{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0001","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-prime-divisor-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,\"prime\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 2, 是, 0, complete_group_and_prime_divisibility_hypotheses_hold)","explanation_zh":"Wave6第六層核對=(6, 2, 是, 0, complete_group_and_prime_divisibility_hypotheses_hold)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-prime-divisor-qualification","seed":202611121001,"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,"prime":2,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-prime-divisor-qualification","semantic_template":"qf-mod43-cauchy-prime-divisor-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,\"prime\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0002","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-prime-divisor-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,\"prime\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, 2, 是, 0, complete_group_and_prime_divisibility_hypotheses_hold)","explanation_zh":"Wave6第六層核對=(8, 2, 是, 0, complete_group_and_prime_divisibility_hypotheses_hold)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-prime-divisor-qualification","seed":202611121002,"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,"prime":2,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-prime-divisor-qualification","semantic_template":"qf-mod43-cauchy-prime-divisor-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,\"prime\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0003","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-prime-divisor-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,\"prime\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, 2, 是, 0, complete_group_and_prime_divisibility_hypotheses_hold)","explanation_zh":"Wave6第六層核對=(10, 2, 是, 0, complete_group_and_prime_divisibility_hypotheses_hold)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-prime-divisor-qualification","seed":202611121003,"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,"prime":2,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-prime-divisor-qualification","semantic_template":"qf-mod43-cauchy-prime-divisor-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,\"prime\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0004","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-prime-divisor-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,\"prime\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, 2, 是, 0, complete_group_and_prime_divisibility_hypotheses_hold)","explanation_zh":"Wave6第六層核對=(12, 2, 是, 0, complete_group_and_prime_divisibility_hypotheses_hold)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-prime-divisor-qualification","seed":202611121004,"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,"prime":2,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-prime-divisor-qualification","semantic_template":"qf-mod43-cauchy-prime-divisor-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,\"prime\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0005","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-prime-divisor-qualification；資料={\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12]],\"identity\":0,\"prime\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(14, 2, 是, 0, complete_group_and_prime_divisibility_hypotheses_hold)","explanation_zh":"Wave6第六層核對=(14, 2, 是, 0, complete_group_and_prime_divisibility_hypotheses_hold)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-prime-divisor-qualification","seed":202611121005,"parameters":{"group_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12]],"identity":0,"prime":2,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-prime-divisor-qualification","semantic_template":"qf-mod43-cauchy-prime-divisor-qualification/{\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12]],\"identity\":0,\"prime\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0006","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-prime-divisor-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,\"prime\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 2, 是, 0, complete_group_and_prime_divisibility_hypotheses_hold)","explanation_zh":"Wave6第六層核對=(6, 2, 是, 0, complete_group_and_prime_divisibility_hypotheses_hold)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-prime-divisor-qualification","seed":202611121006,"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,"prime":2,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-prime-divisor-qualification","semantic_template":"qf-mod43-cauchy-prime-divisor-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,\"prime\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0007","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-prime-divisor-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,\"prime\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, 2, 是, 0, complete_group_and_prime_divisibility_hypotheses_hold)","explanation_zh":"Wave6第六層核對=(8, 2, 是, 0, complete_group_and_prime_divisibility_hypotheses_hold)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-prime-divisor-qualification","seed":202611121007,"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,"prime":2,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-prime-divisor-qualification","semantic_template":"qf-mod43-cauchy-prime-divisor-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,\"prime\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0008","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-prime-divisor-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,\"prime\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, 2, 是, 0, complete_group_and_prime_divisibility_hypotheses_hold)","explanation_zh":"Wave6第六層核對=(10, 2, 是, 0, complete_group_and_prime_divisibility_hypotheses_hold)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-prime-divisor-qualification","seed":202611121008,"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,"prime":2,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-prime-divisor-qualification","semantic_template":"qf-mod43-cauchy-prime-divisor-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,\"prime\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0009","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-prime-divisor-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,\"prime\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, 2, 是, 0, complete_group_and_prime_divisibility_hypotheses_hold)","explanation_zh":"Wave6第六層核對=(12, 2, 是, 0, complete_group_and_prime_divisibility_hypotheses_hold)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-prime-divisor-qualification","seed":202611121009,"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,"prime":2,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-prime-divisor-qualification","semantic_template":"qf-mod43-cauchy-prime-divisor-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,\"prime\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0010","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-prime-divisor-qualification；資料={\"group_table\":[[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11]],\"identity\":1,\"prime\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(14, 2, 是, 0, complete_group_and_prime_divisibility_hypotheses_hold)","explanation_zh":"Wave6第六層核對=(14, 2, 是, 0, complete_group_and_prime_divisibility_hypotheses_hold)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-prime-divisor-qualification","seed":202611121010,"parameters":{"group_table":[[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11]],"identity":1,"prime":2,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-prime-divisor-qualification","semantic_template":"qf-mod43-cauchy-prime-divisor-qualification/{\"group_table\":[[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11]],\"identity\":1,\"prime\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0011","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-prime-divisor-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,\"prime\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 2, 是, 0, complete_group_and_prime_divisibility_hypotheses_hold)","explanation_zh":"Wave6第六層核對=(6, 2, 是, 0, complete_group_and_prime_divisibility_hypotheses_hold)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-prime-divisor-qualification","seed":202611121011,"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,"prime":2,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-prime-divisor-qualification","semantic_template":"qf-mod43-cauchy-prime-divisor-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,\"prime\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0012","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-prime-divisor-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,\"prime\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, 2, 是, 0, complete_group_and_prime_divisibility_hypotheses_hold)","explanation_zh":"Wave6第六層核對=(8, 2, 是, 0, complete_group_and_prime_divisibility_hypotheses_hold)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-prime-divisor-qualification","seed":202611121012,"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,"prime":2,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-prime-divisor-qualification","semantic_template":"qf-mod43-cauchy-prime-divisor-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,\"prime\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0013","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-prime-divisor-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,\"prime\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, 2, 是, 0, complete_group_and_prime_divisibility_hypotheses_hold)","explanation_zh":"Wave6第六層核對=(10, 2, 是, 0, complete_group_and_prime_divisibility_hypotheses_hold)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-prime-divisor-qualification","seed":202611121013,"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,"prime":2,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-prime-divisor-qualification","semantic_template":"qf-mod43-cauchy-prime-divisor-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,\"prime\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0014","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-prime-divisor-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,\"prime\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, 2, 是, 0, complete_group_and_prime_divisibility_hypotheses_hold)","explanation_zh":"Wave6第六層核對=(12, 2, 是, 0, complete_group_and_prime_divisibility_hypotheses_hold)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-prime-divisor-qualification","seed":202611121014,"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,"prime":2,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-prime-divisor-qualification","semantic_template":"qf-mod43-cauchy-prime-divisor-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,\"prime\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0015","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-prime-divisor-qualification；資料={\"group_table\":[[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10]],\"identity\":2,\"prime\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(14, 2, 是, 0, complete_group_and_prime_divisibility_hypotheses_hold)","explanation_zh":"Wave6第六層核對=(14, 2, 是, 0, complete_group_and_prime_divisibility_hypotheses_hold)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-prime-divisor-qualification","seed":202611121015,"parameters":{"group_table":[[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10]],"identity":2,"prime":2,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-prime-divisor-qualification","semantic_template":"qf-mod43-cauchy-prime-divisor-qualification/{\"group_table\":[[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10]],\"identity\":2,\"prime\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0016","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-prime-divisor-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,\"prime\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 2, 是, 0, complete_group_and_prime_divisibility_hypotheses_hold)","explanation_zh":"Wave6第六層核對=(6, 2, 是, 0, complete_group_and_prime_divisibility_hypotheses_hold)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-prime-divisor-qualification","seed":202611121016,"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,"prime":2,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-prime-divisor-qualification","semantic_template":"qf-mod43-cauchy-prime-divisor-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,\"prime\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0017","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-prime-divisor-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,\"prime\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, 2, 是, 0, complete_group_and_prime_divisibility_hypotheses_hold)","explanation_zh":"Wave6第六層核對=(8, 2, 是, 0, complete_group_and_prime_divisibility_hypotheses_hold)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-prime-divisor-qualification","seed":202611121017,"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,"prime":2,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-prime-divisor-qualification","semantic_template":"qf-mod43-cauchy-prime-divisor-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,\"prime\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0018","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-prime-divisor-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,\"prime\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, 2, 是, 0, complete_group_and_prime_divisibility_hypotheses_hold)","explanation_zh":"Wave6第六層核對=(10, 2, 是, 0, complete_group_and_prime_divisibility_hypotheses_hold)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-prime-divisor-qualification","seed":202611121018,"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,"prime":2,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-prime-divisor-qualification","semantic_template":"qf-mod43-cauchy-prime-divisor-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,\"prime\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0019","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-prime-divisor-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,\"prime\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, 2, 是, 0, complete_group_and_prime_divisibility_hypotheses_hold)","explanation_zh":"Wave6第六層核對=(12, 2, 是, 