{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261107-r01-finite-group-action-0001","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-qualification；資料={\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":2,\"action_table\":[[0,1],[1,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 是, 是, 是, 是, 是)","explanation_zh":"Wave6根層核對=(2, 2, 是, 是, 是, 是, 是)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-qualification","seed":202611071001,"parameters":{"group_table":[[0,1],[1,0]],"identity":0,"point_count":2,"action_table":[[0,1],[1,0]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-qualification","semantic_template":"qf-mod38-groupaction-qualification/{\"action_table\":[[0,1],[1,0]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0002","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-qualification；資料={\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":3,\"action_table\":[[0,1,2],[1,2,0],[2,0,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 3, 是, 是, 是, 是, 是)","explanation_zh":"Wave6根層核對=(3, 3, 是, 是, 是, 是, 是)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-qualification","seed":202611071002,"parameters":{"group_table":[[0,1,2],[1,2,0],[2,0,1]],"identity":0,"point_count":3,"action_table":[[0,1,2],[1,2,0],[2,0,1]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-qualification","semantic_template":"qf-mod38-groupaction-qualification/{\"action_table\":[[0,1,2],[1,2,0],[2,0,1]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0003","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-qualification；資料={\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":4,\"action_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 4, 是, 是, 是, 是, 是)","explanation_zh":"Wave6根層核對=(4, 4, 是, 是, 是, 是, 是)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-qualification","seed":202611071003,"parameters":{"group_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"identity":0,"point_count":4,"action_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-qualification","semantic_template":"qf-mod38-groupaction-qualification/{\"action_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0004","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-qualification；資料={\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":5,\"action_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 5, 是, 是, 是, 是, 是)","explanation_zh":"Wave6根層核對=(5, 5, 是, 是, 是, 是, 是)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-qualification","seed":202611071004,"parameters":{"group_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"identity":0,"point_count":5,"action_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-qualification","semantic_template":"qf-mod38-groupaction-qualification/{\"action_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0005","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-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,\"point_count\":6,\"action_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]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 6, 是, 是, 是, 是, 是)","explanation_zh":"Wave6根層核對=(6, 6, 是, 是, 是, 是, 是)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-qualification","seed":202611071005,"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,"point_count":6,"action_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]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-qualification","semantic_template":"qf-mod38-groupaction-qualification/{\"action_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]],\"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,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0006","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-qualification；資料={\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":3,\"action_table\":[[0,1,2],[1,0,2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 3, 是, 是, 是, 是, 是)","explanation_zh":"Wave6根層核對=(2, 3, 是, 是, 是, 是, 是)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-qualification","seed":202611071006,"parameters":{"group_table":[[0,1],[1,0]],"identity":0,"point_count":3,"action_table":[[0,1,2],[1,0,2]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-qualification","semantic_template":"qf-mod38-groupaction-qualification/{\"action_table\":[[0,1,2],[1,0,2]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0007","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-qualification；資料={\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":4,\"action_table\":[[0,1,2,3],[1,2,0,3],[2,0,1,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 4, 是, 是, 是, 是, 是)","explanation_zh":"Wave6根層核對=(3, 4, 是, 是, 是, 是, 是)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-qualification","seed":202611071007,"parameters":{"group_table":[[0,1,2],[1,2,0],[2,0,1]],"identity":0,"point_count":4,"action_table":[[0,1,2,3],[1,2,0,3],[2,0,1,3]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-qualification","semantic_template":"qf-mod38-groupaction-qualification/{\"action_table\":[[0,1,2,3],[1,2,0,3],[2,0,1,3]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0008","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-qualification；資料={\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":5,\"action_table\":[[0,1,2,3,4],[1,2,3,0,4],[2,3,0,1,4],[3,0,1,2,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 5, 是, 是, 是, 是, 是)","explanation_zh":"Wave6根層核對=(4, 5, 是, 是, 是, 是, 是)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-qualification","seed":202611071008,"parameters":{"group_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"identity":0,"point_count":5,"action_table":[[0,1,2,3,4],[1,2,3,0,4],[2,3,0,1,4],[3,0,1,2,4]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-qualification","semantic_template":"qf-mod38-groupaction-qualification/{\"action_table\":[[0,1,2,3,4],[1,2,3,0,4],[2,3,0,1,4],[3,0,1,2,4]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0009","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-qualification；資料={\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":6,\"action_table\":[[0,1,2,3,4,5],[1,2,3,4,0,5],[2,3,4,0,1,5],[3,4,0,1,2,5],[4,0,1,2,3,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 