,3]],\"identity\":0,\"subset\":[0,1],\"witness_g\":5,\"witness_h\":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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0080","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-nonnormal-witness；資料={\"group_table\":[[0,1,2,3,4,5],[1,0,4,5,2,3],[2,3,0,1,5,4],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,4,3,2,1,0]],\"identity\":0,\"subset\":[0,1],\"witness_g\":3,\"witness_h\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1], 3, 1, 4, 5, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)","explanation_zh":"Wave6第七層核對=([0, 1], 3, 1, 4, 5, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-nonnormal-witness","seed":202611131080,"parameters":{"group_table":[[0,1,2,3,4,5],[1,0,4,5,2,3],[2,3,0,1,5,4],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,4,3,2,1,0]],"identity":0,"subset":[0,1],"witness_g":3,"witness_h":1,"domain":"Complete_finite_group_normality","operation":"normal-nonnormal-witness","semantic_template":"qf-mod44-normal-nonnormal-witness/{\"group_table\":[[0,1,2,3,4,5],[1,0,4,5,2,3],[2,3,0,1,5,4],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,4,3,2,1,0]],\"identity\":0,\"subset\":[0,1],\"witness_g\":3,\"witness_h\":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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0081","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-nonnormal-witness；資料={\"group_table\":[[0,1,2,3,4,5],[1,0,3,2,5,4],[2,4,0,5,1,3],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,3,4,1,2,0]],\"identity\":0,\"subset\":[0,1],\"witness_g\":4,\"witness_h\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1], 4, 1, 3, 5, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)","explanation_zh":"Wave6第七層核對=([0, 1], 4, 1, 3, 5, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-nonnormal-witness","seed":202611131081,"parameters":{"group_table":[[0,1,2,3,4,5],[1,0,3,2,5,4],[2,4,0,5,1,3],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,3,4,1,2,0]],"identity":0,"subset":[0,1],"witness_g":4,"witness_h":1,"domain":"Complete_finite_group_normality","operation":"normal-nonnormal-witness","semantic_template":"qf-mod44-normal-nonnormal-witness/{\"group_table\":[[0,1,2,3,4,5],[1,0,3,2,5,4],[2,4,0,5,1,3],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,3,4,1,2,0]],\"identity\":0,\"subset\":[0,1],\"witness_g\":4,\"witness_h\":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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0082","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-nonnormal-witness；資料={\"group_table\":[[0,1,2,3,4,5],[1,0,4,5,2,3],[2,5,0,4,3,1],[3,4,5,0,1,2],[4,3,1,2,5,0],[5,2,3,1,0,4]],\"identity\":0,\"subset\":[0,1],\"witness_g\":4,\"witness_h\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1], 4, 1, 5, 2, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)","explanation_zh":"Wave6第七層核對=([0, 1], 4, 1, 5, 2, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-nonnormal-witness","seed":202611131082,"parameters":{"group_table":[[0,1,2,3,4,5],[1,0,4,5,2,3],[2,5,0,4,3,1],[3,4,5,0,1,2],[4,3,1,2,5,0],[5,2,3,1,0,4]],"identity":0,"subset":[0,1],"witness_g":4,"witness_h":1,"domain":"Complete_finite_group_normality","operation":"normal-nonnormal-witness","semantic_template":"qf-mod44-normal-nonnormal-witness/{\"group_table\":[[0,1,2,3,4,5],[1,0,4,5,2,3],[2,5,0,4,3,1],[3,4,5,0,1,2],[4,3,1,2,5,0],[5,2,3,1,0,4]],\"identity\":0,\"subset\":[0,1],\"witness_g\":4,\"witness_h\":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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0083","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-nonnormal-witness；資料={\"group_table\":[[0,1,2,3,4,5],[1,0,5,4,3,2],[2,4,0,5,1,3],[3,5,4,0,2,1],[4,2,3,1,5,0],[5,3,1,2,0,4]],\"identity\":0,\"subset\":[0,1],\"witness_g\":5,\"witness_h\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1], 5, 1, 4, 2, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)","explanation_zh":"Wave6第七層核對=([0, 