0, complete_group_and_prime_divisibility_hypotheses_hold)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-prime-divisor-qualification","seed":202611121019,"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,"prime":2,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-prime-divisor-qualification","semantic_template":"qf-mod43-cauchy-prime-divisor-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,\"prime\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0020","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-prime-divisor-qualification；資料={\"group_table\":[[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9]],\"identity\":3,\"prime\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(14, 2, 是, 0, complete_group_and_prime_divisibility_hypotheses_hold)","explanation_zh":"Wave6第六層核對=(14, 2, 是, 0, complete_group_and_prime_divisibility_hypotheses_hold)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-prime-divisor-qualification","seed":202611121020,"parameters":{"group_table":[[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9]],"identity":3,"prime":2,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-prime-divisor-qualification","semantic_template":"qf-mod43-cauchy-prime-divisor-qualification/{\"group_table\":[[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9]],\"identity\":3,\"prime\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0021","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-prime-divisor-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,\"prime\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 2, 是, 0, complete_group_and_prime_divisibility_hypotheses_hold)","explanation_zh":"Wave6第六層核對=(6, 2, 是, 0, complete_group_and_prime_divisibility_hypotheses_hold)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-prime-divisor-qualification","seed":202611121021,"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,"prime":2,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-prime-divisor-qualification","semantic_template":"qf-mod43-cauchy-prime-divisor-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,\"prime\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0022","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-prime-divisor-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,\"prime\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, 2, 是, 0, complete_group_and_prime_divisibility_hypotheses_hold)","explanation_zh":"Wave6第六層核對=(8, 2, 是, 0, complete_group_and_prime_divisibility_hypotheses_hold)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-prime-divisor-qualification","seed":202611121022,"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,"prime":2,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-prime-divisor-qualification","semantic_template":"qf-mod43-cauchy-prime-divisor-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,\"prime\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0023","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-prime-divisor-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,\"prime\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, 2, 是, 0, complete_group_and_prime_divisibility_hypotheses_hold)","explanation_zh":"Wave6第六層核對=(10, 2, 是, 0, complete_group_and_prime_divisibility_hypotheses_hold)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-prime-divisor-qualification","seed":202611121023,"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,"prime":2,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-prime-divisor-qualification","semantic_template":"qf-mod43-cauchy-prime-divisor-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,\"prime\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0024","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-prime-divisor-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,\"prime\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, 2, 是, 0, complete_group_and_prime_divisibility_hypotheses_hold)","explanation_zh":"Wave6第六層核對=(12, 2, 是, 0, complete_group_and_prime_divisibility_hypotheses_hold)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-prime-divisor-qualification","seed":202611121024,"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,"prime":2,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-prime-divisor-qualification","semantic_template":"qf-mod43-cauchy-prime-divisor-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,\"prime\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0025","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-prime-divisor-qualification；資料={\"group_table\":[[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8]],\"identity\":4,\"prime\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(14, 2, 是, 0, complete_group_and_prime_divisibility_hypotheses_hold)","explanation_zh":"Wave6第六層核對=(14, 2, 是, 0, complete_group_and_prime_divisibility_hypotheses_hold)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-prime-divisor-qualification","seed":202611121025,"parameters":{"group_table":[[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8]],"identity":4,"prime":2,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-prime-divisor-qualification","semantic_template":"qf-mod43-cauchy-prime-divisor-qualification/{\"group_table\":[[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8]],\"identity\":4,\"prime\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0026","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-element-order-table；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}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, [[0, 1], [1, 6], [2, 3], [3, 2], [4, 3], [5, 6]], 是, complete_element_order_table_for_declared_finite_group)","explanation_zh":"Wave6第六層核對=(6, [[0, 1], [1, 6], [2, 3], [3, 2], [4, 3], [5, 6]], 是, complete_element_order_table_for_declared_finite_group)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-element-order-table","seed":202611121026,"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,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-element-order-table","semantic_template":"qf-mod43-cauchy-element-order-table/{\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0027","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-element-order-table；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}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, [[0, 1], [1, 8], [2, 4], [3, 8], [4, 2], [5, 8], [6, 4], [7, 8]], 是, complete_element_order_table_for_declared_finite_group)","explanation_zh":"Wave6第六層核對=(8, [[0, 1], [1, 8], [2, 4], [3, 8], [4, 2], [5, 8], [6, 4], [7, 8]], 是, complete_element_order_table_for_declared_finite_group)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-element-order-table","seed":202611121027,"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,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-element-order-table","semantic_template":"qf-mod43-cauchy-element-order-table/{\"group_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],\"identity\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0028","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-element-order-table；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}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, [[0, 1], [1, 10], [2, 5], [3, 10], [4, 5], [5, 2], [6, 5], [7, 10], [8, 5], [9, 10]], 是, complete_element_order_table_for_declared_finite_group)","explanation_zh":"Wave6第六層核對=(10, [[0, 1], [1, 10], [2, 5], [3, 10], [4, 5], [5, 2], [6, 5], [7, 10], [8, 5], [9, 10]], 是, complete_element_order_table_for_declared_finite_group)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-element-order-table","seed":202611121028,"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,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-element-order-table","semantic_template":"qf-mod43-cauchy-element-order-table/{\"group_table\":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8]],\"identity\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0029","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-element-order-table；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}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, [[0, 1], [1, 12], [2, 6], [3, 4], [4, 3], [5, 12], [6, 2], [7, 12], [8, 3], [9, 4], [10, 6], [11, 12]], 是, complete_element_order_table_for_declared_finite_group)","explanation_zh":"Wave6第六層核對=(12, [[0, 1], [1, 12], [2, 6], [3, 4], [4, 3], [5, 12], [6, 2], [7, 12], [8, 3], [9, 4], [10, 6], [11, 12]], 是, complete_element_order_table_for_declared_finite_group)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-element-order-table","seed":202611121029,"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,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-element-order-table","semantic_template":"qf-mod43-cauchy-element-order-table/{\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10]],\"identity\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0030","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-element-order-table；資料={\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(14, [[0, 1], [1, 14], [2, 7], [3, 14], [4, 7], [5, 14], [6, 7], [7, 2], [8, 7], [9, 14], [10, 7], [11, 14], [12, 7], [13, 14]], 是, complete_element_order_table_for_declared_finite_group)","explanation_zh":"Wave6第六層核對=(14, [[0, 1], [1, 14], [2, 7], [3, 14], [4, 7], [5, 14], [6, 7], [7, 2], [8, 7], [9, 14], [10, 7], [11, 14], [12, 7], [13, 14]], 是, complete_element_order_table_for_declared_finite_group)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-element-order-table","seed":202611121030,"parameters":{"group_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12]],"identity":0,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-element-order-table","semantic_template":"qf-mod43-cauchy-element-order-table/{\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12]],\"identity\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0031","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-element-order-table；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}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, [[0, 6], [1, 1], [2, 6], [3, 3], [4, 2], [5, 3]], 是, complete_element_order_table_for_declared_finite_group)","explanation_zh":"Wave6第六層核對=(6, [[0, 6], [1, 1], [2, 6], [3, 3], [4, 2], [5, 3]], 是, complete_element_order_table_for_declared_finite_group)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-element-order-table","seed":202611121031,"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,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-element-order-table","semantic_template":"qf-mod43-cauchy-element-order-table/{\"group_table\":[[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3]],\"identity\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0032","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-element-order-table；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}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, [[0, 8], [1, 1], [2, 8], [3, 4], [4, 8], [5, 2], [6, 8], [7, 4]], 是, complete_element_order_table_for_declared_finite_group)","explanation_zh":"Wave6第六層核對=(8, [[0, 8], [1, 1], [2, 8], [3, 4], [4, 8], [5, 2], [6, 8], [7, 4]], 是, complete_element_order_table_for_declared_finite_group)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-element-order-table","seed":202611121032,"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,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-element-order-table","semantic_template":"qf-mod43-cauchy-element-order-table/{\"group_table\":[[7,0,1,2,3,4,5,6],[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5]],\"identity\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0033","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-element-order-table；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}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, [[0, 10], [1, 1], [2, 10], [3, 5], [4, 10], [5, 5], [6, 2], [7, 5], [8, 10], [9, 5]], 是, complete_element_order_table_for_declared_finite_group)","explanation_zh":"Wave6第六層核對=(10, [[0, 10], [1, 1], [2, 10], [3, 5], [4, 10], [5, 5], [6, 2], [7, 5], [8, 10], [9, 5]], 是, complete_element_order_table_for_declared_finite_group)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-element-order-table","seed":202611121033,"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,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-element-order-table","semantic_template":"qf-mod43-cauchy-element-order-table/{\"group_table\":[[9,0,1,2,3,4,5,6,7,8],[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7]],\"identity\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0034","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-element-order-table；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}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, [[0, 12], [1, 1], [2, 12], [3, 6], [4, 4], [5, 3], [6, 12], [7, 2], [8, 12], [9, 3], [10, 4], [11, 6]], 是, complete_element_order_table_for_declared_finite_group)","explanation_zh":"Wave6第六層核對=(12, [[0, 12], [1, 1], [2, 12], [3, 6], [4, 4], [5, 3], [6, 12], [7, 2], [8, 12], [9, 3], [10, 4], [11, 6]], 是, complete_element_order_table_for_declared_finite_group)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-element-order-table","seed":202611121034,"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,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-element-order-table","semantic_template":"qf-mod43-cauchy-element-order-table/{\"group_table\":[[11,0,1,2,3,4,5,6,7,8,9,10],[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9]],\"identity\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0035","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-element-order-table；資料={\"group_table\":[[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11]],\"identity\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(14, [[0, 14], [1, 1], [2, 14], [3, 7], [4, 14], [5, 7], [6, 14], [7, 7], [8, 2], [9, 7], [10, 14], [11, 7], [12, 14], [13, 7]], 是, complete_element_order_table_for_declared_finite_group)","explanation_zh":"Wave6第六層核對=(14, [[0, 14], [1, 1], [2, 14], [3, 7], [4, 14], [5, 7], [6, 14], [7, 7], [8, 2], [9, 7], [10, 14], [11, 7], [12, 14], [13, 7]], 是, complete_element_order_table_for_declared_finite_group)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-element-order-table","seed":202611121035,"parameters":{"group_table":[[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11]],"identity":1,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-element-order-table","semantic_template":"qf-mod43-cauchy-element-order-table/{\"group_table\":[[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11]],\"identity\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0036","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-element-order-table；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}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, [[0, 3], [1, 6], [2, 1], [3, 6], [4, 3], [5, 2]], 是, complete_element_order_table_for_declared_finite_group)","explanation_zh":"Wave6第六層核對=(6, [[0, 3], [1, 6], [2, 1], [3, 6], [4, 3], [5, 2]], 