6, 是, 是, 是, 是, 是)","explanation_zh":"Wave6根層核對=(5, 6, 是, 是, 是, 是, 是)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-qualification","seed":202611071009,"parameters":{"group_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"identity":0,"point_count":6,"action_table":[[0,1,2,3,4,5],[1,2,3,4,0,5],[2,3,4,0,1,5],[3,4,0,1,2,5],[4,0,1,2,3,5]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-qualification","semantic_template":"qf-mod38-groupaction-qualification/{\"action_table\":[[0,1,2,3,4,5],[1,2,3,4,0,5],[2,3,4,0,1,5],[3,4,0,1,2,5],[4,0,1,2,3,5]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0010","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-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,\"point_count\":7,\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,4,5,0,6],[2,3,4,5,0,1,6],[3,4,5,0,1,2,6],[4,5,0,1,2,3,6],[5,0,1,2,3,4,6]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 7, 是, 是, 是, 是, 是)","explanation_zh":"Wave6根層核對=(6, 7, 是, 是, 是, 是, 是)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-qualification","seed":202611071010,"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,"point_count":7,"action_table":[[0,1,2,3,4,5,6],[1,2,3,4,5,0,6],[2,3,4,5,0,1,6],[3,4,5,0,1,2,6],[4,5,0,1,2,3,6],[5,0,1,2,3,4,6]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-qualification","semantic_template":"qf-mod38-groupaction-qualification/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,4,5,0,6],[2,3,4,5,0,1,6],[3,4,5,0,1,2,6],[4,5,0,1,2,3,6],[5,0,1,2,3,4,6]],\"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,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0011","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-qualification；資料={\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":4,\"action_table\":[[0,1,2,3],[1,0,2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 4, 是, 是, 是, 是, 是)","explanation_zh":"Wave6根層核對=(2, 4, 是, 是, 是, 是, 是)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-qualification","seed":202611071011,"parameters":{"group_table":[[0,1],[1,0]],"identity":0,"point_count":4,"action_table":[[0,1,2,3],[1,0,2,3]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-qualification","semantic_template":"qf-mod38-groupaction-qualification/{\"action_table\":[[0,1,2,3],[1,0,2,3]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0012","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-qualification；資料={\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":5,\"action_table\":[[0,1,2,3,4],[1,2,0,3,4],[2,0,1,3,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 5, 是, 是, 是, 是, 是)","explanation_zh":"Wave6根層核對=(3, 5, 是, 是, 是, 是, 是)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-qualification","seed":202611071012,"parameters":{"group_table":[[0,1,2],[1,2,0],[2,0,1]],"identity":0,"point_count":5,"action_table":[[0,1,2,3,4],[1,2,0,3,4],[2,0,1,3,4]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-qualification","semantic_template":"qf-mod38-groupaction-qualification/{\"action_table\":[[0,1,2,3,4],[1,2,0,3,4],[2,0,1,3,4]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0013","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-qualification；資料={\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":6,\"action_table\":[[0,1,2,3,4,5],[1,2,3,0,4,5],[2,3,0,1,4,5],[3,0,1,2,4,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 6, 是, 是, 是, 是, 是)","explanation_zh":"Wave6根層核對=(4, 6, 是, 是, 是, 是, 是)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-qualification","seed":202611071013,"parameters":{"group_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"identity":0,"point_count":6,"action_table":[[0,1,2,3,4,5],[1,2,3,0,4,5],[2,3,0,1,4,5],[3,0,1,2,4,5]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-qualification","semantic_template":"qf-mod38-groupaction-qualification/{\"action_table\":[[0,1,2,3,4,5],[1,2,3,0,4,5],[2,3,0,1,4,5],[3,0,1,2,4,5]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0014","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-qualification；資料={\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":7,\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,4,0,5,6],[2,3,4,0,1,5,6],[3,4,0,1,2,5,6],[4,0,1,2,3,5,6]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 7, 是, 是, 是, 是, 是)","explanation_zh":"Wave6根層核對=(5, 7, 是, 是, 是, 是, 是)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-qualification","seed":202611071014,"parameters":{"group_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"identity":0,"point_count":7,"action_table":[[0,1,2,3,4,5,6],[1,2,3,4,0,5,6],[2,3,4,0,1,5,6],[3,4,0,1,2,5,6],[4,0,1,2,3,5,6]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-qualification","semantic_template":"qf-mod38-groupaction-qualification/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,4,0,5,6],[2,3,4,0,1,5,6],[3,4,0,1,2,5,6],[4,0,1,2,3,5,6]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0015","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-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,\"point_count\":8,\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,0,6,7],[2,3,4,5,0,1,6,7],[3,4,5,0,1,2,6,7],[4,5,0,1,2,3,6,7],[5,0,1,2,3,4,6,7]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 8, 是, 是, 是, 是, 是)","explanation_zh":"Wave6根層核對=(6, 8, 是, 是, 是, 是, 