1], 5, 1, 4, 2, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-nonnormal-witness","seed":202611131083,"parameters":{"group_table":[[0,1,2,3,4,5],[1,0,5,4,3,2],[2,4,0,5,1,3],[3,5,4,0,2,1],[4,2,3,1,5,0],[5,3,1,2,0,4]],"identity":0,"subset":[0,1],"witness_g":5,"witness_h":1,"domain":"Complete_finite_group_normality","operation":"normal-nonnormal-witness","semantic_template":"qf-mod44-normal-nonnormal-witness/{\"group_table\":[[0,1,2,3,4,5],[1,0,5,4,3,2],[2,4,0,5,1,3],[3,5,4,0,2,1],[4,2,3,1,5,0],[5,3,1,2,0,4]],\"identity\":0,\"subset\":[0,1],\"witness_g\":5,\"witness_h\":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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0084","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-nonnormal-witness；資料={\"group_table\":[[0,1,2,3,4,5],[1,0,4,5,2,3],[2,3,5,4,1,0],[3,2,1,0,5,4],[4,5,3,2,0,1],[5,4,0,1,3,2]],\"identity\":0,\"subset\":[0,1],\"witness_g\":2,\"witness_h\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1], 2, 1, 5, 4, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)","explanation_zh":"Wave6第七層核對=([0, 1], 2, 1, 5, 4, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-nonnormal-witness","seed":202611131084,"parameters":{"group_table":[[0,1,2,3,4,5],[1,0,4,5,2,3],[2,3,5,4,1,0],[3,2,1,0,5,4],[4,5,3,2,0,1],[5,4,0,1,3,2]],"identity":0,"subset":[0,1],"witness_g":2,"witness_h":1,"domain":"Complete_finite_group_normality","operation":"normal-nonnormal-witness","semantic_template":"qf-mod44-normal-nonnormal-witness/{\"group_table\":[[0,1,2,3,4,5],[1,0,4,5,2,3],[2,3,5,4,1,0],[3,2,1,0,5,4],[4,5,3,2,0,1],[5,4,0,1,3,2]],\"identity\":0,\"subset\":[0,1],\"witness_g\":2,\"witness_h\":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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0085","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-nonnormal-witness；資料={\"group_table\":[[0,1,2,3,4,5],[1,0,3,2,5,4],[2,4,5,1,3,0],[3,5,4,0,2,1],[4,2,1,5,0,3],[5,3,0,4,1,2]],\"identity\":0,\"subset\":[0,1],\"witness_g\":5,\"witness_h\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1], 5, 1, 2, 4, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)","explanation_zh":"Wave6第七層核對=([0, 1], 5, 1, 2, 4, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-nonnormal-witness","seed":202611131085,"parameters":{"group_table":[[0,1,2,3,4,5],[1,0,3,2,5,4],[2,4,5,1,3,0],[3,5,4,0,2,1],[4,2,1,5,0,3],[5,3,0,4,1,2]],"identity":0,"subset":[0,1],"witness_g":5,"witness_h":1,"domain":"Complete_finite_group_normality","operation":"normal-nonnormal-witness","semantic_template":"qf-mod44-normal-nonnormal-witness/{\"group_table\":[[0,1,2,3,4,5],[1,0,3,2,5,4],[2,4,5,1,3,0],[3,5,4,0,2,1],[4,2,1,5,0,3],[5,3,0,4,1,2]],\"identity\":0,\"subset\":[0,1],\"witness_g\":5,\"witness_h\":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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0086","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-nonnormal-witness；資料={\"group_table\":[[0,1,2,3,4,5],[1,0,5,4,3,2],[2,3,4,5,0,1],[3,2,1,0,5,4],[4,5,0,1,2,3],[5,4,3,2,1,0]],\"identity\":0,\"subset\":[0,1],\"witness_g\":2,\"witness_h\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1], 2, 1, 4, 5, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)","explanation_zh":"Wave6第七層核對=([0, 1], 2, 1, 4, 5, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-nonnormal-witness","seed":202611131086,"parameters":{"group_table":[[0,1,2,3,4,5],[1,0,5,4,3,2],[2,3,4,5,0,1],[3,2,1,0,5,4],[4,5,0,1,2,3],[5,4,3,2,1,0]],"identity":0,"subset":[0,1],"witness_g":2,"witness_h":1,"domain":"Complete_finite_group_normality","operation":"normal-nonnormal-witness","semantic_template":"qf-mod44-normal-nonnormal-witness/{\"group_table\":[[0,1,2,3,4,5],[1,0,5,4,3,2],[2,3,4,5,0,1],[3,2,1,0,5,4],[4,5,0,1,2,3],[5,4,3,2,1,0]],\"identity\":0,\"subset\":[0,1],\"witness_g\":2,\"witness_h\":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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0087","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-nonnormal-witness；資料={\"group_table\":[[0,1,2,3,4,5],[1,0,3,2,5,4],[2,5,4,1,0,3],[3,4,5,0,1,2],[4,3,0,5,2,1],[5,2,1,4,3,0]],\"identity\":0,\"subset\":[0,1],\"witness_g\":4,\"witness_h\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1], 