是, complete_element_order_table_for_declared_finite_group)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-element-order-table","seed":202611121036,"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,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-element-order-table","semantic_template":"qf-mod43-cauchy-element-order-table/{\"group_table\":[[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2]],\"identity\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0037","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-element-order-table；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}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, [[0, 4], [1, 8], [2, 1], [3, 8], [4, 4], [5, 8], [6, 2], [7, 8]], 是, complete_element_order_table_for_declared_finite_group)","explanation_zh":"Wave6第六層核對=(8, [[0, 4], [1, 8], [2, 1], [3, 8], [4, 4], [5, 8], [6, 2], [7, 8]], 是, complete_element_order_table_for_declared_finite_group)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-element-order-table","seed":202611121037,"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,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-element-order-table","semantic_template":"qf-mod43-cauchy-element-order-table/{\"group_table\":[[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6],[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4]],\"identity\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0038","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-element-order-table；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}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, [[0, 5], [1, 10], [2, 1], [3, 10], [4, 5], [5, 10], [6, 5], [7, 2], [8, 5], [9, 10]], 是, complete_element_order_table_for_declared_finite_group)","explanation_zh":"Wave6第六層核對=(10, [[0, 5], [1, 10], [2, 1], [3, 10], [4, 5], [5, 10], [6, 5], [7, 2], [8, 5], [9, 10]], 是, complete_element_order_table_for_declared_finite_group)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-element-order-table","seed":202611121038,"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,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-element-order-table","semantic_template":"qf-mod43-cauchy-element-order-table/{\"group_table\":[[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8],[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6]],\"identity\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0039","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-element-order-table；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}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, [[0, 6], [1, 12], [2, 1], [3, 12], [4, 6], [5, 4], [6, 3], [7, 12], [8, 2], [9, 12], [10, 3], [11, 4]], 是, complete_element_order_table_for_declared_finite_group)","explanation_zh":"Wave6第六層核對=(12, [[0, 6], [1, 12], [2, 1], [3, 12], [4, 6], [5, 4], [6, 3], [7, 12], [8, 2], [9, 12], [10, 3], [11, 4]], 是, complete_element_order_table_for_declared_finite_group)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-element-order-table","seed":202611121039,"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,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-element-order-table","semantic_template":"qf-mod43-cauchy-element-order-table/{\"group_table\":[[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10],[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8]],\"identity\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0040","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-element-order-table；資料={\"group_table\":[[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10]],\"identity\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(14, [[0, 7], [1, 14], [2, 1], [3, 14], [4, 7], [5, 14], [6, 7], [7, 14], [8, 7], [9, 2], [10, 7], [11, 14], [12, 7], [13, 14]], 是, complete_element_order_table_for_declared_finite_group)","explanation_zh":"Wave6第六層核對=(14, [[0, 7], [1, 14], [2, 1], [3, 14], [4, 7], [5, 14], [6, 7], [7, 14], [8, 7], [9, 2], [10, 7], [11, 14], [12, 7], [13, 14]], 是, complete_element_order_table_for_declared_finite_group)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-element-order-table","seed":202611121040,"parameters":{"group_table":[[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10]],"identity":2,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-element-order-table","semantic_template":"qf-mod43-cauchy-element-order-table/{\"group_table\":[[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10]],\"identity\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0041","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-element-order-table；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}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, [[0, 2], [1, 3], [2, 6], [3, 1], [4, 6], [5, 3]], 是, complete_element_order_table_for_declared_finite_group)","explanation_zh":"Wave6第六層核對=(6, [[0, 2], [1, 3], [2, 6], [3, 1], [4, 6], [5, 3]], 是, complete_element_order_table_for_declared_finite_group)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-element-order-table","seed":202611121041,"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,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-element-order-table","semantic_template":"qf-mod43-cauchy-element-order-table/{\"group_table\":[[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1]],\"identity\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0042","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-element-order-table；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}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, [[0, 8], [1, 4], [2, 8], [3, 1], [4, 8], [5, 4], [6, 8], [7, 2]], 是, complete_element_order_table_for_declared_finite_group)","explanation_zh":"Wave6第六層核對=(8, [[0, 8], [1, 4], [2, 8], [3, 1], [4, 8], [5, 4], [6, 8], [7, 2]], 是, complete_element_order_table_for_declared_finite_group)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-element-order-table","seed":202611121042,"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,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-element-order-table","semantic_template":"qf-mod43-cauchy-element-order-table/{\"group_table\":[[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6],[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3]],\"identity\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0043","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-element-order-table；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}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, [[0, 10], [1, 5], [2, 10], [3, 1], [4, 10], [5, 5], [6, 10], [7, 5], [8, 2], [9, 5]], 是, complete_element_order_table_for_declared_finite_group)","explanation_zh":"Wave6第六層核對=(10, [[0, 10], [1, 5], [2, 10], [3, 1], [4, 10], [5, 5], [6, 10], [7, 5], [8, 2], [9, 5]], 是, complete_element_order_table_for_declared_finite_group)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-element-order-table","seed":202611121043,"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,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-element-order-table","semantic_template":"qf-mod43-cauchy-element-order-table/{\"group_table\":[[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8],[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5]],\"identity\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0044","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-element-order-table；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}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, [[0, 4], [1, 6], [2, 12], [3, 1], [4, 12], [5, 6], [6, 4], [7, 3], [8, 12], [9, 2], [10, 12], [11, 3]], 是, complete_element_order_table_for_declared_finite_group)","explanation_zh":"Wave6第六層核對=(12, [[0, 4], [1, 6], [2, 12], [3, 1], [4, 12], [5, 6], [6, 4], [7, 3], [8, 12], [9, 2], [10, 12], [11, 3]], 是, complete_element_order_table_for_declared_finite_group)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-element-order-table","seed":202611121044,"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,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-element-order-table","semantic_template":"qf-mod43-cauchy-element-order-table/{\"group_table\":[[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10],[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7]],\"identity\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0045","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-element-order-table；資料={\"group_table\":[[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9]],\"identity\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(14, [[0, 14], [1, 7], [2, 14], [3, 1], [4, 14], [5, 7], [6, 14], [7, 7], [8, 14], [9, 7], [10, 2], [11, 7], [12, 14], [13, 7]], 是, complete_element_order_table_for_declared_finite_group)","explanation_zh":"Wave6第六層核對=(14, [[0, 14], [1, 7], [2, 14], [3, 1], [4, 14], [5, 7], [6, 14], [7, 7], [8, 14], [9, 7], [10, 2], [11, 7], [12, 14], [13, 7]], 是, complete_element_order_table_for_declared_finite_group)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-element-order-table","seed":202611121045,"parameters":{"group_table":[[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9]],"identity":3,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-element-order-table","semantic_template":"qf-mod43-cauchy-element-order-table/{\"group_table\":[[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9]],\"identity\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0046","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-element-order-table；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}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, [[0, 3], [1, 2], [2, 3], [3, 6], [4, 1], [5, 6]], 是, complete_element_order_table_for_declared_finite_group)","explanation_zh":"Wave6第六層核對=(6, [[0, 3], [1, 2], [2, 3], [3, 6], [4, 1], [5, 6]], 是, complete_element_order_table_for_declared_finite_group)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-element-order-table","seed":202611121046,"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,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-element-order-table","semantic_template":"qf-mod43-cauchy-element-order-table/{\"group_table\":[[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0]],\"identity\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0047","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-element-order-table；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}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, [[0, 2], [1, 8], [2, 4], [3, 8], [4, 1], [5, 8], [6, 4], [7, 8]], 是, complete_element_order_table_for_declared_finite_group)","explanation_zh":"Wave6第六層核對=(8, [[0, 2], [1, 8], [2, 4], [3, 8], [4, 1], [5, 8], [6, 4], [7, 8]], 是, complete_element_order_table_for_declared_finite_group)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-element-order-table","seed":202611121047,"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,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-element-order-table","semantic_template":"qf-mod43-cauchy-element-order-table/{\"group_table\":[[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6],[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2]],\"identity\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0048","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-element-order-table；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}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, [[0, 5], [1, 10], [2, 5], [3, 10], [4, 1], [5, 10], [6, 5], [7, 10], [8, 5], [9, 2]], 是, complete_element_order_table_for_declared_finite_group)","explanation_zh":"Wave6第六層核對=(10, [[0, 5], [1, 10], [2, 5], [3, 10], [4, 1], [5, 10], [6, 5], [7, 10], [8, 5], [9, 2]], 是, complete_element_order_table_for_declared_finite_group)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-element-order-table","seed":202611121048,"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,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-element-order-table","semantic_template":"qf-mod43-cauchy-element-order-table/{\"group_table\":[[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8],[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4]],\"identity\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0049","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-element-order-table；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}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, [[0, 3], [1, 4], [2, 6], [3, 12], [4, 1], [5, 12], [6, 6], [7, 4], [8, 3], [9, 12], [10, 2], [11, 12]], 是, complete_element_order_table_for_declared_finite_group)","explanation_zh":"Wave6第六層核對=(12, [[0, 3], [1, 4], [2, 6], [3, 12], [4, 1], [5, 12], [6, 6], [7, 4], [8, 3], [9, 12], [10, 2], [11, 12]], 是, complete_element_order_table_for_declared_finite_group)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-element-order-table","seed":202611121049,"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,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-element-order-table","semantic_template":"qf-mod43-cauchy-element-order-table/{\"group_table\":[[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10],[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6]],\"identity\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0050","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-element-order-table；資料={\"group_table\":[[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8]],\"identity\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(14, [[0, 7], [1, 14], [2, 7], [3, 14], [4, 1], [5, 14], [6, 7], [7, 14], [8, 7], [9, 14], [10, 7], [11, 2], [12, 7], [13, 14]], 是, complete_element_order_table_for_declared_finite_group)","explanation_zh":"Wave6第六層核對=(14, [[0, 7], [1, 14], [2, 7], [3, 14], [4, 1], [5, 14], [6, 7], [7, 14], [8, 7], [9, 14], [10, 7], [11, 2], [12, 7], [13, 14]], 是, complete_element_order_table_for_declared_finite_group)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-element-order-table","seed":202611121050,"parameters":{"group_table":[[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8]],"identity":4,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-element-order-table","semantic_template":"qf-mod43-cauchy-element-order-table/{\"group_table\":[[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8]],\"identity\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0051","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-order-p-witness；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,\"prime\":2,\"element\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 2, 3, [[1, 3], [2, 0]], 是, 是, g_to_p_is_identity_and_no_smaller_positive_power_is)","explanation_zh":"Wave6第六層核對=(6, 2, 3, [[1, 3], [2, 0]], 是, 是, g_to_p_is_identity_and_no_smaller_positive_power_is)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-order-p-witness","seed":202611121051,"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,"prime":2,"element":3,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-order-p-witness","semantic_template":"qf-mod43-cauchy-order-p-witness/{\"element\":3,\"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,\"prime\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0052","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-order-p-witness；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,\"prime\":2,\"element\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, 