是)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-qualification","seed":202611071015,"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,"point_count":8,"action_table":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,0,6,7],[2,3,4,5,0,1,6,7],[3,4,5,0,1,2,6,7],[4,5,0,1,2,3,6,7],[5,0,1,2,3,4,6,7]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-qualification","semantic_template":"qf-mod38-groupaction-qualification/{\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,0,6,7],[2,3,4,5,0,1,6,7],[3,4,5,0,1,2,6,7],[4,5,0,1,2,3,6,7],[5,0,1,2,3,4,6,7]],\"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,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0016","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-qualification；資料={\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":5,\"action_table\":[[0,1,2,3,4],[1,0,2,3,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 5, 是, 是, 是, 是, 是)","explanation_zh":"Wave6根層核對=(2, 5, 是, 是, 是, 是, 是)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-qualification","seed":202611071016,"parameters":{"group_table":[[0,1],[1,0]],"identity":0,"point_count":5,"action_table":[[0,1,2,3,4],[1,0,2,3,4]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-qualification","semantic_template":"qf-mod38-groupaction-qualification/{\"action_table\":[[0,1,2,3,4],[1,0,2,3,4]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0017","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-qualification；資料={\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":6,\"action_table\":[[0,1,2,3,4,5],[1,2,0,3,4,5],[2,0,1,3,4,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 6, 是, 是, 是, 是, 是)","explanation_zh":"Wave6根層核對=(3, 6, 是, 是, 是, 是, 是)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-qualification","seed":202611071017,"parameters":{"group_table":[[0,1,2],[1,2,0],[2,0,1]],"identity":0,"point_count":6,"action_table":[[0,1,2,3,4,5],[1,2,0,3,4,5],[2,0,1,3,4,5]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-qualification","semantic_template":"qf-mod38-groupaction-qualification/{\"action_table\":[[0,1,2,3,4,5],[1,2,0,3,4,5],[2,0,1,3,4,5]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0018","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-qualification；資料={\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":7,\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,0,4,5,6],[2,3,0,1,4,5,6],[3,0,1,2,4,5,6]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 7, 是, 是, 是, 是, 是)","explanation_zh":"Wave6根層核對=(4, 7, 是, 是, 是, 是, 是)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-qualification","seed":202611071018,"parameters":{"group_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"identity":0,"point_count":7,"action_table":[[0,1,2,3,4,5,6],[1,2,3,0,4,5,6],[2,3,0,1,4,5,6],[3,0,1,2,4,5,6]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-qualification","semantic_template":"qf-mod38-groupaction-qualification/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,0,4,5,6],[2,3,0,1,4,5,6],[3,0,1,2,4,5,6]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0019","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-qualification；資料={\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":8,\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,0,5,6,7],[2,3,4,0,1,5,6,7],[3,4,0,1,2,5,6,7],[4,0,1,2,3,5,6,7]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 8, 是, 是, 是, 是, 是)","explanation_zh":"Wave6根層核對=(5, 8, 是, 是, 是, 是, 是)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-qualification","seed":202611071019,"parameters":{"group_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"identity":0,"point_count":8,"action_table":[[0,1,2,3,4,5,6,7],[1,2,3,4,0,5,6,7],[2,3,4,0,1,5,6,7],[3,4,0,1,2,5,6,7],[4,0,1,2,3,5,6,7]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-qualification","semantic_template":"qf-mod38-groupaction-qualification/{\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,0,5,6,7],[2,3,4,0,1,5,6,7],[3,4,0,1,2,5,6,7],[4,0,1,2,3,5,6,7]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0020","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-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,\"point_count\":9,\"action_table\":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,5,0,6,7,8],[2,3,4,5,0,1,6,7,8],[3,4,5,0,1,2,6,7,8],[4,5,0,1,2,3,6,7,8],[5,0,1,2,3,4,6,7,8]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 9, 是, 是, 是, 是, 是)","explanation_zh":"Wave6根層核對=(6, 9, 是, 是, 是, 是, 是)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-qualification","seed":202611071020,"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,"point_count":9,"action_table":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,5,0,6,7,8],[2,3,4,5,0,1,6,7,8],[3,4,5,0,1,2,6,7,8],[4,5,0,1,2,3,6,7,8],[5,0,1,2,3,4,6,7,8]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-qualification","semantic_template":"qf-mod38-groupaction-qualification/{\"action_table\":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,5,0,6,7,8],[2,3,4,5,0,1,6,7,8],[3,4,5,0,1,2,6,7,8],[4,5,0,1,2,3,6,7,8],[5,0,1,2,3,4,6,7,8]],\"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,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0021","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-qualification；資料={\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":6,\"action_table\":[[0,1,2,3,4,5],[1,0,2,3,4,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 6, 是, 是, 是, 是, 是)","explanation_zh":"Wave6根層核對=(2, 6, 是, 是, 是, 是, 是)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-qualification","seed":202611071021,"parameters":{"group_table":[[0,1],[1,0]],"identity":0,"point_count":6,"action_table":[[0,1,2,3,4,5],[1,0,2,3,4,5]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-qualification","semantic_template":"qf-mod38-groupaction-qualification/{\"action_table\":[[0,1,2,3,4,5],[1,0,2,3,4,5]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0022","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-qualification；資料={\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":7,\"action_table\":[[0,1,2,3,4,5,6],[1,2,0,3,4,5,6],[2,0,1,3,4,5,6]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 