4, 1, 2, 5, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)","explanation_zh":"Wave6第七層核對=([0, 1], 4, 1, 2, 5, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-nonnormal-witness","seed":202611131087,"parameters":{"group_table":[[0,1,2,3,4,5],[1,0,3,2,5,4],[2,5,4,1,0,3],[3,4,5,0,1,2],[4,3,0,5,2,1],[5,2,1,4,3,0]],"identity":0,"subset":[0,1],"witness_g":4,"witness_h":1,"domain":"Complete_finite_group_normality","operation":"normal-nonnormal-witness","semantic_template":"qf-mod44-normal-nonnormal-witness/{\"group_table\":[[0,1,2,3,4,5],[1,0,3,2,5,4],[2,5,4,1,0,3],[3,4,5,0,1,2],[4,3,0,5,2,1],[5,2,1,4,3,0]],\"identity\":0,\"subset\":[0,1],\"witness_g\":4,\"witness_h\":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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0088","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-nonnormal-witness；資料={\"group_table\":[[0,1,2,3,4,5],[1,0,3,2,5,4],[2,5,0,4,3,1],[3,4,1,5,2,0],[4,3,5,1,0,2],[5,2,4,0,1,3]],\"identity\":0,\"subset\":[0,1],\"witness_g\":3,\"witness_h\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1], 3, 1, 5, 2, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)","explanation_zh":"Wave6第七層核對=([0, 1], 3, 1, 5, 2, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-nonnormal-witness","seed":202611131088,"parameters":{"group_table":[[0,1,2,3,4,5],[1,0,3,2,5,4],[2,5,0,4,3,1],[3,4,1,5,2,0],[4,3,5,1,0,2],[5,2,4,0,1,3]],"identity":0,"subset":[0,1],"witness_g":3,"witness_h":1,"domain":"Complete_finite_group_normality","operation":"normal-nonnormal-witness","semantic_template":"qf-mod44-normal-nonnormal-witness/{\"group_table\":[[0,1,2,3,4,5],[1,0,3,2,5,4],[2,5,0,4,3,1],[3,4,1,5,2,0],[4,3,5,1,0,2],[5,2,4,0,1,3]],\"identity\":0,\"subset\":[0,1],\"witness_g\":3,\"witness_h\":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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0089","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-nonnormal-witness；資料={\"group_table\":[[0,1,2,3,4,5],[1,0,5,4,3,2],[2,3,0,1,5,4],[3,2,4,5,1,0],[4,5,3,2,0,1],[5,4,1,0,2,3]],\"identity\":0,\"subset\":[0,1],\"witness_g\":5,\"witness_h\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1], 5, 1, 3, 2, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)","explanation_zh":"Wave6第七層核對=([0, 1], 5, 1, 3, 2, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-nonnormal-witness","seed":202611131089,"parameters":{"group_table":[[0,1,2,3,4,5],[1,0,5,4,3,2],[2,3,0,1,5,4],[3,2,4,5,1,0],[4,5,3,2,0,1],[5,4,1,0,2,3]],"identity":0,"subset":[0,1],"witness_g":5,"witness_h":1,"domain":"Complete_finite_group_normality","operation":"normal-nonnormal-witness","semantic_template":"qf-mod44-normal-nonnormal-witness/{\"group_table\":[[0,1,2,3,4,5],[1,0,5,4,3,2],[2,3,0,1,5,4],[3,2,4,5,1,0],[4,5,3,2,0,1],[5,4,1,0,2,3]],\"identity\":0,\"subset\":[0,1],\"witness_g\":5,\"witness_h\":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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0090","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-nonnormal-witness；資料={\"group_table\":[[0,1,2,3,4,5],[1,0,3,2,5,4],[2,4,5,1,3,0],[3,5,4,0,2,1],[4,2,1,5,0,3],[5,3,0,4,1,2]],\"identity\":0,\"subset\":[0,1],\"witness_g\":2,\"witness_h\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1], 2, 1, 5, 3, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)","explanation_zh":"Wave6第七層核對=([0, 1], 2, 1, 5, 3, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-nonnormal-witness","seed":202611131090,"parameters":{"group_table":[[0,1,2,3,4,5],[1,0,3,2,5,4],[2,4,5,1,3,0],[3,5,4,0,2,1],[4,2,1,5,0,3],[5,3,0,4,1,2]],"identity":0,"subset":[0,1],"witness_g":2,"witness_h":1,"domain":"Complete_finite_group_normality","operation":"normal-nonnormal-witness","semantic_template":"qf-mod44-normal-nonnormal-witness/{\"group_table\":[[0,1,2,3,4,5],[1,0,3,2,5,4],[2,4,5,1,3,0],[3,5,4,0,2,1],[4,2,1,5,0,3],[5,3,0,4,1,2]],\"identity\":0,\"subset\":[0,1],\"witness_g\":2,\"witness_h\":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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0091","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-nonnormal-witness；資料={\"group_table\":[[0,1,2,3,4,5],[1,0,4,5,2,3],[2,3,5,4,1,0],[3,2,1,0,5,4],[4,5,3,2,0,1],[5,4,0,1,3,2]],\"identity\":0,\"subset\":[0,1],\"witness_g\":5,\"witness_h\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1], 5, 1, 2, 3, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)","explanation_zh":"Wave6第七層核對=([0, 1], 5, 1, 2, 3, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-nonnormal-witness","seed":202611131091,"parameters":{"group_table":[[0,1,2,3,4,5],[1,0,4,5,2,3],[2,3,5,4,1,0],[3,2,1,0,5,4],[4,5,3,2,0,1],[5,4,0,1,3,2]],"identity":0,"subset":[0,1],"witness_g":5,"witness_h":1,"domain":"Complete_finite_group_normality","operation":"normal-nonnormal-witness","semantic_template":"qf-mod44-normal-nonnormal-witness/{\"group_table\":[[0,1,2,3,4,5],[1,0,4,5,2,3],[2,3,5,4,1,0],[3,2,1,0,5,4],[4,5,3,2,0,1],[5,4,0,1,3,2]],\"identity\":0,\"subset\":[0,1],\"witness_g\":5,\"witness_h\":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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0092","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-nonnormal-witness；資料={\"group_table\":[[0,1,2,3,4,5],[1,0,5,4,3,2],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,2,1,5,0,3],[5,3,4,1,2,0]],\"identity\":0,\"subset\":[0,1],\"witness_g\":2,\"witness_h\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1], 2, 1, 3, 5, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)","explanation_zh":"Wave6第七層核對=([0, 1], 2, 1, 3, 5, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-nonnormal-witness","seed":202611131092,"parameters":{"group_table":[[0,1,2,3,4,5],[1,0,5,4,3,2],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,2,1,5,0,3],[5,3,4,1,2,0]],"identity":0,"subset":[0,1],"witness_g":2,"witness_h":1,"domain":"Complete_finite_group_normality","operation":"normal-nonnormal-witness","semantic_template":"qf-mod44-normal-nonnormal-witness/{\"group_table\":[[0,1,2,3,4,5],[1,0,5,4,3,2],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,2,1,5,0,3],[5,3,4,1,2,0]],\"identity\":0,\"subset\":[0,1],\"witness_g\":2,\"witness_h\":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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0093","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-nonnormal-witness；資料={\"group_table\":[[0,1,2,3,4,5],[1,0,4,5,2,3],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,3,5,1,0,2],[5,2,1,4,3,0]],\"identity\":0,\"subset\":[0,1],\"witness_g\":3,\"witness_h\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1], 3, 1, 2, 5, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)","explanation_zh":"Wave6第七層核對=([0, 1], 3, 1, 2, 5, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-nonnormal-witness","seed":202611131093,"parameters":{"group_table":[[0,1,2,3,4,5],[1,0,4,5,2,3],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,3,5,1,0,2],[5,2,1,4,3,0]],"identity":0,"subset":[0,1],"witness_g":3,"witness_h":1,"domain":"Complete_finite_group_normality","operation":"normal-nonnormal-witness","semantic_template":"qf-mod44-normal-nonnormal-witness/{\"group_table\":[[0,1,2,3,4,5],[1,0,4,5,2,3],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,3,5,1,0,2],[5,2,1,4,3,0]],\"identity\":0,\"subset\":[0,1],\"witness_g\":3,\"witness_h\":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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0094","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-nonnormal-witness；資料={\"group_table\":[[0,1,2,3,4,5],[1,0,3,2,5,4],[2,4,0,5,1,3],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,3,4,1,2,0]],\"identity\":0,\"subset\":[0,1],\"witness_g\":3,\"witness_h\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1], 