2, 4, [[1, 4], [2, 0]], 是, 是, g_to_p_is_identity_and_no_smaller_positive_power_is)","explanation_zh":"Wave6第六層核對=(8, 2, 4, [[1, 4], [2, 0]], 是, 是, g_to_p_is_identity_and_no_smaller_positive_power_is)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-order-p-witness","seed":202611121052,"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,"prime":2,"element":4,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-order-p-witness","semantic_template":"qf-mod43-cauchy-order-p-witness/{\"element\":4,\"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,\"prime\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0053","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-order-p-witness；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,\"prime\":2,\"element\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, 2, 5, [[1, 5], [2, 0]], 是, 是, g_to_p_is_identity_and_no_smaller_positive_power_is)","explanation_zh":"Wave6第六層核對=(10, 2, 5, [[1, 5], [2, 0]], 是, 是, g_to_p_is_identity_and_no_smaller_positive_power_is)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-order-p-witness","seed":202611121053,"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,"prime":2,"element":5,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-order-p-witness","semantic_template":"qf-mod43-cauchy-order-p-witness/{\"element\":5,\"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,\"prime\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0054","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-order-p-witness；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,\"prime\":2,\"element\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, 2, 6, [[1, 6], [2, 0]], 是, 是, g_to_p_is_identity_and_no_smaller_positive_power_is)","explanation_zh":"Wave6第六層核對=(12, 2, 6, [[1, 6], [2, 0]], 是, 是, g_to_p_is_identity_and_no_smaller_positive_power_is)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-order-p-witness","seed":202611121054,"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,"prime":2,"element":6,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-order-p-witness","semantic_template":"qf-mod43-cauchy-order-p-witness/{\"element\":6,\"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,\"prime\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0055","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-order-p-witness；資料={\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12]],\"identity\":0,\"prime\":2,\"element\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(14, 2, 7, [[1, 7], [2, 0]], 是, 是, g_to_p_is_identity_and_no_smaller_positive_power_is)","explanation_zh":"Wave6第六層核對=(14, 2, 7, [[1, 7], [2, 0]], 是, 是, g_to_p_is_identity_and_no_smaller_positive_power_is)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-order-p-witness","seed":202611121055,"parameters":{"group_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12]],"identity":0,"prime":2,"element":7,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-order-p-witness","semantic_template":"qf-mod43-cauchy-order-p-witness/{\"element\":7,\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12]],\"identity\":0,\"prime\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0056","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-order-p-witness；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,\"prime\":2,\"element\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 2, 4, [[1, 4], [2, 1]], 是, 是, g_to_p_is_identity_and_no_smaller_positive_power_is)","explanation_zh":"Wave6第六層核對=(6, 2, 4, [[1, 4], [2, 1]], 是, 是, g_to_p_is_identity_and_no_smaller_positive_power_is)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-order-p-witness","seed":202611121056,"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,"prime":2,"element":4,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-order-p-witness","semantic_template":"qf-mod43-cauchy-order-p-witness/{\"element\":4,\"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,\"prime\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0057","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-order-p-witness；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,\"prime\":2,\"element\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, 2, 5, [[1, 5], [2, 1]], 是, 是, g_to_p_is_identity_and_no_smaller_positive_power_is)","explanation_zh":"Wave6第六層核對=(8, 2, 5, [[1, 5], [2, 1]], 是, 是, g_to_p_is_identity_and_no_smaller_positive_power_is)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-order-p-witness","seed":202611121057,"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,"prime":2,"element":5,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-order-p-witness","semantic_template":"qf-mod43-cauchy-order-p-witness/{\"element\":5,\"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,\"prime\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0058","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-order-p-witness；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,\"prime\":2,\"element\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, 2, 6, [[1, 6], [2, 1]], 是, 是, g_to_p_is_identity_and_no_smaller_positive_power_is)","explanation_zh":"Wave6第六層核對=(10, 2, 6, [[1, 6], [2, 1]], 是, 是, g_to_p_is_identity_and_no_smaller_positive_power_is)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-order-p-witness","seed":202611121058,"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,"prime":2,"element":6,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-order-p-witness","semantic_template":"qf-mod43-cauchy-order-p-witness/{\"element\":6,\"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,\"prime\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0059","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-order-p-witness；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,\"prime\":2,\"element\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, 2, 7, [[1, 7], [2, 1]], 是, 是, g_to_p_is_identity_and_no_smaller_positive_power_is)","explanation_zh":"Wave6第六層核對=(12, 2, 7, [[1, 7], [2, 1]], 是, 是, g_to_p_is_identity_and_no_smaller_positive_power_is)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-order-p-witness","seed":202611121059,"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,"prime":2,"element":7,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-order-p-witness","semantic_template":"qf-mod43-cauchy-order-p-witness/{\"element\":7,\"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,\"prime\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0060","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-order-p-witness；資料={\"group_table\":[[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11]],\"identity\":1,\"prime\":2,\"element\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(14, 2, 8, [[1, 8], [2, 1]], 是, 是, g_to_p_is_identity_and_no_smaller_positive_power_is)","explanation_zh":"Wave6第六層核對=(14, 2, 8, [[1, 8], [2, 1]], 是, 是, g_to_p_is_identity_and_no_smaller_positive_power_is)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-order-p-witness","seed":202611121060,"parameters":{"group_table":[[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11]],"identity":1,"prime":2,"element":8,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-order-p-witness","semantic_template":"qf-mod43-cauchy-order-p-witness/{\"element\":8,\"group_table\":[[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11]],\"identity\":1,\"prime\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0061","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-order-p-witness；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,\"prime\":2,\"element\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 2, 5, [[1, 5], [2, 2]], 是, 是, g_to_p_is_identity_and_no_smaller_positive_power_is)","explanation_zh":"Wave6第六層核對=(6, 2, 5, [[1, 5], [2, 2]], 是, 是, g_to_p_is_identity_and_no_smaller_positive_power_is)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-order-p-witness","seed":202611121061,"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,"prime":2,"element":5,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-order-p-witness","semantic_template":"qf-mod43-cauchy-order-p-witness/{\"element\":5,\"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,\"prime\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0062","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-order-p-witness；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,\"prime\":2,\"element\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, 2, 6, [[1, 6], [2, 2]], 是, 是, g_to_p_is_identity_and_no_smaller_positive_power_is)","explanation_zh":"Wave6第六層核對=(8, 2, 6, [[1, 6], [2, 2]], 是, 是, g_to_p_is_identity_and_no_smaller_positive_power_is)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-order-p-witness","seed":202611121062,"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,"prime":2,"element":6,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-order-p-witness","semantic_template":"qf-mod43-cauchy-order-p-witness/{\"element\":6,\"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,\"prime\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0063","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-order-p-witness；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,\"prime\":2,\"element\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, 2, 7, [[1, 7], [2, 2]], 是, 是, g_to_p_is_identity_and_no_smaller_positive_power_is)","explanation_zh":"Wave6第六層核對=(10, 2, 7, [[1, 7], [2, 2]], 是, 是, g_to_p_is_identity_and_no_smaller_positive_power_is)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-order-p-witness","seed":202611121063,"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,"prime":2,"element":7,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-order-p-witness","semantic_template":"qf-mod43-cauchy-order-p-witness/{\"element\":7,\"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,\"prime\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0064","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-order-p-witness；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,\"prime\":2,\"element\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, 2, 8, [[1, 8], [2, 2]], 是, 是, g_to_p_is_identity_and_no_smaller_positive_power_is)","explanation_zh":"Wave6第六層核對=(12, 2, 8, [[1, 8], [2, 2]], 是, 是, g_to_p_is_identity_and_no_smaller_positive_power_is)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-order-p-witness","seed":202611121064,"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,"prime":2,"element":8,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-order-p-witness","semantic_template":"qf-mod43-cauchy-order-p-witness/{\"element\":8,\"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,\"prime\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0065","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-order-p-witness；資料={\"group_table\":[[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10]],\"identity\":2,\"prime\":2,\"element\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(14, 2, 9, [[1, 9], [2, 2]], 是, 是, g_to_p_is_identity_and_no_smaller_positive_power_is)","explanation_zh":"Wave6第六層核對=(14, 2, 9, [[1, 9], [2, 2]], 是, 是, g_to_p_is_identity_and_no_smaller_positive_power_is)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-order-p-witness","seed":202611121065,"parameters":{"group_table":[[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10]],"identity":2,"prime":2,"element":9,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-order-p-witness","semantic_template":"qf-mod43-cauchy-order-p-witness/{\"element\":9,\"group_table\":[[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10]],\"identity\":2,\"prime\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0066","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-order-p-witness；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,\"prime\":2,\"element\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 2, 0, [[1, 0], [2, 3]], 是, 是, g_to_p_is_identity_and_no_smaller_positive_power_is)","explanation_zh":"Wave6第六層核對=(6, 2, 0, [[1, 0], [2, 3]], 是, 是, g_to_p_is_identity_and_no_smaller_positive_power_is)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-order-p-witness","seed":202611121066,"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,"prime":2,"element":0,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-order-p-witness","semantic_template":"qf-mod43-cauchy-order-p-witness/{\"element\":0,\"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,\"prime\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0067","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-order-p-witness；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,\"prime\":2,\"element\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, 2, 7, [[1, 7], [2, 3]], 是, 是, g_to_p_is_identity_and_no_smaller_positive_power_is)","explanation_zh":"Wave6第六層核對=(8, 2, 7, [[1, 7], [2, 3]], 是, 是, g_to_p_is_identity_and_no_smaller_positive_power_is)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-order-p-witness","seed":202611121067,"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,"prime":2,"element":7,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-order-p-witness","semantic_template":"qf-mod43-cauchy-order-p-witness/{\"element\":7,\"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,\"prime\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0068","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-order-p-witness；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,\"prime\":2,\"element\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, 2, 8, [[1, 8], [2, 3]], 是, 是, g_to_p_is_identity_and_no_smaller_positive_power_is)","explanation_zh":"Wave6第六層核對=(10, 2, 8, [[1, 8], [2, 3]], 是, 是, g_to_p_is_identity_and_no_smaller_positive_power_is)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-order-p-witness","seed":202611121068,"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,"prime":2,"element":8,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-order-p-witness","semantic_template":"qf-mod43-cauchy-order-p-witness/{\"element\":8,\"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,\"prime\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0069","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-order-p-witness；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,\"prime\":2,\"element\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, 2, 9, [[1, 9], [2, 3]], 是, 是, g_to_p_is_identity_and_no_smaller_positive_power_is)","explanation_zh":"Wave6第六層核對=(12, 2, 9, [[1, 9], [2, 3]], 是, 