7, 是, 是, 是, 是, 是)","explanation_zh":"Wave6根層核對=(3, 7, 是, 是, 是, 是, 是)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-qualification","seed":202611071022,"parameters":{"group_table":[[0,1,2],[1,2,0],[2,0,1]],"identity":0,"point_count":7,"action_table":[[0,1,2,3,4,5,6],[1,2,0,3,4,5,6],[2,0,1,3,4,5,6]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-qualification","semantic_template":"qf-mod38-groupaction-qualification/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,0,3,4,5,6],[2,0,1,3,4,5,6]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0023","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-qualification；資料={\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":8,\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,0,4,5,6,7],[2,3,0,1,4,5,6,7],[3,0,1,2,4,5,6,7]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 8, 是, 是, 是, 是, 是)","explanation_zh":"Wave6根層核對=(4, 8, 是, 是, 是, 是, 是)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-qualification","seed":202611071023,"parameters":{"group_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"identity":0,"point_count":8,"action_table":[[0,1,2,3,4,5,6,7],[1,2,3,0,4,5,6,7],[2,3,0,1,4,5,6,7],[3,0,1,2,4,5,6,7]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-qualification","semantic_template":"qf-mod38-groupaction-qualification/{\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,0,4,5,6,7],[2,3,0,1,4,5,6,7],[3,0,1,2,4,5,6,7]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0024","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-qualification；資料={\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":9,\"action_table\":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,0,5,6,7,8],[2,3,4,0,1,5,6,7,8],[3,4,0,1,2,5,6,7,8],[4,0,1,2,3,5,6,7,8]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 9, 是, 是, 是, 是, 是)","explanation_zh":"Wave6根層核對=(5, 9, 是, 是, 是, 是, 是)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-qualification","seed":202611071024,"parameters":{"group_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"identity":0,"point_count":9,"action_table":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,0,5,6,7,8],[2,3,4,0,1,5,6,7,8],[3,4,0,1,2,5,6,7,8],[4,0,1,2,3,5,6,7,8]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-qualification","semantic_template":"qf-mod38-groupaction-qualification/{\"action_table\":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,0,5,6,7,8],[2,3,4,0,1,5,6,7,8],[3,4,0,1,2,5,6,7,8],[4,0,1,2,3,5,6,7,8]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0025","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-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,\"point_count\":10,\"action_table\":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,0,6,7,8,9],[2,3,4,5,0,1,6,7,8,9],[3,4,5,0,1,2,6,7,8,9],[4,5,0,1,2,3,6,7,8,9],[5,0,1,2,3,4,6,7,8,9]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 10, 是, 是, 是, 是, 是)","explanation_zh":"Wave6根層核對=(6, 10, 是, 是, 是, 是, 是)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-qualification","seed":202611071025,"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,"point_count":10,"action_table":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,0,6,7,8,9],[2,3,4,5,0,1,6,7,8,9],[3,4,5,0,1,2,6,7,8,9],[4,5,0,1,2,3,6,7,8,9],[5,0,1,2,3,4,6,7,8,9]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-qualification","semantic_template":"qf-mod38-groupaction-qualification/{\"action_table\":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,0,6,7,8,9],[2,3,4,5,0,1,6,7,8,9],[3,4,5,0,1,2,6,7,8,9],[4,5,0,1,2,3,6,7,8,9],[5,0,1,2,3,4,6,7,8,9]],\"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,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0026","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-identity-audit；資料={\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":2,\"action_table\":[[0,1],[1,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [[0, 0, 是], [1, 1, 是]], 是)","explanation_zh":"Wave6根層核對=(0, [[0, 0, 是], [1, 1, 是]], 是)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-identity-audit","seed":202611071026,"parameters":{"group_table":[[0,1],[1,0]],"identity":0,"point_count":2,"action_table":[[0,1],[1,0]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-identity-audit","semantic_template":"qf-mod38-groupaction-identity-audit/{\"action_table\":[[0,1],[1,0]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0027","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-identity-audit；資料={\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":3,\"action_table\":[[0,1,2],[1,2,0],[2,0,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [[0, 0, 是], [1, 1, 是], [2, 2, 是]], 是)","explanation_zh":"Wave6根層核對=(0, [[0, 0, 是], [1, 1, 是], [2, 2, 是]], 是)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-identity-audit","seed":202611071027,"parameters":{"group_table":[[0,1,2],[1,2,0],[2,0,1]],"identity":0,"point_count":3,"action_table":[[0,1,2],[1,2,0],[2,0,1]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-identity-audit","semantic_template":"qf-mod38-groupaction-identity-audit/{\"action_table\":[[0,1,2],[1,2,0],[2,0,1]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0028","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-identity-audit；資料={\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":4,\"action_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [[0, 0, 是], [1, 1, 是], [2, 2, 是], [3, 3, 是]], 是)","explanation_zh":"Wave6根層核對=(0, [[0, 0, 是], [1, 1, 是], [2, 2, 是], [3, 3, 是]], 