3, 1, 4, 2, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)","explanation_zh":"Wave6第七層核對=([0, 1], 3, 1, 4, 2, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-nonnormal-witness","seed":202611131094,"parameters":{"group_table":[[0,1,2,3,4,5],[1,0,3,2,5,4],[2,4,0,5,1,3],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,3,4,1,2,0]],"identity":0,"subset":[0,1],"witness_g":3,"witness_h":1,"domain":"Complete_finite_group_normality","operation":"normal-nonnormal-witness","semantic_template":"qf-mod44-normal-nonnormal-witness/{\"group_table\":[[0,1,2,3,4,5],[1,0,3,2,5,4],[2,4,0,5,1,3],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,3,4,1,2,0]],\"identity\":0,\"subset\":[0,1],\"witness_g\":3,\"witness_h\":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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0095","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-nonnormal-witness；資料={\"group_table\":[[0,1,2,3,4,5],[1,0,4,5,2,3],[2,3,0,1,5,4],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,4,3,2,1,0]],\"identity\":0,\"subset\":[0,1],\"witness_g\":4,\"witness_h\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1], 4, 1, 3, 2, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)","explanation_zh":"Wave6第七層核對=([0, 1], 4, 1, 3, 2, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-nonnormal-witness","seed":202611131095,"parameters":{"group_table":[[0,1,2,3,4,5],[1,0,4,5,2,3],[2,3,0,1,5,4],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,4,3,2,1,0]],"identity":0,"subset":[0,1],"witness_g":4,"witness_h":1,"domain":"Complete_finite_group_normality","operation":"normal-nonnormal-witness","semantic_template":"qf-mod44-normal-nonnormal-witness/{\"group_table\":[[0,1,2,3,4,5],[1,0,4,5,2,3],[2,3,0,1,5,4],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,4,3,2,1,0]],\"identity\":0,\"subset\":[0,1],\"witness_g\":4,\"witness_h\":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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0096","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-nonnormal-witness；資料={\"group_table\":[[0,1,2,3,4,5],[1,0,3,2,5,4],[2,5,4,1,0,3],[3,4,5,0,1,2],[4,3,0,5,2,1],[5,2,1,4,3,0]],\"identity\":0,\"subset\":[0,1],\"witness_g\":2,\"witness_h\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1], 2, 1, 4, 3, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)","explanation_zh":"Wave6第七層核對=([0, 1], 2, 1, 4, 3, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-nonnormal-witness","seed":202611131096,"parameters":{"group_table":[[0,1,2,3,4,5],[1,0,3,2,5,4],[2,5,4,1,0,3],[3,4,5,0,1,2],[4,3,0,5,2,1],[5,2,1,4,3,0]],"identity":0,"subset":[0,1],"witness_g":2,"witness_h":1,"domain":"Complete_finite_group_normality","operation":"normal-nonnormal-witness","semantic_template":"qf-mod44-normal-nonnormal-witness/{\"group_table\":[[0,1,2,3,4,5],[1,0,3,2,5,4],[2,5,4,1,0,3],[3,4,5,0,1,2],[4,3,0,5,2,1],[5,2,1,4,3,0]],\"identity\":0,\"subset\":[0,1],\"witness_g\":2,\"witness_h\":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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0097","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-nonnormal-witness；資料={\"group_table\":[[0,1,2,3,4,5],[1,0,5,4,3,2],[2,3,4,5,0,1],[3,2,1,0,5,4],[4,5,0,1,2,3],[5,4,3,2,1,0]],\"identity\":0,\"subset\":[0,1],\"witness_g\":4,\"witness_h\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1], 4, 1, 2, 3, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)","explanation_zh":"Wave6第七層核對=([0, 1], 4, 1, 2, 3, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-nonnormal-witness","seed":202611131097,"parameters":{"group_table":[[0,1,2,3,4,5],[1,0,5,4,3,2],[2,3,4,5,0,1],[3,2,1,0,5,4],[4,5,0,1,2,3],[5,4,3,2,1,0]],"identity":0,"subset":[0,1],"witness_g":4,"witness_h":1,"domain":"Complete_finite_group_normality","operation":"normal-nonnormal-witness","semantic_template":"qf-mod44-normal-nonnormal-witness/{\"group_table\":[[0,1,2,3,4,5],[1,0,5,4,3,2],[2,3,4,5,0,1],[3,2,1,0,5,4],[4,5,0,1,2,3],[5,4,3,2,1,0]],\"identity\":0,\"subset\":[0,1],\"witness_g\":4,\"witness_h\":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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0098","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-nonnormal-witness；資料={\"group_table\":[[0,1,2,3,4,5],[1,0,4,5,2,3],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,3,5,1,0,2],[5,2,1,4,3,0]],\"identity\":0,\"subset\":[0,1],\"witness_g\":2,\"witness_h\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1], 2, 1, 3, 4, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)","explanation_zh":"Wave6第七層核對=([0, 1], 2, 1, 3, 4, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-nonnormal-witness","seed":202611131098,"parameters":{"group_table":[[0,1,2,3,4,5],[1,0,4,5,2,3],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,3,5,1,0,2],[5,2,1,4,3,0]],"identity":0,"subset":[0,1],"witness_g":2,"witness_h":1,"domain":"Complete_finite_group_normality","operation":"normal-nonnormal-witness","semantic_template":"qf-mod44-normal-nonnormal-witness/{\"group_table\":[[0,1,2,3,4,5],[1,0,4,5,2,3],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,3,5,1,0,2],[5,2,1,4,3,0]],\"identity\":0,\"subset\":[0,1],\"witness_g\":2,\"witness_h\":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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0099","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-nonnormal-witness；資料={\"group_table\":[[0,1,2,3,4,5],[1,0,5,4,3,2],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,2,1,5,0,3],[5,3,4,1,2,0]],\"identity\":0,\"subset\":[0,1],\"witness_g\":3,\"witness_h\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1], 3, 1, 2, 4, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)","explanation_zh":"Wave6第七層核對=([0, 1], 3, 1, 2, 4, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-nonnormal-witness","seed":202611131099,"parameters":{"group_table":[[0,1,2,3,4,5],[1,0,5,4,3,2],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,2,1,5,0,3],[5,3,4,1,2,0]],"identity":0,"subset":[0,1],"witness_g":3,"witness_h":1,"domain":"Complete_finite_group_normality","operation":"normal-nonnormal-witness","semantic_template":"qf-mod44-normal-nonnormal-witness/{\"group_table\":[[0,1,2,3,4,5],[1,0,5,4,3,2],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,2,1,5,0,3],[5,3,4,1,2,0]],\"identity\":0,\"subset\":[0,1],\"witness_g\":3,\"witness_h\":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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r01-finite-normal-subgroup-0100","batch_id":"batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-normal-subgroup"],"title_zh":"有限正規子群","stem_zh":"完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-nonnormal-witness；資料={\"group_table\":[[0,1,2,3,4,5],[1,0,4,5,2,3],[2,5,0,4,3,1],[3,4,5,0,1,2],[4,3,1,2,5,0],[5,2,3,1,0,4]],\"identity\":0,\"subset\":[0,2],\"witness_g\":4,\"witness_h\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 2], 4, 2, 5, 3, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)","explanation_zh":"Wave6第七層核對=([0, 2], 4, 2, 5, 3, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限正規子群"]},"variant":{"family_id":"qf-mod44-normal-nonnormal-witness","seed":202611131100,"parameters":{"group_table":[[0,1,2,3,4,5],[1,0,4,5,2,3],[2,5,0,4,3,1],[3,4,5,0,1,2],[4,3,1,2,5,0],[5,2,3,1,0,4]],"identity":0,"subset":[0,2],"witness_g":4,"witness_h":2,"domain":"Complete_finite_group_normality","operation":"normal-nonnormal-witness","semantic_template":"qf-mod44-normal-nonnormal-witness/{\"group_table\":[[0,1,2,3,4,5],[1,0,4,5,2,3],[2,5,0,4,3,1],[3,4,5,0,1,2],[4,3,1,2,5,0],[5,2,3,1,0,4]],\"identity\":0,\"subset\":[0,2],\"witness_g\":4,\"witness_h\":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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}