是, g_to_p_is_identity_and_no_smaller_positive_power_is)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-order-p-witness","seed":202611121069,"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,"prime":2,"element":9,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-order-p-witness","semantic_template":"qf-mod43-cauchy-order-p-witness/{\"element\":9,\"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,\"prime\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0070","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-order-p-witness；資料={\"group_table\":[[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9]],\"identity\":3,\"prime\":2,\"element\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(14, 2, 10, [[1, 10], [2, 3]], 是, 是, g_to_p_is_identity_and_no_smaller_positive_power_is)","explanation_zh":"Wave6第六層核對=(14, 2, 10, [[1, 10], [2, 3]], 是, 是, g_to_p_is_identity_and_no_smaller_positive_power_is)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-order-p-witness","seed":202611121070,"parameters":{"group_table":[[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9]],"identity":3,"prime":2,"element":10,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-order-p-witness","semantic_template":"qf-mod43-cauchy-order-p-witness/{\"element\":10,\"group_table\":[[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9]],\"identity\":3,\"prime\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0071","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-order-p-witness；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,\"prime\":2,\"element\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 2, 1, [[1, 1], [2, 4]], 是, 是, g_to_p_is_identity_and_no_smaller_positive_power_is)","explanation_zh":"Wave6第六層核對=(6, 2, 1, [[1, 1], [2, 4]], 是, 是, g_to_p_is_identity_and_no_smaller_positive_power_is)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-order-p-witness","seed":202611121071,"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,"prime":2,"element":1,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-order-p-witness","semantic_template":"qf-mod43-cauchy-order-p-witness/{\"element\":1,\"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,\"prime\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0072","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-order-p-witness；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,\"prime\":2,\"element\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, 2, 0, [[1, 0], [2, 4]], 是, 是, g_to_p_is_identity_and_no_smaller_positive_power_is)","explanation_zh":"Wave6第六層核對=(8, 2, 0, [[1, 0], [2, 4]], 是, 是, g_to_p_is_identity_and_no_smaller_positive_power_is)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-order-p-witness","seed":202611121072,"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,"prime":2,"element":0,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-order-p-witness","semantic_template":"qf-mod43-cauchy-order-p-witness/{\"element\":0,\"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,\"prime\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0073","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-order-p-witness；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,\"prime\":2,\"element\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, 2, 9, [[1, 9], [2, 4]], 是, 是, g_to_p_is_identity_and_no_smaller_positive_power_is)","explanation_zh":"Wave6第六層核對=(10, 2, 9, [[1, 9], [2, 4]], 是, 是, g_to_p_is_identity_and_no_smaller_positive_power_is)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-order-p-witness","seed":202611121073,"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,"prime":2,"element":9,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-order-p-witness","semantic_template":"qf-mod43-cauchy-order-p-witness/{\"element\":9,\"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,\"prime\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0074","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-order-p-witness；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,\"prime\":2,\"element\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, 2, 10, [[1, 10], [2, 4]], 是, 是, g_to_p_is_identity_and_no_smaller_positive_power_is)","explanation_zh":"Wave6第六層核對=(12, 2, 10, [[1, 10], [2, 4]], 是, 是, g_to_p_is_identity_and_no_smaller_positive_power_is)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-order-p-witness","seed":202611121074,"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,"prime":2,"element":10,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-order-p-witness","semantic_template":"qf-mod43-cauchy-order-p-witness/{\"element\":10,\"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,\"prime\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0075","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-order-p-witness；資料={\"group_table\":[[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8]],\"identity\":4,\"prime\":2,\"element\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(14, 2, 11, [[1, 11], [2, 4]], 是, 是, g_to_p_is_identity_and_no_smaller_positive_power_is)","explanation_zh":"Wave6第六層核對=(14, 2, 11, [[1, 11], [2, 4]], 是, 是, g_to_p_is_identity_and_no_smaller_positive_power_is)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-order-p-witness","seed":202611121075,"parameters":{"group_table":[[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8]],"identity":4,"prime":2,"element":11,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-order-p-witness","semantic_template":"qf-mod43-cauchy-order-p-witness/{\"element\":11,\"group_table\":[[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8]],\"identity\":4,\"prime\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0076","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-composite-divisor-boundary；資料={\"group_table\":[[0,1,2,3],[1,0,3,2],[2,3,0,1],[3,2,1,0]],\"identity\":0,\"divisor\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 4, [[0, 1], [1, 2], [2, 2], [3, 2]], 是, 否, composite_divisor_need_not_be_an_element_order)","explanation_zh":"Wave6第六層核對=(4, 4, [[0, 1], [1, 2], [2, 2], [3, 2]], 是, 否, composite_divisor_need_not_be_an_element_order)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-composite-divisor-boundary","seed":202611121076,"parameters":{"group_table":[[0,1,2,3],[1,0,3,2],[2,3,0,1],[3,2,1,0]],"identity":0,"divisor":4,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-composite-divisor-boundary","semantic_template":"qf-mod43-cauchy-composite-divisor-boundary/{\"divisor\":4,\"group_table\":[[0,1,2,3],[1,0,3,2],[2,3,0,1],[3,2,1,0]],\"identity\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0077","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-composite-divisor-boundary；資料={\"group_table\":[[1,0,3,2],[0,1,2,3],[3,2,1,0],[2,3,0,1]],\"identity\":1,\"divisor\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 4, [[0, 2], [1, 1], [2, 2], [3, 2]], 是, 否, composite_divisor_need_not_be_an_element_order)","explanation_zh":"Wave6第六層核對=(4, 4, [[0, 2], [1, 1], [2, 2], [3, 2]], 是, 否, composite_divisor_need_not_be_an_element_order)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-composite-divisor-boundary","seed":202611121077,"parameters":{"group_table":[[1,0,3,2],[0,1,2,3],[3,2,1,0],[2,3,0,1]],"identity":1,"divisor":4,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-composite-divisor-boundary","semantic_template":"qf-mod43-cauchy-composite-divisor-boundary/{\"divisor\":4,\"group_table\":[[1,0,3,2],[0,1,2,3],[3,2,1,0],[2,3,0,1]],\"identity\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0078","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-composite-divisor-boundary；資料={\"group_table\":[[2,3,0,1],[3,2,1,0],[0,1,2,3],[1,0,3,2]],\"identity\":2,\"divisor\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 4, [[0, 2], [1, 2], [2, 1], [3, 2]], 是, 否, composite_divisor_need_not_be_an_element_order)","explanation_zh":"Wave6第六層核對=(4, 4, [[0, 2], [1, 2], [2, 1], [3, 2]], 是, 否, composite_divisor_need_not_be_an_element_order)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-composite-divisor-boundary","seed":202611121078,"parameters":{"group_table":[[2,3,0,1],[3,2,1,0],[0,1,2,3],[1,0,3,2]],"identity":2,"divisor":4,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-composite-divisor-boundary","semantic_template":"qf-mod43-cauchy-composite-divisor-boundary/{\"divisor\":4,\"group_table\":[[2,3,0,1],[3,2,1,0],[0,1,2,3],[1,0,3,2]],\"identity\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0079","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-composite-divisor-boundary；資料={\"group_table\":[[3,2,1,0],[2,3,0,1],[1,0,3,2],[0,1,2,3]],\"identity\":3,\"divisor\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 4, [[0, 2], [1, 2], [2, 2], [3, 1]], 是, 否, composite_divisor_need_not_be_an_element_order)","explanation_zh":"Wave6第六層核對=(4, 4, [[0, 2], [1, 2], [2, 2], [3, 1]], 是, 否, composite_divisor_need_not_be_an_element_order)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-composite-divisor-boundary","seed":202611121079,"parameters":{"group_table":[[3,2,1,0],[2,3,0,1],[1,0,3,2],[0,1,2,3]],"identity":3,"divisor":4,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-composite-divisor-boundary","semantic_template":"qf-mod43-cauchy-composite-divisor-boundary/{\"divisor\":4,\"group_table\":[[3,2,1,0],[2,3,0,1],[1,0,3,2],[0,1,2,3]],\"identity\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0080","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-composite-divisor-boundary；資料={\"group_table\":[[0,1,2,3,4,5,6,7],[1,0,3,2,5,4,7,6],[2,3,0,1,6,7,4,5],[3,2,1,0,7,6,5,4],[4,5,6,7,0,1,2,3],[5,4,7,6,1,0,3,2],[6,7,4,5,2,3,0,1],[7,6,5,4,3,2,1,0]],\"identity\":0,\"divisor\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, 8, [[0, 1], [1, 2], [2, 2], [3, 2], [4, 2], [5, 2], [6, 2], [7, 2]], 是, 否, composite_divisor_need_not_be_an_element_order)","explanation_zh":"Wave6第六層核對=(8, 8, [[0, 1], [1, 2], [2, 2], [3, 2], [4, 2], [5, 2], [6, 2], [7, 2]], 是, 否, composite_divisor_need_not_be_an_element_order)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-composite-divisor-boundary","seed":202611121080,"parameters":{"group_table":[[0,1,2,3,4,5,6,7],[1,0,3,2,5,4,7,6],[2,3,0,1,6,7,4,5],[3,2,1,0,7,6,5,4],[4,5,6,7,0,1,2,3],[5,4,7,6,1,0,3,2],[6,7,4,5,2,3,0,1],[7,6,5,4,3,2,1,0]],"identity":0,"divisor":8,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-composite-divisor-boundary","semantic_template":"qf-mod43-cauchy-composite-divisor-boundary/{\"divisor\":8,\"group_table\":[[0,1,2,3,4,5,6,7],[1,0,3,2,5,4,7,6],[2,3,0,1,6,7,4,5],[3,2,1,0,7,6,5,4],[4,5,6,7,0,1,2,3],[5,4,7,6,1,0,3,2],[6,7,4,5,2,3,0,1],[7,6,5,4,3,2,1,0]],\"identity\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0081","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-composite-divisor-boundary；資料={\"group_table\":[[1,0,3,2,5,4,7,6],[0,1,2,3,4,5,6,7],[3,2,1,0,7,6,5,4],[2,3,0,1,6,7,4,5],[5,4,7,6,1,0,3,2],[4,5,6,7,0,1,2,3],[7,6,5,4,3,2,1,0],[6,7,4,5,2,3,0,1]],\"identity\":1,\"divisor\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, 8, [[0, 2], [1, 1], [2, 2], [3, 2], [4, 2], [5, 2], [6, 2], [7, 2]], 是, 否, composite_divisor_need_not_be_an_element_order)","explanation_zh":"Wave6第六層核對=(8, 8, [[0, 2], [1, 1], [2, 2], [3, 2], [4, 2], [5, 2], [6, 2], [7, 2]], 是, 否, composite_divisor_need_not_be_an_element_order)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-composite-divisor-boundary","seed":202611121081,"parameters":{"group_table":[[1,0,3,2,5,4,7,6],[0,1,2,3,4,5,6,7],[3,2,1,0,7,6,5,4],[2,3,0,1,6,7,4,5],[5,4,7,6,1,0,3,2],[4,5,6,7,0,1,2,3],[7,6,5,4,3,2,1,0],[6,7,4,5,2,3,0,1]],"identity":1,"divisor":8,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-composite-divisor-boundary","semantic_template":"qf-mod43-cauchy-composite-divisor-boundary/{\"divisor\":8,\"group_table\":[[1,0,3,2,5,4,7,6],[0,1,2,3,4,5,6,7],[3,2,1,0,7,6,5,4],[2,3,0,1,6,7,4,5],[5,4,7,6,1,0,3,2],[4,5,6,7,0,1,2,3],[7,6,5,4,3,2,1,0],[6,7,4,5,2,3,0,1]],\"identity\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0082","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-composite-divisor-boundary；資料={\"group_table\":[[2,3,0,1,6,7,4,5],[3,2,1,0,7,6,5,4],[0,1,2,3,4,5,6,7],[1,0,3,2,5,4,7,6],[6,7,4,5,2,3,0,1],[7,6,5,4,3,2,1,0],[4,5,6,7,0,1,2,3],[5,4,7,6,1,0,3,2]],\"identity\":2,\"divisor\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, 8, [[0, 2], [1, 2], [2, 1], [3, 2], [4, 2], [5, 2], [6, 2], [7, 2]], 是, 否, composite_divisor_need_not_be_an_element_order)","explanation_zh":"Wave6第六層核對=(8, 8, [[0, 2], [1, 2], [2, 1], [3, 2], [4, 2], [5, 2], [6, 2], [7, 2]], 是, 否, composite_divisor_need_not_be_an_element_order)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-composite-divisor-boundary","seed":202611121082,"parameters":{"group_table":[[2,3,0,1,6,7,4,5],[3,2,1,0,7,6,5,4],[0,1,2,3,4,5,6,7],[1,0,3,2,5,4,7,6],[6,7,4,5,2,3,0,1],[7,6,5,4,3,2,1,0],[4,5,6,7,0,1,2,3],[5,4,7,6,1,0,3,2]],"identity":2,"divisor":8,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-composite-divisor-boundary","semantic_template":"qf-mod43-cauchy-composite-divisor-boundary/{\"divisor\":8,\"group_table\":[[2,3,0,1,6,7,4,5],[3,2,1,0,7,6,5,4],[0,1,2,3,4,5,6,7],[1,0,3,2,5,4,7,6],[6,7,4,5,2,3,0,1],[7,6,5,4,3,2,1,0],[4,5,6,7,0,1,2,3],[5,4,7,6,1,0,3,2]],\"identity\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0083","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-composite-divisor-boundary；資料={\"group_table\":[[3,2,1,0,7,6,5,4],[2,3,0,1,6,7,4,5],[1,0,3,2,5,4,7,6],[0,1,2,3,4,5,6,7],[7,6,5,4,3,2,1,0],[6,7,4,5,2,3,0,1],[5,4,7,6,1,0,3,2],[4,5,6,7,0,1,2,3]],\"identity\":3,\"divisor\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, 8, [[0, 2], [1, 2], [2, 2], [3, 1], [4, 2], [5, 2], [6, 2], [7, 2]], 是, 否, composite_divisor_need_not_be_an_element_order)","explanation_zh":"Wave6第六層核對=(8, 8, [[0, 2], [1, 2], [2, 2], [3, 1], [4, 2], [5, 2], [6, 2], [7, 2]], 是, 否, composite_divisor_need_not_be_an_element_order)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-composite-divisor-boundary","seed":202611121083,"parameters":{"group_table":[[3,2,1,0,7,6,5,4],[2,3,0,1,6,7,4,5],[1,0,3,2,5,4,7,6],[0,1,2,3,4,5,6,7],[7,6,5,4,3,2,1,0],[6,7,4,5,2,3,0,1],[5,4,7,6,1,0,3,2],[4,5,6,7,0,1,2,3]],"identity":3,"divisor":8,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-composite-divisor-boundary","semantic_template":"qf-mod43-cauchy-composite-divisor-boundary/{\"divisor\":8,\"group_table\":[[3,2,1,0,7,6,5,4],[2,3,0,1,6,7,4,5],[1,0,3,2,5,4,7,6],[0,1,2,3,4,5,6,7],[7,6,5,4,3,2,1,0],[6,7,4,5,2,3,0,1],[5,4,7,6,1,0,3,2],[4,5,6,7,0,1,2,3]],\"identity\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0084","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-composite-divisor-boundary；資料={\"group_table\":[[4,5,6,7,0,1,2,3],[5,4,7,6,1,0,3,2],[6,7,4,5,2,3,0,1],[7,6,5,4,3,2,1,0],[0,1,2,3,4,5,6,7],[1,0,3,2,5,4,7,6],[2,3,0,1,6,7,4,5],[3,2,1,0,7,6,5,4]],\"identity\":4,\"divisor\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, 8, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 1], [5, 2], [6, 2], [7, 2]], 是, 