是)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-identity-audit","seed":202611071028,"parameters":{"group_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"identity":0,"point_count":4,"action_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-identity-audit","semantic_template":"qf-mod38-groupaction-identity-audit/{\"action_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0029","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-identity-audit；資料={\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":5,\"action_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [[0, 0, 是], [1, 1, 是], [2, 2, 是], [3, 3, 是], [4, 4, 是]], 是)","explanation_zh":"Wave6根層核對=(0, [[0, 0, 是], [1, 1, 是], [2, 2, 是], [3, 3, 是], [4, 4, 是]], 是)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-identity-audit","seed":202611071029,"parameters":{"group_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"identity":0,"point_count":5,"action_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-identity-audit","semantic_template":"qf-mod38-groupaction-identity-audit/{\"action_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0030","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-identity-audit；資料={\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point_count\":6,\"action_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]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [[0, 0, 是], [1, 1, 是], [2, 2, 是], [3, 3, 是], [4, 4, 是], [5, 5, 是]], 是)","explanation_zh":"Wave6根層核對=(0, [[0, 0, 是], [1, 1, 是], [2, 2, 是], [3, 3, 是], [4, 4, 是], [5, 5, 是]], 是)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-identity-audit","seed":202611071030,"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,"point_count":6,"action_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]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-identity-audit","semantic_template":"qf-mod38-groupaction-identity-audit/{\"action_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]],\"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,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0031","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-identity-audit；資料={\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":3,\"action_table\":[[0,1,2],[1,0,2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [[0, 0, 是], [1, 1, 是], [2, 2, 是]], 是)","explanation_zh":"Wave6根層核對=(0, [[0, 0, 是], [1, 1, 是], [2, 2, 是]], 是)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-identity-audit","seed":202611071031,"parameters":{"group_table":[[0,1],[1,0]],"identity":0,"point_count":3,"action_table":[[0,1,2],[1,0,2]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-identity-audit","semantic_template":"qf-mod38-groupaction-identity-audit/{\"action_table\":[[0,1,2],[1,0,2]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0032","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-identity-audit；資料={\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":4,\"action_table\":[[0,1,2,3],[1,2,0,3],[2,0,1,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [[0, 0, 是], [1, 1, 是], [2, 2, 是], [3, 3, 是]], 是)","explanation_zh":"Wave6根層核對=(0, [[0, 0, 是], [1, 1, 是], [2, 2, 是], [3, 3, 是]], 是)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-identity-audit","seed":202611071032,"parameters":{"group_table":[[0,1,2],[1,2,0],[2,0,1]],"identity":0,"point_count":4,"action_table":[[0,1,2,3],[1,2,0,3],[2,0,1,3]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-identity-audit","semantic_template":"qf-mod38-groupaction-identity-audit/{\"action_table\":[[0,1,2,3],[1,2,0,3],[2,0,1,3]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0033","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-identity-audit；資料={\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":5,\"action_table\":[[0,1,2,3,4],[1,2,3,0,4],[2,3,0,1,4],[3,0,1,2,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [[0, 0, 是], [1, 1, 是], [2, 2, 是], [3, 3, 是], [4, 4, 是]], 是)","explanation_zh":"Wave6根層核對=(0, [[0, 0, 是], [1, 1, 是], [2, 2, 是], [3, 3, 是], [4, 4, 是]], 是)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-identity-audit","seed":202611071033,"parameters":{"group_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"identity":0,"point_count":5,"action_table":[[0,1,2,3,4],[1,2,3,0,4],[2,3,0,1,4],[3,0,1,2,4]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-identity-audit","semantic_template":"qf-mod38-groupaction-identity-audit/{\"action_table\":[[0,1,2,3,4],[1,2,3,0,4],[2,3,0,1,4],[3,0,1,2,4]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0034","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-identity-audit；資料={\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":6,\"action_table\":[[0,1,2,3,4,5],[1,2,3,4,0,5],[2,3,4,0,1,5],[3,4,0,1,2,5],[4,0,1,2,3,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [[0, 0, 是], [1, 1, 是], [2, 2, 是], [3, 3, 是], [4, 4, 是], [5, 5, 是]], 是)","explanation_zh":"Wave6根層核對=(0, [[0, 0, 是], [1, 1, 是], [2, 2, 是], [3, 3, 是], [4, 4, 是], [5, 5, 是]], 是)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-identity-audit","seed":202611071034,"parameters":{"group_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"identity":0,"point_count":6,"action_table":[[0,1,2,3,4,5],[1,2,3,4,0,5],[2,3,4,0,1,5],[3,4,0,1,2,5],[4,0,1,2,3,5]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-identity-audit","semantic_template":"qf-mod38-groupaction-identity-audit/{\"action_table\":[[0,1,2,3,4,5],[1,2,3,4,0,5],[2,3,4,0,1,5],[3,4,0,1,2,5],[4,0,1,2,3,5]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0035","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-identity-audit；資料={\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point_count\":7,\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,4,5,0,6],[2,3,4,5,0,1,6],[3,4,5,0,1,2,6],[4,5,0,1,2,3,6],[5,0,1,2,3,4,6]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [[0, 0, 是], [1, 1, 是], [2, 2, 是], [3, 3, 是], [4, 4, 是], [5, 5, 是], [6, 6, 是]], 是)","explanation_zh":"Wave6根層核對=(0, [[0, 0, 是], [1, 1, 是], [2, 2, 是], [3, 3, 是], [4, 4, 是], [5, 5, 是], [6, 6, 