否, composite_divisor_need_not_be_an_element_order)","explanation_zh":"Wave6第六層核對=(8, 8, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 1], [5, 2], [6, 2], [7, 2]], 是, 否, composite_divisor_need_not_be_an_element_order)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-composite-divisor-boundary","seed":202611121084,"parameters":{"group_table":[[4,5,6,7,0,1,2,3],[5,4,7,6,1,0,3,2],[6,7,4,5,2,3,0,1],[7,6,5,4,3,2,1,0],[0,1,2,3,4,5,6,7],[1,0,3,2,5,4,7,6],[2,3,0,1,6,7,4,5],[3,2,1,0,7,6,5,4]],"identity":4,"divisor":8,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-composite-divisor-boundary","semantic_template":"qf-mod43-cauchy-composite-divisor-boundary/{\"divisor\":8,\"group_table\":[[4,5,6,7,0,1,2,3],[5,4,7,6,1,0,3,2],[6,7,4,5,2,3,0,1],[7,6,5,4,3,2,1,0],[0,1,2,3,4,5,6,7],[1,0,3,2,5,4,7,6],[2,3,0,1,6,7,4,5],[3,2,1,0,7,6,5,4]],\"identity\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0085","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-composite-divisor-boundary；資料={\"group_table\":[[5,4,7,6,1,0,3,2],[4,5,6,7,0,1,2,3],[7,6,5,4,3,2,1,0],[6,7,4,5,2,3,0,1],[1,0,3,2,5,4,7,6],[0,1,2,3,4,5,6,7],[3,2,1,0,7,6,5,4],[2,3,0,1,6,7,4,5]],\"identity\":5,\"divisor\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, 8, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 2], [5, 1], [6, 2], [7, 2]], 是, 否, composite_divisor_need_not_be_an_element_order)","explanation_zh":"Wave6第六層核對=(8, 8, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 2], [5, 1], [6, 2], [7, 2]], 是, 否, composite_divisor_need_not_be_an_element_order)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-composite-divisor-boundary","seed":202611121085,"parameters":{"group_table":[[5,4,7,6,1,0,3,2],[4,5,6,7,0,1,2,3],[7,6,5,4,3,2,1,0],[6,7,4,5,2,3,0,1],[1,0,3,2,5,4,7,6],[0,1,2,3,4,5,6,7],[3,2,1,0,7,6,5,4],[2,3,0,1,6,7,4,5]],"identity":5,"divisor":8,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-composite-divisor-boundary","semantic_template":"qf-mod43-cauchy-composite-divisor-boundary/{\"divisor\":8,\"group_table\":[[5,4,7,6,1,0,3,2],[4,5,6,7,0,1,2,3],[7,6,5,4,3,2,1,0],[6,7,4,5,2,3,0,1],[1,0,3,2,5,4,7,6],[0,1,2,3,4,5,6,7],[3,2,1,0,7,6,5,4],[2,3,0,1,6,7,4,5]],\"identity\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0086","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-composite-divisor-boundary；資料={\"group_table\":[[6,7,4,5,2,3,0,1],[7,6,5,4,3,2,1,0],[4,5,6,7,0,1,2,3],[5,4,7,6,1,0,3,2],[2,3,0,1,6,7,4,5],[3,2,1,0,7,6,5,4],[0,1,2,3,4,5,6,7],[1,0,3,2,5,4,7,6]],\"identity\":6,\"divisor\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, 8, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 2], [5, 2], [6, 1], [7, 2]], 是, 否, composite_divisor_need_not_be_an_element_order)","explanation_zh":"Wave6第六層核對=(8, 8, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 2], [5, 2], [6, 1], [7, 2]], 是, 否, composite_divisor_need_not_be_an_element_order)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-composite-divisor-boundary","seed":202611121086,"parameters":{"group_table":[[6,7,4,5,2,3,0,1],[7,6,5,4,3,2,1,0],[4,5,6,7,0,1,2,3],[5,4,7,6,1,0,3,2],[2,3,0,1,6,7,4,5],[3,2,1,0,7,6,5,4],[0,1,2,3,4,5,6,7],[1,0,3,2,5,4,7,6]],"identity":6,"divisor":8,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-composite-divisor-boundary","semantic_template":"qf-mod43-cauchy-composite-divisor-boundary/{\"divisor\":8,\"group_table\":[[6,7,4,5,2,3,0,1],[7,6,5,4,3,2,1,0],[4,5,6,7,0,1,2,3],[5,4,7,6,1,0,3,2],[2,3,0,1,6,7,4,5],[3,2,1,0,7,6,5,4],[0,1,2,3,4,5,6,7],[1,0,3,2,5,4,7,6]],\"identity\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0087","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-composite-divisor-boundary；資料={\"group_table\":[[7,6,5,4,3,2,1,0],[6,7,4,5,2,3,0,1],[5,4,7,6,1,0,3,2],[4,5,6,7,0,1,2,3],[3,2,1,0,7,6,5,4],[2,3,0,1,6,7,4,5],[1,0,3,2,5,4,7,6],[0,1,2,3,4,5,6,7]],\"identity\":7,\"divisor\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, 8, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 2], [5, 2], [6, 2], [7, 1]], 是, 否, composite_divisor_need_not_be_an_element_order)","explanation_zh":"Wave6第六層核對=(8, 8, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 2], [5, 2], [6, 2], [7, 1]], 是, 否, composite_divisor_need_not_be_an_element_order)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-composite-divisor-boundary","seed":202611121087,"parameters":{"group_table":[[7,6,5,4,3,2,1,0],[6,7,4,5,2,3,0,1],[5,4,7,6,1,0,3,2],[4,5,6,7,0,1,2,3],[3,2,1,0,7,6,5,4],[2,3,0,1,6,7,4,5],[1,0,3,2,5,4,7,6],[0,1,2,3,4,5,6,7]],"identity":7,"divisor":8,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-composite-divisor-boundary","semantic_template":"qf-mod43-cauchy-composite-divisor-boundary/{\"divisor\":8,\"group_table\":[[7,6,5,4,3,2,1,0],[6,7,4,5,2,3,0,1],[5,4,7,6,1,0,3,2],[4,5,6,7,0,1,2,3],[3,2,1,0,7,6,5,4],[2,3,0,1,6,7,4,5],[1,0,3,2,5,4,7,6],[0,1,2,3,4,5,6,7]],\"identity\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0088","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-composite-divisor-boundary；資料={\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,0,3,2,5,4,7,6,9,8,11,10,13,12,15,14],[2,3,0,1,6,7,4,5,10,11,8,9,14,15,12,13],[3,2,1,0,7,6,5,4,11,10,9,8,15,14,13,12],[4,5,6,7,0,1,2,3,12,13,14,15,8,9,10,11],[5,4,7,6,1,0,3,2,13,12,15,14,9,8,11,10],[6,7,4,5,2,3,0,1,14,15,12,13,10,11,8,9],[7,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[9,8,11,10,13,12,15,14,1,0,3,2,5,4,7,6],[10,11,8,9,14,15,12,13,2,3,0,1,6,7,4,5],[11,10,9,8,15,14,13,12,3,2,1,0,7,6,5,4],[12,13,14,15,8,9,10,11,4,5,6,7,0,1,2,3],[13,12,15,14,9,8,11,10,5,4,7,6,1,0,3,2],[14,15,12,13,10,11,8,9,6,7,4,5,2,3,0,1],[15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0,\"divisor\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(16, 16, [[0, 1], [1, 2], [2, 2], [3, 2], [4, 2], [5, 2], [6, 2], [7, 2], [8, 2], [9, 2], [10, 2], [11, 2], [12, 2], [13, 2], [14, 2], [15, 2]], 是, 否, composite_divisor_need_not_be_an_element_order)","explanation_zh":"Wave6第六層核對=(16, 16, [[0, 1], [1, 2], [2, 2], [3, 2], [4, 2], [5, 2], [6, 2], [7, 2], [8, 2], [9, 2], [10, 2], [11, 2], [12, 2], [13, 2], [14, 2], [15, 2]], 是, 否, composite_divisor_need_not_be_an_element_order)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-composite-divisor-boundary","seed":202611121088,"parameters":{"group_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,0,3,2,5,4,7,6,9,8,11,10,13,12,15,14],[2,3,0,1,6,7,4,5,10,11,8,9,14,15,12,13],[3,2,1,0,7,6,5,4,11,10,9,8,15,14,13,12],[4,5,6,7,0,1,2,3,12,13,14,15,8,9,10,11],[5,4,7,6,1,0,3,2,13,12,15,14,9,8,11,10],[6,7,4,5,2,3,0,1,14,15,12,13,10,11,8,9],[7,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[9,8,11,10,13,12,15,14,1,0,3,2,5,4,7,6],[10,11,8,9,14,15,12,13,2,3,0,1,6,7,4,5],[11,10,9,8,15,14,13,12,3,2,1,0,7,6,5,4],[12,13,14,15,8,9,10,11,4,5,6,7,0,1,2,3],[13,12,15,14,9,8,11,10,5,4,7,6,1,0,3,2],[14,15,12,13,10,11,8,9,6,7,4,5,2,3,0,1],[15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"divisor":16,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-composite-divisor-boundary","semantic_template":"qf-mod43-cauchy-composite-divisor-boundary/{\"divisor\":16,\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,0,3,2,5,4,7,6,9,8,11,10,13,12,15,14],[2,3,0,1,6,7,4,5,10,11,8,9,14,15,12,13],[3,2,1,0,7,6,5,4,11,10,9,8,15,14,13,12],[4,5,6,7,0,1,2,3,12,13,14,15,8,9,10,11],[5,4,7,6,1,0,3,2,13,12,15,14,9,8,11,10],[6,7,4,5,2,3,0,1,14,15,12,13,10,11,8,9],[7,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[9,8,11,10,13,12,15,14,1,0,3,2,5,4,7,6],[10,11,8,9,14,15,12,13,2,3,0,1,6,7,4,5],[11,10,9,8,15,14,13,12,3,2,1,0,7,6,5,4],[12,13,14,15,8,9,10,11,4,5,6,7,0,1,2,3],[13,12,15,14,9,8,11,10,5,4,7,6,1,0,3,2],[14,15,12,13,10,11,8,9,6,7,4,5,2,3,0,1],[15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0089","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-composite-divisor-boundary；資料={\"group_table\":[[1,0,3,2,5,4,7,6,9,8,11,10,13,12,15,14],[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[3,2,1,0,7,6,5,4,11,10,9,8,15,14,13,12],[2,3,0,1,6,7,4,5,10,11,8,9,14,15,12,13],[5,4,7,6,1,0,3,2,13,12,15,14,9,8,11,10],[4,5,6,7,0,1,2,3,12,13,14,15,8,9,10,11],[7,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8],[6,7,4,5,2,3,0,1,14,15,12,13,10,11,8,9],[9,8,11,10,13,12,15,14,1,0,3,2,5,4,7,6],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[11,10,9,8,15,14,13,12,3,2,1,0,7,6,5,4],[10,11,8,9,14,15,12,13,2,3,0,1,6,7,4,5],[13,12,15,14,9,8,11,10,5,4,7,6,1,0,3,2],[12,13,14,15,8,9,10,11,4,5,6,7,0,1,2,3],[15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0],[14,15,12,13,10,11,8,9,6,7,4,5,2,3,0,1]],\"identity\":1,\"divisor\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(16, 16, [[0, 2], [1, 1], [2, 2], [3, 2], [4, 2], [5, 2], [6, 2], [7, 2], [8, 2], [9, 2], [10, 2], [11, 2], [12, 2], [13, 2], [14, 2], [15, 2]], 是, 否, composite_divisor_need_not_be_an_element_order)","explanation_zh":"Wave6第六層核對=(16, 16, [[0, 2], [1, 1], [2, 2], [3, 2], [4, 2], [5, 2], [6, 2], [7, 2], [8, 2], [9, 2], [10, 2], [11, 2], [12, 2], [13, 2], [14, 2], [15, 2]], 是, 否, composite_divisor_need_not_be_an_element_order)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-composite-divisor-boundary","seed":202611121089,"parameters":{"group_table":[[1,0,3,2,5,4,7,6,9,8,11,10,13,12,15,14],[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[3,2,1,0,7,6,5,4,11,10,9,8,15,14,13,12],[2,3,0,1,6,7,4,5,10,11,8,9,14,15,12,13],[5,4,7,6,1,0,3,2,13,12,15,14,9,8,11,10],[4,5,6,7,0,1,2,3,12,13,14,15,8,9,10,11],[7,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8],[6,7,4,5,2,3,0,1,14,15,12,13,10,11,8,9],[9,8,11,10,13,12,15,14,1,0,3,2,5,4,7,6],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[11,10,9,8,15,14,13,12,3,2,1,0,7,6,5,4],[10,11,8,9,14,15,12,13,2,3,0,1,6,7,4,5],[13,12,15,14,9,8,11,10,5,4,7,6,1,0,3,2],[12,13,14,15,8,9,10,11,4,5,6,7,0,1,2,3],[15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0],[14,15,12,13,10,11,8,9,6,7,4,5,2,3,0,1]],"identity":1,"divisor":16,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-composite-divisor-boundary","semantic_template":"qf-mod43-cauchy-composite-divisor-boundary/{\"divisor\":16,\"group_table\":[[1,0,3,2,5,4,7,6,9,8,11,10,13,12,15,14],[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[3,2,1,0,7,6,5,4,11,10,9,8,15,14,13,12],[2,3,0,1,6,7,4,5,10,11,8,9,14,15,12,13],[5,4,7,6,1,0,3,2,13,12,15,14,9,8,11,10],[4,5,6,7,0,1,2,3,12,13,14,15,8,9,10,11],[7,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8],[6,7,4,5,2,3,0,1,14,15,12,13,10,11,8,9],[9,8,11,10,13,12,15,14,1,0,3,2,5,4,7,6],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[11,10,9,8,15,14,13,12,3,2,1,0,7,6,5,4],[10,11,8,9,14,15,12,13,2,3,0,1,6,7,4,5],[13,12,15,14,9,8,11,10,5,4,7,6,1,0,3,2],[12,13,14,15,8,9,10,11,4,5,6,7,0,1,2,3],[15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0],[14,15,12,13,10,11,8,9,6,7,4,5,2,3,0,1]],\"identity\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0090","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-composite-divisor-boundary；資料={\"group_table\":[[2,3,0,1,6,7,4,5,10,11,8,9,14,15,12,13],[3,2,1,0,7,6,5,4,11,10,9,8,15,14,13,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,0,3,2,5,4,7,6,9,8,11,10,13,12,15,14],[6,7,4,5,2,3,0,1,14,15,12,13,10,11,8,9],[7,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8],[4,5,6,7,0,1,2,3,12,13,14,15,8,9,10,11],[5,4,7,6,1,0,3,2,13,12,15,14,9,8,11,10],[10,11,8,9,14,15,12,13,2,3,0,1,6,7,4,5],[11,10,9,8,15,14,13,12,3,2,1,0,7,6,5,4],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[9,8,11,10,13,12,15,14,1,0,3,2,5,4,7,6],[14,15,12,13,10,11,8,9,6,7,4,5,2,3,0,1],[15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0],[12,13,14,15,8,9,10,11,4,5,6,7,0,1,2,3],[13,12,15,14,9,8,11,10,5,4,7,6,1,0,3,2]],\"identity\":2,\"divisor\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(16, 16, [[0, 2], [1, 2], [2, 1], [3, 2], [4, 2], [5, 2], [6, 2], [7, 2], [8, 2], [9, 2], [10, 2], [11, 2], [12, 2], [13, 2], [14, 2], [15, 2]], 是, 否, composite_divisor_need_not_be_an_element_order)","explanation_zh":"Wave6第六層核對=(16, 16, [[0, 2], [1, 2], [2, 1], [3, 2], [4, 2], [5, 2], [6, 2], [7, 2], [8, 2], [9, 2], [10, 2], [11, 2], [12, 2], [13, 2], [14, 2], [15, 2]], 是, 否, composite_divisor_need_not_be_an_element_order)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-composite-divisor-boundary","seed":202611121090,"parameters":{"group_table":[[2,3,0,1,6,7,4,5,10,11,8,9,14,15,12,13],[3,2,1,0,7,6,5,4,11,10,9,8,15,14,13,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,0,3,2,5,4,7,6,9,8,11,10,13,12,15,14],[6,7,4,5,2,3,0,1,14,15,12,13,10,11,8,9],[7,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8],[4,5,6,7,0,1,2,3,12,13,14,15,8,9,10,11],[5,4,7,6,1,0,3,2,13,12,15,14,9,8,11,10],[10,11,8,9,14,15,12,13,2,3,0,1,6,7,4,5],[11,10,9,8,15,14,13,12,3,2,1,0,7,6,5,4],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[9,8,11,10,13,12,15,14,1,0,3,2,5,4,7,6],[14,15,12,13,10,11,8,9,6,7,4,5,2,3,0,1],[15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0],[12,13,14,15,8,9,10,11,4,5,6,7,0,1,2,3],[13,12,15,14,9,8,11,10,5,4,7,6,1,0,3,2]],"identity":2,"divisor":16,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-composite-divisor-boundary","semantic_template":"qf-mod43-cauchy-composite-divisor-boundary/{\"divisor\":16,\"group_table\":[[2,3,0,1,6,7,4,5,10,11,8,9,14,15,12,13],[3,2,1,0,7,6,5,4,11,10,9,8,15,14,13,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,0,3,2,5,4,7,6,9,8,11,10,13,12,15,14],[6,7,4,5,2,3,0,1,14,15,12,13,10,11,8,9],[7,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8],[4,5,6,7,0,1,2,3,12,13,14,15,8,9,10,11],[5,4,7,6,1,0,3,2,13,12,15,14,9,8,11,10],[10,11,8,9,14,15,12,13,2,3,0,1,6,7,4,5],[11,10,9,8,15,14,13,12,3,2,1,0,7,6,5,4],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[9,8,11,10,13,12,15,14,1,0,3,2,5,4,7,6],[14,15,12,13,10,11,8,9,6,7,4,5,2,3,0,1],[15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0],[12,13,14,15,8,9,10,11,4,5,6,7,0,1,2,3],[13,12,15,14,9,8,11,10,5,4,7,6,1,0,3,2]],\"identity\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0091","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-composite-divisor-boundary；資料={\"group_table\":[[3,2,1,0,7,6,5,4,11,10,9,8,15,14,13,12],[2,3,0,1,6,7,4,5,10,11,8,9,14,15,12,13],[1,0,3,2,5,4,7,6,9,8,11,10,13,12,15,14],[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[7,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8],[6,7,4,5,2,3,0,1,14,15,12,13,10,11,8,9],[5,4,7,6,1,0,3,2,13,12,15,14,9,8,11,10],[4,5,6,7,0,1,2,3,12,13,14,15,8,9,10,11],[11,10,9,8,15,14,13,12,3,2,1,0,7,6,5,4],[10,11,8,9,14,15,12,13,2,3,0,1,6,7,4,5],[9,8,11,10,13,12,15,14,1,0,3,2,5,4,7,6],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0],[14,15,12,13,10,11,8,9,6,7,4,5,2,3,0,1],[13,12,15,14,9,8,11,10,5,4,7,6,1,0,3,2],[12,13,14,15,8,9,10,11,4,5,6,7,0,1,2,3]],\"identity\":3,\"divisor\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(16, 16, [[0, 2], [1, 2], [2, 2], [3, 1], [4, 2], [5, 2], [6, 2], [7, 2], [8, 2], [9, 2], [10, 2], [11, 2], [12, 2], [13, 2], [14, 2], [15, 2]], 是, 否, composite_divisor_need_not_be_an_element_order)","explanation_zh":"Wave6第六層核對=(16, 