是]], 是)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-identity-audit","seed":202611071035,"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,"point_count":7,"action_table":[[0,1,2,3,4,5,6],[1,2,3,4,5,0,6],[2,3,4,5,0,1,6],[3,4,5,0,1,2,6],[4,5,0,1,2,3,6],[5,0,1,2,3,4,6]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-identity-audit","semantic_template":"qf-mod38-groupaction-identity-audit/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,4,5,0,6],[2,3,4,5,0,1,6],[3,4,5,0,1,2,6],[4,5,0,1,2,3,6],[5,0,1,2,3,4,6]],\"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,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0036","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-identity-audit；資料={\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":4,\"action_table\":[[0,1,2,3],[1,0,2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [[0, 0, 是], [1, 1, 是], [2, 2, 是], [3, 3, 是]], 是)","explanation_zh":"Wave6根層核對=(0, [[0, 0, 是], [1, 1, 是], [2, 2, 是], [3, 3, 是]], 是)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-identity-audit","seed":202611071036,"parameters":{"group_table":[[0,1],[1,0]],"identity":0,"point_count":4,"action_table":[[0,1,2,3],[1,0,2,3]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-identity-audit","semantic_template":"qf-mod38-groupaction-identity-audit/{\"action_table\":[[0,1,2,3],[1,0,2,3]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0037","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-identity-audit；資料={\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":5,\"action_table\":[[0,1,2,3,4],[1,2,0,3,4],[2,0,1,3,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [[0, 0, 是], [1, 1, 是], [2, 2, 是], [3, 3, 是], [4, 4, 是]], 是)","explanation_zh":"Wave6根層核對=(0, [[0, 0, 是], [1, 1, 是], [2, 2, 是], [3, 3, 是], [4, 4, 是]], 是)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-identity-audit","seed":202611071037,"parameters":{"group_table":[[0,1,2],[1,2,0],[2,0,1]],"identity":0,"point_count":5,"action_table":[[0,1,2,3,4],[1,2,0,3,4],[2,0,1,3,4]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-identity-audit","semantic_template":"qf-mod38-groupaction-identity-audit/{\"action_table\":[[0,1,2,3,4],[1,2,0,3,4],[2,0,1,3,4]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0038","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-identity-audit；資料={\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":6,\"action_table\":[[0,1,2,3,4,5],[1,2,3,0,4,5],[2,3,0,1,4,5],[3,0,1,2,4,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [[0, 0, 是], [1, 1, 是], [2, 2, 是], [3, 3, 是], [4, 4, 是], [5, 5, 是]], 是)","explanation_zh":"Wave6根層核對=(0, [[0, 0, 是], [1, 1, 是], [2, 2, 是], [3, 3, 是], [4, 4, 是], [5, 5, 是]], 是)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-identity-audit","seed":202611071038,"parameters":{"group_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"identity":0,"point_count":6,"action_table":[[0,1,2,3,4,5],[1,2,3,0,4,5],[2,3,0,1,4,5],[3,0,1,2,4,5]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-identity-audit","semantic_template":"qf-mod38-groupaction-identity-audit/{\"action_table\":[[0,1,2,3,4,5],[1,2,3,0,4,5],[2,3,0,1,4,5],[3,0,1,2,4,5]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0039","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-identity-audit；資料={\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":7,\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,4,0,5,6],[2,3,4,0,1,5,6],[3,4,0,1,2,5,6],[4,0,1,2,3,5,6]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [[0, 0, 是], [1, 1, 是], [2, 2, 是], [3, 3, 是], [4, 4, 是], [5, 5, 是], [6, 6, 是]], 是)","explanation_zh":"Wave6根層核對=(0, [[0, 0, 是], [1, 1, 是], [2, 2, 是], [3, 3, 是], [4, 4, 是], [5, 5, 是], [6, 6, 是]], 是)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-identity-audit","seed":202611071039,"parameters":{"group_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"identity":0,"point_count":7,"action_table":[[0,1,2,3,4,5,6],[1,2,3,4,0,5,6],[2,3,4,0,1,5,6],[3,4,0,1,2,5,6],[4,0,1,2,3,5,6]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-identity-audit","semantic_template":"qf-mod38-groupaction-identity-audit/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,4,0,5,6],[2,3,4,0,1,5,6],[3,4,0,1,2,5,6],[4,0,1,2,3,5,6]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0040","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-identity-audit；資料={\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point_count\":8,\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,0,6,7],[2,3,4,5,0,1,6,7],[3,4,5,0,1,2,6,7],[4,5,0,1,2,3,6,7],[5,0,1,2,3,4,6,7]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [[0, 0, 是], [1, 1, 是], [2, 2, 是], [3, 3, 是], [4, 4, 是], [5, 5, 是], [6, 6, 是], [7, 7, 是]], 是)","explanation_zh":"Wave6根層核對=(0, [[0, 0, 是], [1, 1, 是], [2, 2, 是], [3, 3, 是], [4, 4, 是], [5, 5, 是], [6, 6, 是], [7, 7, 是]], 是)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-identity-audit","seed":202611071040,"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,"point_count":8,"action_table":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,0,6,7],[2,3,4,5,0,1,6,7],[3,4,5,0,1,2,6,7],[4,5,0,1,2,3,6,7],[5,0,1,2,3,4,6,7]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-identity-audit","semantic_template":"qf-mod38-groupaction-identity-audit/{\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,0,6,7],[2,3,4,5,0,1,6,7],[3,4,5,0,1,2,6,7],[4,5,0,1,2,3,6,7],[5,0,1,2,3,4,6,7]],\"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,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0041","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-identity-audit；資料={\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":5,\"action_table\":[[0,1,2,3,4],[1,0,2,3,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [[0, 0, 是], [1, 1, 是], [2, 2, 是], [3, 3, 是], [4, 4, 是]], 是)","explanation_zh":"Wave6根層核對=(0, [[0, 0, 是], [1, 1, 是], [2, 2, 是], [3, 3, 是], [4, 4, 是]], 是)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-identity-audit","seed":202611071041,"parameters":{"group_table":[[0,1],[1,0]],"identity":0,"point_count":5,"action_table":[[0,1,2,3,4],[1,0,2,3,4]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-identity-audit","semantic_template":"qf-mod38-groupaction-identity-audit/{\"action_table\":[[0,1,2,3,4],[1,0,2,3,4]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0042","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-identity-audit；資料={\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":6,\"action_table\":[[0,1,2,3,4,5],[1,2,0,3,4,5],[2,0,1,3,4,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [[0, 0, 是], [1, 1, 是], [2, 2, 是], [3, 3, 是], [4, 4, 是], [5, 5, 是]], 是)","explanation_zh":"Wave6根層核對=(0, [[0, 0, 是], [1, 1, 是], [2, 2, 是], [3, 3, 是], [4, 4, 是], [5, 5, 是]], 是)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-identity-audit","seed":202611071042,"parameters":{"group_table":[[0,1,2],[1,2,0],[2,0,1]],"identity":0,"point_count":6,"action_table":[[0,1,2,3,4,5],[1,2,0,3,4,5],[2,0,1,3,4,5]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-identity-audit","semantic_template":"qf-mod38-groupaction-identity-audit/{\"action_table\":[[0,1,2,3,4,5],[1,2,0,3,4,5],[2,0,1,3,4,5]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0043","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-identity-audit；資料={\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":7,\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,0,4,5,6],[2,3,0,1,4,5,6],[3,0,1,2,4,5,6]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [[0, 0, 是], [1, 1, 是], [2, 2, 是], [3, 3, 是], [4, 4, 是], [5, 5, 是], [6, 6, 是]], 是)","explanation_zh":"Wave6根層核對=(0, [[0, 0, 是], [1, 1, 是], [2, 2, 是], [3, 3, 是], [4, 4, 是], [5, 5, 是], [6, 6, 是]], 是)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-identity-audit","seed":202611071043,"parameters":{"group_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"identity":0,"point_count":7,"action_table":[[0,1,2,3,4,5,6],[1,2,3,0,4,5,6],[2,3,0,1,4,5,6],[3,0,1,2,4,5,6]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-identity-audit","semantic_template":"qf-mod38-groupaction-identity-audit/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,0,4,5,6],[2,3,0,1,4,5,6],[3,0,1,2,4,5,6]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0044","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-identity-audit；資料={\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":8,\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,0,5,6,7],[2,3,4,0,1,5,6,7],[3,4,0,1,2,5,6,7],[4,0,1,2,3,5,6,7]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [[0, 0, 是], [1, 1, 是], [2, 2, 是], [3, 3, 是], [4, 4, 是], [5, 5, 是], [6, 6, 是], [7, 7, 是]], 是)","explanation_zh":"Wave6根層核對=(0, [[0, 0, 是], [1, 1, 是], [2, 2, 是], [3, 3, 是], [4, 4, 是], [5, 5, 是], [6, 6, 是], [7, 7, 是]], 是)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-identity-audit","seed":202611071044,"parameters":{"group_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"identity":0,"point_count":8,"action_table":[[0,1,2,3,4,5,6,7],[1,2,3,4,0,5,6,7],[2,3,4,0,1,5,6,7],[3,4,0,1,2,5,6,7],[4,0,1,2,3,5,6,7]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-identity-audit","semantic_template":"qf-mod38-groupaction-identity-audit/{\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,0,5,6,7],[2,3,4,0,1,5,6,7],[3,4,0,1,2,5,6,7],[4,0,1,2,3,5,6,7]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0045","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-identity-audit；資料={\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point_count\":9,\"action_table\":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,5,0,6,7,8],[2,3,4,5,0,1,6,7,8],[3,4,5,0,1,2,6,7,8],[4,5,0,1,2,3,6,7,8],[5,0,1,2,3,4,6,7,8]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [[0, 0, 是], [1, 1, 是], [2, 2, 是], [3, 3, 是], [4, 4, 是], [5, 5, 是], [6, 6, 是], [7, 7, 是], [8, 8, 是]], 是)","explanation_zh":"Wave6根層核對=(0, [[0, 0, 是], [1, 1, 是], [2, 2, 是], [3, 3, 是], [4, 4, 是], [5, 5, 是], [6, 6, 是], [7, 7, 是], [8, 8, 是]], 是)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-identity-audit","seed":202611071045,"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,"point_count":9,"action_table":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,5,0,6,7,8],[2,3,4,5,0,1,6,7,8],[3,4,5,0,1,2,6,7,8],[4,5,0,1,2,3,6,7,8],[5,0,1,2,3,4,6,7,8]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-identity-audit","semantic_template":"qf-mod38-groupaction-identity-audit/{\"action_table\":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,5,0,6,7,8],[2,3,4,5,0,1,6,7,8],[3,4,5,0,1,2,6,7,8],[4,5,0,1,2,3,6,7,8],[5,0,1,2,3,4,6,7,8]],\"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,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0046","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-identity-audit；資料={\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":6,\"action_table\":[[0,1,2,3,4,5],[1,0,2,3,4,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [[0, 0, 是], [1, 1, 是], [2, 2, 是], [3, 3, 是], [4, 4, 是], [5, 5, 是]], 是)","explanation_zh":"Wave6根層核對=(0, [[0, 0, 是], [1, 1, 是], [2, 2, 是], [3, 3, 是], [4, 4, 是], [5, 5, 是]], 是)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-identity-audit","seed":202611071046,"parameters":{"group_table":[[0,1],[1,0]],"identity":0,"point_count":6,"action_table":[[0,1,2,3,4,5],[1,0,2,3,4,5]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-identity-audit","semantic_template":"qf-mod38-groupaction-identity-audit/{\"action_table\":[[0,1,2,3,4,5],[1,0,2,3,4,5]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0047","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-identity-audit；資料={\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":7,\"action_table\":[[0,1,2,3,4,5,6],[1,2,0,3,4,5,6],[2,0,1,3,4,5,6]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [[0, 