16, [[0, 2], [1, 2], [2, 2], [3, 1], [4, 2], [5, 2], [6, 2], [7, 2], [8, 2], [9, 2], [10, 2], [11, 2], [12, 2], [13, 2], [14, 2], [15, 2]], 是, 否, composite_divisor_need_not_be_an_element_order)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-composite-divisor-boundary","seed":202611121091,"parameters":{"group_table":[[3,2,1,0,7,6,5,4,11,10,9,8,15,14,13,12],[2,3,0,1,6,7,4,5,10,11,8,9,14,15,12,13],[1,0,3,2,5,4,7,6,9,8,11,10,13,12,15,14],[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[7,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8],[6,7,4,5,2,3,0,1,14,15,12,13,10,11,8,9],[5,4,7,6,1,0,3,2,13,12,15,14,9,8,11,10],[4,5,6,7,0,1,2,3,12,13,14,15,8,9,10,11],[11,10,9,8,15,14,13,12,3,2,1,0,7,6,5,4],[10,11,8,9,14,15,12,13,2,3,0,1,6,7,4,5],[9,8,11,10,13,12,15,14,1,0,3,2,5,4,7,6],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0],[14,15,12,13,10,11,8,9,6,7,4,5,2,3,0,1],[13,12,15,14,9,8,11,10,5,4,7,6,1,0,3,2],[12,13,14,15,8,9,10,11,4,5,6,7,0,1,2,3]],"identity":3,"divisor":16,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-composite-divisor-boundary","semantic_template":"qf-mod43-cauchy-composite-divisor-boundary/{\"divisor\":16,\"group_table\":[[3,2,1,0,7,6,5,4,11,10,9,8,15,14,13,12],[2,3,0,1,6,7,4,5,10,11,8,9,14,15,12,13],[1,0,3,2,5,4,7,6,9,8,11,10,13,12,15,14],[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[7,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8],[6,7,4,5,2,3,0,1,14,15,12,13,10,11,8,9],[5,4,7,6,1,0,3,2,13,12,15,14,9,8,11,10],[4,5,6,7,0,1,2,3,12,13,14,15,8,9,10,11],[11,10,9,8,15,14,13,12,3,2,1,0,7,6,5,4],[10,11,8,9,14,15,12,13,2,3,0,1,6,7,4,5],[9,8,11,10,13,12,15,14,1,0,3,2,5,4,7,6],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0],[14,15,12,13,10,11,8,9,6,7,4,5,2,3,0,1],[13,12,15,14,9,8,11,10,5,4,7,6,1,0,3,2],[12,13,14,15,8,9,10,11,4,5,6,7,0,1,2,3]],\"identity\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0092","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-composite-divisor-boundary；資料={\"group_table\":[[4,5,6,7,0,1,2,3,12,13,14,15,8,9,10,11],[5,4,7,6,1,0,3,2,13,12,15,14,9,8,11,10],[6,7,4,5,2,3,0,1,14,15,12,13,10,11,8,9],[7,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8],[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,0,3,2,5,4,7,6,9,8,11,10,13,12,15,14],[2,3,0,1,6,7,4,5,10,11,8,9,14,15,12,13],[3,2,1,0,7,6,5,4,11,10,9,8,15,14,13,12],[12,13,14,15,8,9,10,11,4,5,6,7,0,1,2,3],[13,12,15,14,9,8,11,10,5,4,7,6,1,0,3,2],[14,15,12,13,10,11,8,9,6,7,4,5,2,3,0,1],[15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[9,8,11,10,13,12,15,14,1,0,3,2,5,4,7,6],[10,11,8,9,14,15,12,13,2,3,0,1,6,7,4,5],[11,10,9,8,15,14,13,12,3,2,1,0,7,6,5,4]],\"identity\":4,\"divisor\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(16, 16, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 1], [5, 2], [6, 2], [7, 2], [8, 2], [9, 2], [10, 2], [11, 2], [12, 2], [13, 2], [14, 2], [15, 2]], 是, 否, composite_divisor_need_not_be_an_element_order)","explanation_zh":"Wave6第六層核對=(16, 16, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 1], [5, 2], [6, 2], [7, 2], [8, 2], [9, 2], [10, 2], [11, 2], [12, 2], [13, 2], [14, 2], [15, 2]], 是, 否, composite_divisor_need_not_be_an_element_order)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-composite-divisor-boundary","seed":202611121092,"parameters":{"group_table":[[4,5,6,7,0,1,2,3,12,13,14,15,8,9,10,11],[5,4,7,6,1,0,3,2,13,12,15,14,9,8,11,10],[6,7,4,5,2,3,0,1,14,15,12,13,10,11,8,9],[7,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8],[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,0,3,2,5,4,7,6,9,8,11,10,13,12,15,14],[2,3,0,1,6,7,4,5,10,11,8,9,14,15,12,13],[3,2,1,0,7,6,5,4,11,10,9,8,15,14,13,12],[12,13,14,15,8,9,10,11,4,5,6,7,0,1,2,3],[13,12,15,14,9,8,11,10,5,4,7,6,1,0,3,2],[14,15,12,13,10,11,8,9,6,7,4,5,2,3,0,1],[15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[9,8,11,10,13,12,15,14,1,0,3,2,5,4,7,6],[10,11,8,9,14,15,12,13,2,3,0,1,6,7,4,5],[11,10,9,8,15,14,13,12,3,2,1,0,7,6,5,4]],"identity":4,"divisor":16,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-composite-divisor-boundary","semantic_template":"qf-mod43-cauchy-composite-divisor-boundary/{\"divisor\":16,\"group_table\":[[4,5,6,7,0,1,2,3,12,13,14,15,8,9,10,11],[5,4,7,6,1,0,3,2,13,12,15,14,9,8,11,10],[6,7,4,5,2,3,0,1,14,15,12,13,10,11,8,9],[7,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8],[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,0,3,2,5,4,7,6,9,8,11,10,13,12,15,14],[2,3,0,1,6,7,4,5,10,11,8,9,14,15,12,13],[3,2,1,0,7,6,5,4,11,10,9,8,15,14,13,12],[12,13,14,15,8,9,10,11,4,5,6,7,0,1,2,3],[13,12,15,14,9,8,11,10,5,4,7,6,1,0,3,2],[14,15,12,13,10,11,8,9,6,7,4,5,2,3,0,1],[15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[9,8,11,10,13,12,15,14,1,0,3,2,5,4,7,6],[10,11,8,9,14,15,12,13,2,3,0,1,6,7,4,5],[11,10,9,8,15,14,13,12,3,2,1,0,7,6,5,4]],\"identity\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0093","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-composite-divisor-boundary；資料={\"group_table\":[[5,4,7,6,1,0,3,2,13,12,15,14,9,8,11,10],[4,5,6,7,0,1,2,3,12,13,14,15,8,9,10,11],[7,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8],[6,7,4,5,2,3,0,1,14,15,12,13,10,11,8,9],[1,0,3,2,5,4,7,6,9,8,11,10,13,12,15,14],[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[3,2,1,0,7,6,5,4,11,10,9,8,15,14,13,12],[2,3,0,1,6,7,4,5,10,11,8,9,14,15,12,13],[13,12,15,14,9,8,11,10,5,4,7,6,1,0,3,2],[12,13,14,15,8,9,10,11,4,5,6,7,0,1,2,3],[15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0],[14,15,12,13,10,11,8,9,6,7,4,5,2,3,0,1],[9,8,11,10,13,12,15,14,1,0,3,2,5,4,7,6],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[11,10,9,8,15,14,13,12,3,2,1,0,7,6,5,4],[10,11,8,9,14,15,12,13,2,3,0,1,6,7,4,5]],\"identity\":5,\"divisor\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(16, 16, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 2], [5, 1], [6, 2], [7, 2], [8, 2], [9, 2], [10, 2], [11, 2], [12, 2], [13, 2], [14, 2], [15, 2]], 是, 否, composite_divisor_need_not_be_an_element_order)","explanation_zh":"Wave6第六層核對=(16, 16, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 2], [5, 1], [6, 2], [7, 2], [8, 2], [9, 2], [10, 2], [11, 2], [12, 2], [13, 2], [14, 2], [15, 2]], 是, 否, composite_divisor_need_not_be_an_element_order)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-composite-divisor-boundary","seed":202611121093,"parameters":{"group_table":[[5,4,7,6,1,0,3,2,13,12,15,14,9,8,11,10],[4,5,6,7,0,1,2,3,12,13,14,15,8,9,10,11],[7,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8],[6,7,4,5,2,3,0,1,14,15,12,13,10,11,8,9],[1,0,3,2,5,4,7,6,9,8,11,10,13,12,15,14],[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[3,2,1,0,7,6,5,4,11,10,9,8,15,14,13,12],[2,3,0,1,6,7,4,5,10,11,8,9,14,15,12,13],[13,12,15,14,9,8,11,10,5,4,7,6,1,0,3,2],[12,13,14,15,8,9,10,11,4,5,6,7,0,1,2,3],[15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0],[14,15,12,13,10,11,8,9,6,7,4,5,2,3,0,1],[9,8,11,10,13,12,15,14,1,0,3,2,5,4,7,6],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[11,10,9,8,15,14,13,12,3,2,1,0,7,6,5,4],[10,11,8,9,14,15,12,13,2,3,0,1,6,7,4,5]],"identity":5,"divisor":16,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-composite-divisor-boundary","semantic_template":"qf-mod43-cauchy-composite-divisor-boundary/{\"divisor\":16,\"group_table\":[[5,4,7,6,1,0,3,2,13,12,15,14,9,8,11,10],[4,5,6,7,0,1,2,3,12,13,14,15,8,9,10,11],[7,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8],[6,7,4,5,2,3,0,1,14,15,12,13,10,11,8,9],[1,0,3,2,5,4,7,6,9,8,11,10,13,12,15,14],[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[3,2,1,0,7,6,5,4,11,10,9,8,15,14,13,12],[2,3,0,1,6,7,4,5,10,11,8,9,14,15,12,13],[13,12,15,14,9,8,11,10,5,4,7,6,1,0,3,2],[12,13,14,15,8,9,10,11,4,5,6,7,0,1,2,3],[15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0],[14,15,12,13,10,11,8,9,6,7,4,5,2,3,0,1],[9,8,11,10,13,12,15,14,1,0,3,2,5,4,7,6],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[11,10,9,8,15,14,13,12,3,2,1,0,7,6,5,4],[10,11,8,9,14,15,12,13,2,3,0,1,6,7,4,5]],\"identity\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0094","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-composite-divisor-boundary；資料={\"group_table\":[[6,7,4,5,2,3,0,1,14,15,12,13,10,11,8,9],[7,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8],[4,5,6,7,0,1,2,3,12,13,14,15,8,9,10,11],[5,4,7,6,1,0,3,2,13,12,15,14,9,8,11,10],[2,3,0,1,6,7,4,5,10,11,8,9,14,15,12,13],[3,2,1,0,7,6,5,4,11,10,9,8,15,14,13,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,0,3,2,5,4,7,6,9,8,11,10,13,12,15,14],[14,15,12,13,10,11,8,9,6,7,4,5,2,3,0,1],[15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0],[12,13,14,15,8,9,10,11,4,5,6,7,0,1,2,3],[13,12,15,14,9,8,11,10,5,4,7,6,1,0,3,2],[10,11,8,9,14,15,12,13,2,3,0,1,6,7,4,5],[11,10,9,8,15,14,13,12,3,2,1,0,7,6,5,4],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[9,8,11,10,13,12,15,14,1,0,3,2,5,4,7,6]],\"identity\":6,\"divisor\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(16, 16, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 2], [5, 2], [6, 1], [7, 2], [8, 2], [9, 2], [10, 2], [11, 2], [12, 2], [13, 2], [14, 2], [15, 2]], 是, 否, composite_divisor_need_not_be_an_element_order)","explanation_zh":"Wave6第六層核對=(16, 16, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 2], [5, 2], [6, 1], [7, 2], [8, 2], [9, 2], [10, 2], [11, 2], [12, 2], [13, 2], [14, 2], [15, 2]], 是, 否, composite_divisor_need_not_be_an_element_order)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-composite-divisor-boundary","seed":202611121094,"parameters":{"group_table":[[6,7,4,5,2,3,0,1,14,15,12,13,10,11,8,9],[7,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8],[4,5,6,7,0,1,2,3,12,13,14,15,8,9,10,11],[5,4,7,6,1,0,3,2,13,12,15,14,9,8,11,10],[2,3,0,1,6,7,4,5,10,11,8,9,14,15,12,13],[3,2,1,0,7,6,5,4,11,10,9,8,15,14,13,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,0,3,2,5,4,7,6,9,8,11,10,13,12,15,14],[14,15,12,13,10,11,8,9,6,7,4,5,2,3,0,1],[15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0],[12,13,14,15,8,9,10,11,4,5,6,7,0,1,2,3],[13,12,15,14,9,8,11,10,5,4,7,6,1,0,3,2],[10,11,8,9,14,15,12,13,2,3,0,1,6,7,4,5],[11,10,9,8,15,14,13,12,3,2,1,0,7,6,5,4],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[9,8,11,10,13,12,15,14,1,0,3,2,5,4,7,6]],"identity":6,"divisor":16,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-composite-divisor-boundary","semantic_template":"qf-mod43-cauchy-composite-divisor-boundary/{\"divisor\":16,\"group_table\":[[6,7,4,5,2,3,0,1,14,15,12,13,10,11,8,9],[7,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8],[4,5,6,7,0,1,2,3,12,13,14,15,8,9,10,11],[5,4,7,6,1,0,3,2,13,12,15,14,9,8,11,10],[2,3,0,1,6,7,4,5,10,11,8,9,14,15,12,13],[3,2,1,0,7,6,5,4,11,10,9,8,15,14,13,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,0,3,2,5,4,7,6,9,8,11,10,13,12,15,14],[14,15,12,13,10,11,8,9,6,7,4,5,2,3,0,1],[15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0],[12,13,14,15,8,9,10,11,4,5,6,7,0,1,2,3],[13,12,15,14,9,8,11,10,5,4,7,6,1,0,3,2],[10,11,8,9,14,15,12,13,2,3,0,1,6,7,4,5],[11,10,9,8,15,14,13,12,3,2,1,0,7,6,5,4],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[9,8,11,10,13,12,15,14,1,0,3,2,5,4,7,6]],\"identity\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0095","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-composite-divisor-boundary；資料={\"group_table\":[[7,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8],[6,7,4,5,2,3,0,1,14,15,12,13,10,11,8,9],[5,4,7,6,1,0,3,2,13,12,15,14,9,8,11,10],[4,5,6,7,0,1,2,3,12,13,14,15,8,9,10,11],[3,2,1,0,7,6,5,4,11,10,9,8,15,14,13,12],[2,3,0,1,6,7,4,5,10,11,8,9,14,15,12,13],[1,0,3,2,5,4,7,6,9,8,11,10,13,12,15,14],[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0],[14,15,12,13,10,11,8,9,6,7,4,5,2,3,0,1],[13,12,15,14,9,8,11,10,5,4,7,6,1,0,3,2],[12,13,14,15,8,9,10,11,4,5,6,7,0,1,2,3],[11,10,9,8,15,14,13,12,3,2,1,0,7,6,5,4],[10,11,8,9,14,15,12,13,2,3,0,1,6,7,4,5],[9,8,11,10,13,12,15,14,1,0,3,2,5,4,7,6],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7]],\"identity\":7,\"divisor\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(16, 16, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 2], [5, 2], [6, 2], [7, 1], [8, 2], [9, 2], [10, 2], [11, 2], [12, 2], [13, 2], [14, 2], [15, 2]], 是, 否, composite_divisor_need_not_be_an_element_order)","explanation_zh":"Wave6第六層核對=(16, 16, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 2], [5, 2], [6, 2], [7, 1], [8, 2], [9, 2], [10, 2], [11, 2], [12, 2], [13, 2], [14, 2], [15, 2]], 是, 否, composite_divisor_need_not_be_an_element_order)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-composite-divisor-boundary","seed":202611121095,"parameters":{"group_table":[[7,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8],[6,7,4,5,2,3,0,1,14,15,12,13,10,11,8,9],[5,4,7,6,1,0,3,2,13,12,15,14,9,8,11,10],[4,5,6,7,0,1,2,3,12,13,14,15,8,9,10,11],[3,2,1,0,7,6,5,4,11,10,9,8,15,14,13,12],[2,3,0,1,6,7,4,5,10,11,8,9,14,15,12,13],[1,0,3,2,5,4,7,6,9,8,11,10,13,12,15,14],[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0],[14,15,12,13,10,11,8,9,6,7,4,5,2,3,0,1],[13,12,15,14,9,8,11,10,5,4,7,6,1,0,3,2],[12,13,14,15,8,9,10,11,4,5,6,7,0,1,2,3],[11,10,9,8,15,14,13,12,3,2,1,0,7,6,5,4],[10,11,8,9,14,15,12,13,2,3,0,1,6,7,4,5],[9,8,11,10,13,12,15,14,1,0,3,2,5,4,7,6],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7]],"identity":7,"divisor":16,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-composite-divisor-boundary","semantic_template":"qf-mod43-cauchy-composite-divisor-boundary/{\"divisor\":16,\"group_table\":[[7,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8],[6,7,4,5,2,3,0,1,14,15,12,13,10,11,8,9],[5,4,7,6,1,0,3,2,13,12,15,14,9,8,11,10],[4,5,6,7,0,1,2,3,12,13,14,15,8,9,10,11],[3,2,1,0,7,6,5,4,11,10,9,8,15,14,13,12],[2,3,0,1,6,7,4,5,10,11,8,9,14,15,12,13],[1,0,3,2,5,4,7,6,9,8,11,10,13,12,15,14],[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0],[14,15,12,13,10,11,8,9,6,7,4,5,2,3,0,1],[13,12,15,14,9,8,11,10,5,4,7,6,1,0,3,2],[12,13,14,15,8,9,10,11,4,5,6,7,0,1,2,3],[11,10,9,8,15,14,13,12,3,2,1,0,7,6,5,4],[10,11,8,9,14,15,12,13,2,3,0,1,6,7,4,5],[9,8,11,10,13,12,15,14,1,0,3,2,5,4,7,6],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7]],\"identity\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0096","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-composite-divisor-boundary；資料={\"group_table\":[[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[9,8,11,10,13,12,15,14,1,0,3,2,5,4,7,6],[10,11,8,9,14,15,12,13,2,3,0,1,6,7,4,5],[11,10,9,8,15,14,13,12,3,2,1,0,7,6,5,4],[12,13,14,15,8,9,10,11,4,5,6,7,0,1,2,3],[13,12,15,14,9,8,11,10,5,4,7,6,1,0,3,2],[14,15,12,13,10,11,8,9,6,7,4,5,2,3,0,1],[15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0],[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,0,3,2,5,4,7,6,9,8,11,10,13,12,15,14],[2,3,0,1,6,7,4,5,10,11,8,9,14,15,12,13],[3,2,1,0,7,6,5,4,11,10,9,8,15,14,13,12],[4,5,6,7,0,1,2,3,12,13,14,15,8,9,10,11],[5,4,7,6,1,0,3,2,13,12,15,14,9,8,11,10],[6,7,4,5,2,3,0,1,14,15,12,13,10,11,8,9],[7,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8]],\"identity\":8,\"divisor\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(16, 16, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 2], [5, 2], [6, 2], [7, 2], [8, 1], [9, 2], [10, 2], [11, 2], [12, 2], [13, 2], [14, 2], [15, 2]], 是, 否, composite_divisor_need_not_be_an_element_order)","explanation_zh":"Wave6第六層核對=(16, 16, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 2], [5, 2], [6, 2], [7, 2], [8, 1], [9, 2], [10, 2], [11, 2], [12, 2], [13, 2], [14, 2], [15, 2]], 是, 否, composite_divisor_need_not_be_an_element_order)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-composite-divisor-boundary","seed":202611121096,"parameters":{"group_table":[[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[9,8,11,10,13,12,15,14,1,0,3,2,5,4,7,6],[10,11,8,9,14,15,12,13,2,3,0,1,6,7,4,5],[11,10,9,8,15,14,13,12,3,2,1,0,7,6,5,4],[12,13,14,15,8,9,10,11,4,5,6,7,0,1,2,3],[13,12,15,14,9,8,11,10,5,4,7,6,1,0,3,2],[14,15,12,13,10,11,8,9,6,7,4,5,2,3,0,1],[15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0],[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,0,3,2,5,4,7,6,9,8,11,10,13,12,15,14],[2,3,0,1,6,7,4,5,10,11,8,9,14,15,12,13],[3,2,1,0,7,6,5,4,11,10,9,8,15,14,13,12],[4,5,6,7,0,1,2,3,12,13,14,15,8,9,10,11],[5,4,7,6,1,0,3,2,13,12,15,14,9,8,11,10],[6,7,4,5,2,3,0,1,14,15,12,13,10,11,8,9],[7,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8]],"identity":8,"divisor":16,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-composite-divisor-boundary","semantic_template":"qf-mod43-cauchy-composite-divisor-boundary/{\"divisor\":16,\"group_table\":[[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[9,8,11,10,13,12,15,14,1,0,3,2,5,4,7,6],[10,11,8,9,14,15,12,13,2,3,0,1,6,7,4,5],[11,10,9,8,15,14,13,12,3,2,1,0,7,6,5,4],[12,13,14,15,8,9,10,11,4,5,6,7,0,1,2,3],[13,12,15,14,9,8,11,10,5,4,7,6,1,0,3,2],[14,15,12,13,10,11,8,9,6,7,4,5,2,3,0,1],[15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0],[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,0,3,2,5,4,7,6,9,8,11,10,13,12,15,14],[2,3,0,1,6,7,4,5,10,11,8,9,14,15,12,13],[3,2,1,0,7,6,5,4,11,10,9,8,15,14,13,12],[4,5,6,7,0,1,2,3,12,13,14,15,8,9,10,11],[5,4,7,6,1,0,3,2,13,12,15,14,9,8,11,10],[6,7,4,5,2,3,0,1,14,15,12,13,10,11,8,9],[7,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8]],\"identity\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0097","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-composite-divisor-boundary；資料={\"group_table\":[[9,8,11,10,13,12,15,14,1,0,3,2,5,4,7,6],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[11,10,9,8,15,14,13,12,3,2,1,0,7,6,5,4],[10,11,8,9,14,15,12,13,2,3,0,1,6,7,4,5],[13,12,15,14,9,8,11,10,5,4,7,6,1,0,3,2],[12,13,14,15,8,9,10,11,4,5,6,7,0,1,2,3],[15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0],[14,15,12,13,10,11,8,9,6,7,4,5,2,3,0,1],[1,0,3,2,5,4,7,6,9,8,11,10,13,12,15,14],[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[3,2,1,0,7,6,5,4,11,10,9,8,15,14,13,12],[2,3,0,1,6,7,4,5,10,11,8,9,14,15,12,13],[5,4,7,6,1,0,3,2,13,12,15,14,9,8,11,10],[4,5,6,7,0,1,2,3,12,13,14,15,8,9,10,11],[7,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8],[6,7,4,5,2,3,0,1,14,15,12,13,10,11,8,9]],\"identity\":9,\"divisor\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(16, 16, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 2], [5, 2], [6, 2], [7, 2], [8, 2], [9, 1], [10, 2], [11, 2], [12, 2], [13, 2], [14, 2], [15, 2]], 是, 否, composite_divisor_need_not_be_an_element_order)","explanation_zh":"Wave6第六層核對=(16, 16, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 2], [5, 2], [6, 2], [7, 2], [8, 2], [9, 1], [10, 2], [11, 2], [12, 2], [13, 2], [14, 2], [15, 2]], 是, 否, composite_divisor_need_not_be_an_element_order)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-composite-divisor-boundary","seed":202611121097,"parameters":{"group_table":[[9,8,11,10,13,12,15,14,1,0,3,2,5,4,7,6],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[11,10,9,8,15,14,13,12,3,2,1,0,7,6,5,4],[10,11,8,9,14,15,12,13,2,3,0,1,6,7,4,5],[13,12,15,14,9,8,11,10,5,4,7,6,1,0,3,2],[12,13,14,15,8,9,10,11,4,5,6,7,0,1,2,3],[15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0],[14,15,12,13,10,11,8,9,6,7,4,5,2,3,0,1],[1,0,3,2,5,4,7,6,9,8,11,10,13,12,15,14],[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[3,2,1,0,7,6,5,4,11,10,9,8,15,14,13,12],[2,3,0,1,6,7,4,5,10,11,8,9,14,15,12,13],[5,4,7,6,1,0,3,2,13,12,15,14,9,8,11,10],[4,5,6,7,0,1,2,3,12,13,14,15,8,9,10,11],[7,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8],[6,7,4,5,2,3,0,1,14,15,12,13,10,11,8,9]],"identity":9,"divisor":16,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-composite-divisor-boundary","semantic_template":"qf-mod43-cauchy-composite-divisor-boundary/{\"divisor\":16,\"group_table\":[[9,8,11,10,13,12,15,14,1,0,3,2,5,4,7,6],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[11,10,9,8,15,14,13,12,3,2,1,0,7,6,5,4],[10,11,8,9,14,15,12,13,2,3,0,1,6,7,4,5],[13,12,15,14,9,8,11,10,5,4,7,6,1,0,3,2],[12,13,14,15,8,9,10,11,4,5,6,7,0,1,2,3],[15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0],[14,15,12,13,10,11,8,9,6,7,4,5,2,3,0,1],[1,0,3,2,5,4,7,6,9,8,11,10,13,12,15,14],[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[3,2,1,0,7,6,5,4,11,10,9,8,15,14,13,12],[2,3,0,1,6,7,4,5,10,11,8,9,14,15,12,13],[5,4,7,6,1,0,3,2,13,12,15,14,9,8,11,10],[4,5,6,7,0,1,2,3,12,13,14,15,8,9,10,11],[7,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8],[6,7,4,5,2,3,0,1,14,15,12,13,10,11,8,9]],\"identity\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0098","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-composite-divisor-boundary；資料={\"group_table\":[[10,11,8,9,14,15,12,13,2,3,0,1,6,7,4,5],[11,10,9,8,15,14,13,12,3,2,1,0,7,6,5,4],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[9,8,11,10,13,12,15,14,1,0,3,2,5,4,7,6],[14,15,12,13,10,11,8,9,6,7,4,5,2,3,0,1],[15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0],[12,13,14,15,8,9,10,11,4,5,6,7,0,1,2,3],[13,12,15,14,9,8,11,10,5,4,7,6,1,0,3,2],[2,3,0,1,6,7,4,5,10,11,8,9,14,15,12,13],[3,2,1,0,7,6,5,4,11,10,9,8,15,14,13,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,0,3,2,5,4,7,6,9,8,11,10,13,12,15,14],[6,7,4,5,2,3,0,1,14,15,12,13,10,11,8,9],[7,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8],[4,5,6,7,0,1,2,3,12,13,14,15,8,9,10,11],[5,4,7,6,1,0,3,2,13,12,15,14,9,8,11,10]],\"identity\":10,\"divisor\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(16, 16, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 2], [5, 2], [6, 2], [7, 2], [8, 2], [9, 2], [10, 1], [11, 2], [12, 2], [13, 2], [14, 2], [15, 2]], 是, 否, composite_divisor_need_not_be_an_element_order)","explanation_zh":"Wave6第六層核對=(16, 16, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 2], [5, 2], [6, 2], [7, 2], [8, 2], [9, 2], [10, 1], [11, 2], [12, 2], [13, 2], [14, 2], [15, 2]], 是, 否, composite_divisor_need_not_be_an_element_order)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-composite-divisor-boundary","seed":202611121098,"parameters":{"group_table":[[10,11,8,9,14,15,12,13,2,3,0,1,6,7,4,5],[11,10,9,8,15,14,13,12,3,2,1,0,7,6,5,4],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[9,8,11,10,13,12,15,14,1,0,3,2,5,4,7,6],[14,15,12,13,10,11,8,9,6,7,4,5,2,3,0,1],[15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0],[12,13,14,15,8,9,10,11,4,5,6,7,0,1,2,3],[13,12,15,14,9,8,11,10,5,4,7,6,1,0,3,2],[2,3,0,1,6,7,4,5,10,11,8,9,14,15,12,13],[3,2,1,0,7,6,5,4,11,10,9,8,15,14,13,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,0,3,2,5,4,7,6,9,8,11,10,13,12,15,14],[6,7,4,5,2,3,0,1,14,15,12,13,10,11,8,9],[7,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8],[4,5,6,7,0,1,2,3,12,13,14,15,8,9,10,11],[5,4,7,6,1,0,3,2,13,12,15,14,9,8,11,10]],"identity":10,"divisor":16,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-composite-divisor-boundary","semantic_template":"qf-mod43-cauchy-composite-divisor-boundary/{\"divisor\":16,\"group_table\":[[10,11,8,9,14,15,12,13,2,3,0,1,6,7,4,5],[11,10,9,8,15,14,13,12,3,2,1,0,7,6,5,4],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[9,8,11,10,13,12,15,14,1,0,3,2,5,4,7,6],[14,15,12,13,10,11,8,9,6,7,4,5,2,3,0,1],[15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0],[12,13,14,15,8,9,10,11,4,5,6,7,0,1,2,3],[13,12,15,14,9,8,11,10,5,4,7,6,1,0,3,2],[2,3,0,1,6,7,4,5,10,11,8,9,14,15,12,13],[3,2,1,0,7,6,5,4,11,10,9,8,15,14,13,12],[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,0,3,2,5,4,7,6,9,8,11,10,13,12,15,14],[6,7,4,5,2,3,0,1,14,15,12,13,10,11,8,9],[7,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8],[4,5,6,7,0,1,2,3,12,13,14,15,8,9,10,11],[5,4,7,6,1,0,3,2,13,12,15,14,9,8,11,10]],\"identity\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0099","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-composite-divisor-boundary；資料={\"group_table\":[[11,10,9,8,15,14,13,12,3,2,1,0,7,6,5,4],[10,11,8,9,14,15,12,13,2,3,0,1,6,7,4,5],[9,8,11,10,13,12,15,14,1,0,3,2,5,4,7,6],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0],[14,15,12,13,10,11,8,9,6,7,4,5,2,3,0,1],[13,12,15,14,9,8,11,10,5,4,7,6,1,0,3,2],[12,13,14,15,8,9,10,11,4,5,6,7,0,1,2,3],[3,2,1,0,7,6,5,4,11,10,9,8,15,14,13,12],[2,3,0,1,6,7,4,5,10,11,8,9,14,15,12,13],[1,0,3,2,5,4,7,6,9,8,11,10,13,12,15,14],[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[7,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8],[6,7,4,5,2,3,0,1,14,15,12,13,10,11,8,9],[5,4,7,6,1,0,3,2,13,12,15,14,9,8,11,10],[4,5,6,7,0,1,2,3,12,13,14,15,8,9,10,11]],\"identity\":11,\"divisor\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(16, 16, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 2], [5, 2], [6, 2], [7, 2], [8, 2], [9, 2], [10, 2], [11, 1], [12, 2], [13, 2], [14, 2], [15, 2]], 是, 否, composite_divisor_need_not_be_an_element_order)","explanation_zh":"Wave6第六層核對=(16, 16, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 2], [5, 2], [6, 2], [7, 2], [8, 2], [9, 2], [10, 2], [11, 1], [12, 2], [13, 2], [14, 2], [15, 2]], 是, 否, composite_divisor_need_not_be_an_element_order)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-composite-divisor-boundary","seed":202611121099,"parameters":{"group_table":[[11,10,9,8,15,14,13,12,3,2,1,0,7,6,5,4],[10,11,8,9,14,15,12,13,2,3,0,1,6,7,4,5],[9,8,11,10,13,12,15,14,1,0,3,2,5,4,7,6],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0],[14,15,12,13,10,11,8,9,6,7,4,5,2,3,0,1],[13,12,15,14,9,8,11,10,5,4,7,6,1,0,3,2],[12,13,14,15,8,9,10,11,4,5,6,7,0,1,2,3],[3,2,1,0,7,6,5,4,11,10,9,8,15,14,13,12],[2,3,0,1,6,7,4,5,10,11,8,9,14,15,12,13],[1,0,3,2,5,4,7,6,9,8,11,10,13,12,15,14],[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[7,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8],[6,7,4,5,2,3,0,1,14,15,12,13,10,11,8,9],[5,4,7,6,1,0,3,2,13,12,15,14,9,8,11,10],[4,5,6,7,0,1,2,3,12,13,14,15,8,9,10,11]],"identity":11,"divisor":16,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-composite-divisor-boundary","semantic_template":"qf-mod43-cauchy-composite-divisor-boundary/{\"divisor\":16,\"group_table\":[[11,10,9,8,15,14,13,12,3,2,1,0,7,6,5,4],[10,11,8,9,14,15,12,13,2,3,0,1,6,7,4,5],[9,8,11,10,13,12,15,14,1,0,3,2,5,4,7,6],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0],[14,15,12,13,10,11,8,9,6,7,4,5,2,3,0,1],[13,12,15,14,9,8,11,10,5,4,7,6,1,0,3,2],[12,13,14,15,8,9,10,11,4,5,6,7,0,1,2,3],[3,2,1,0,7,6,5,4,11,10,9,8,15,14,13,12],[2,3,0,1,6,7,4,5,10,11,8,9,14,15,12,13],[1,0,3,2,5,4,7,6,9,8,11,10,13,12,15,14],[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[7,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8],[6,7,4,5,2,3,0,1,14,15,12,13,10,11,8,9],[5,4,7,6,1,0,3,2,13,12,15,14,9,8,11,10],[4,5,6,7,0,1,2,3,12,13,14,15,8,9,10,11]],\"identity\":11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0100","batch_id":"batch-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-theorem-finite-groups"],"title_zh":"有限群的Cauchy定理","stem_zh":"完整有限群、prime divisor與exact element-order證書。操作=cauchy-composite-divisor-boundary；資料={\"group_table\":[[12,13,14,15,8,9,10,11,4,5,6,7,0,1,2,3],[13,12,15,14,9,8,11,10,5,4,7,6,1,0,3,2],[14,15,12,13,10,11,8,9,6,7,4,5,2,3,0,1],[15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[9,8,11,10,13,12,15,14,1,0,3,2,5,4,7,6],[10,11,8,9,14,15,12,13,2,3,0,1,6,7,4,5],[11,10,9,8,15,14,13,12,3,2,1,0,7,6,5,4],[4,5,6,7,0,1,2,3,12,13,14,15,8,9,10,11],[5,4,7,6,1,0,3,2,13,12,15,14,9,8,11,10],[6,7,4,5,2,3,0,1,14,15,12,13,10,11,8,9],[7,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8],[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,0,3,2,5,4,7,6,9,8,11,10,13,12,15,14],[2,3,0,1,6,7,4,5,10,11,8,9,14,15,12,13],[3,2,1,0,7,6,5,4,11,10,9,8,15,14,13,12]],\"identity\":12,\"divisor\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(16, 16, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 2], [5, 2], [6, 2], [7, 2], [8, 2], [9, 2], [10, 2], [11, 2], [12, 1], [13, 2], [14, 2], [15, 2]], 是, 否, composite_divisor_need_not_be_an_element_order)","explanation_zh":"Wave6第六層核對=(16, 16, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 2], [5, 2], [6, 2], [7, 2], [8, 2], [9, 2], [10, 2], [11, 2], [12, 1], [13, 2], [14, 2], [15, 2]], 是, 否, composite_divisor_need_not_be_an_element_order)；依完整明示資料獨立重算；order-p witness必須同時驗g^p=e與所有1<=k<p皆有g^k!=e；完整有限枚舉只證指定model；composite divisor不保證是某元素的階。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群的Cauchy定理"]},"variant":{"family_id":"qf-mod43-cauchy-composite-divisor-boundary","seed":202611121100,"parameters":{"group_table":[[12,13,14,15,8,9,10,11,4,5,6,7,0,1,2,3],[13,12,15,14,9,8,11,10,5,4,7,6,1,0,3,2],[14,15,12,13,10,11,8,9,6,7,4,5,2,3,0,1],[15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[9,8,11,10,13,12,15,14,1,0,3,2,5,4,7,6],[10,11,8,9,14,15,12,13,2,3,0,1,6,7,4,5],[11,10,9,8,15,14,13,12,3,2,1,0,7,6,5,4],[4,5,6,7,0,1,2,3,12,13,14,15,8,9,10,11],[5,4,7,6,1,0,3,2,13,12,15,14,9,8,11,10],[6,7,4,5,2,3,0,1,14,15,12,13,10,11,8,9],[7,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8],[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,0,3,2,5,4,7,6,9,8,11,10,13,12,15,14],[2,3,0,1,6,7,4,5,10,11,8,9,14,15,12,13],[3,2,1,0,7,6,5,4,11,10,9,8,15,14,13,12]],"identity":12,"divisor":16,"domain":"Complete_finite_group_Cauchy_witness","operation":"cauchy-composite-divisor-boundary","semantic_template":"qf-mod43-cauchy-composite-divisor-boundary/{\"divisor\":16,\"group_table\":[[12,13,14,15,8,9,10,11,4,5,6,7,0,1,2,3],[13,12,15,14,9,8,11,10,5,4,7,6,1,0,3,2],[14,15,12,13,10,11,8,9,6,7,4,5,2,3,0,1],[15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[9,8,11,10,13,12,15,14,1,0,3,2,5,4,7,6],[10,11,8,9,14,15,12,13,2,3,0,1,6,7,4,5],[11,10,9,8,15,14,13,12,3,2,1,0,7,6,5,4],[4,5,6,7,0,1,2,3,12,13,14,15,8,9,10,11],[5,4,7,6,1,0,3,2,13,12,15,14,9,8,11,10],[6,7,4,5,2,3,0,1,14,15,12,13,10,11,8,9],[7,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8],[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,0,3,2,5,4,7,6,9,8,11,10,13,12,15,14],[2,3,0,1,6,7,4,5,10,11,8,9,14,15,12,13],[3,2,1,0,7,6,5,4,11,10,9,8,15,14,13,12]],\"identity\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}