0, 是], [1, 1, 是], [2, 2, 是], [3, 3, 是], [4, 4, 是], [5, 5, 是], [6, 6, 是]], 是)","explanation_zh":"Wave6根層核對=(0, [[0, 0, 是], [1, 1, 是], [2, 2, 是], [3, 3, 是], [4, 4, 是], [5, 5, 是], [6, 6, 是]], 是)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-identity-audit","seed":202611071047,"parameters":{"group_table":[[0,1,2],[1,2,0],[2,0,1]],"identity":0,"point_count":7,"action_table":[[0,1,2,3,4,5,6],[1,2,0,3,4,5,6],[2,0,1,3,4,5,6]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-identity-audit","semantic_template":"qf-mod38-groupaction-identity-audit/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,0,3,4,5,6],[2,0,1,3,4,5,6]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0048","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-identity-audit；資料={\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":8,\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,0,4,5,6,7],[2,3,0,1,4,5,6,7],[3,0,1,2,4,5,6,7]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [[0, 0, 是], [1, 1, 是], [2, 2, 是], [3, 3, 是], [4, 4, 是], [5, 5, 是], [6, 6, 是], [7, 7, 是]], 是)","explanation_zh":"Wave6根層核對=(0, [[0, 0, 是], [1, 1, 是], [2, 2, 是], [3, 3, 是], [4, 4, 是], [5, 5, 是], [6, 6, 是], [7, 7, 是]], 是)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-identity-audit","seed":202611071048,"parameters":{"group_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"identity":0,"point_count":8,"action_table":[[0,1,2,3,4,5,6,7],[1,2,3,0,4,5,6,7],[2,3,0,1,4,5,6,7],[3,0,1,2,4,5,6,7]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-identity-audit","semantic_template":"qf-mod38-groupaction-identity-audit/{\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,0,4,5,6,7],[2,3,0,1,4,5,6,7],[3,0,1,2,4,5,6,7]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0049","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-identity-audit；資料={\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":9,\"action_table\":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,0,5,6,7,8],[2,3,4,0,1,5,6,7,8],[3,4,0,1,2,5,6,7,8],[4,0,1,2,3,5,6,7,8]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [[0, 0, 是], [1, 1, 是], [2, 2, 是], [3, 3, 是], [4, 4, 是], [5, 5, 是], [6, 6, 是], [7, 7, 是], [8, 8, 是]], 是)","explanation_zh":"Wave6根層核對=(0, [[0, 0, 是], [1, 1, 是], [2, 2, 是], [3, 3, 是], [4, 4, 是], [5, 5, 是], [6, 6, 是], [7, 7, 是], [8, 8, 是]], 是)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-identity-audit","seed":202611071049,"parameters":{"group_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"identity":0,"point_count":9,"action_table":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,0,5,6,7,8],[2,3,4,0,1,5,6,7,8],[3,4,0,1,2,5,6,7,8],[4,0,1,2,3,5,6,7,8]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-identity-audit","semantic_template":"qf-mod38-groupaction-identity-audit/{\"action_table\":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,0,5,6,7,8],[2,3,4,0,1,5,6,7,8],[3,4,0,1,2,5,6,7,8],[4,0,1,2,3,5,6,7,8]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0050","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-identity-audit；資料={\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point_count\":10,\"action_table\":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,0,6,7,8,9],[2,3,4,5,0,1,6,7,8,9],[3,4,5,0,1,2,6,7,8,9],[4,5,0,1,2,3,6,7,8,9],[5,0,1,2,3,4,6,7,8,9]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [[0, 0, 是], [1, 1, 是], [2, 2, 是], [3, 3, 是], [4, 4, 是], [5, 5, 是], [6, 6, 是], [7, 7, 是], [8, 8, 是], [9, 9, 是]], 是)","explanation_zh":"Wave6根層核對=(0, [[0, 0, 是], [1, 1, 是], [2, 2, 是], [3, 3, 是], [4, 4, 是], [5, 5, 是], [6, 6, 是], [7, 7, 是], [8, 8, 是], [9, 9, 是]], 是)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-identity-audit","seed":202611071050,"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,"point_count":10,"action_table":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,0,6,7,8,9],[2,3,4,5,0,1,6,7,8,9],[3,4,5,0,1,2,6,7,8,9],[4,5,0,1,2,3,6,7,8,9],[5,0,1,2,3,4,6,7,8,9]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-identity-audit","semantic_template":"qf-mod38-groupaction-identity-audit/{\"action_table\":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,0,6,7,8,9],[2,3,4,5,0,1,6,7,8,9],[3,4,5,0,1,2,6,7,8,9],[4,5,0,1,2,3,6,7,8,9],[5,0,1,2,3,4,6,7,8,9]],\"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,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0051","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-compatibility-audit；資料={\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":2,\"action_table\":[[0,1],[1,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, null, 是, all_g_h_x_checked)","explanation_zh":"Wave6根層核對=(8, null, 是, all_g_h_x_checked)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-compatibility-audit","seed":202611071051,"parameters":{"group_table":[[0,1],[1,0]],"identity":0,"point_count":2,"action_table":[[0,1],[1,0]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-compatibility-audit","semantic_template":"qf-mod38-groupaction-compatibility-audit/{\"action_table\":[[0,1],[1,0]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0052","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-compatibility-audit；資料={\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":3,\"action_table\":[[0,1,2],[1,2,0],[2,0,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(27, null, 是, all_g_h_x_checked)","explanation_zh":"Wave6根層核對=(27, null, 是, all_g_h_x_checked)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-compatibility-audit","seed":202611071052,"parameters":{"group_table":[[0,1,2],[1,2,0],[2,0,1]],"identity":0,"point_count":3,"action_table":[[0,1,2],[1,2,0],[2,0,1]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-compatibility-audit","semantic_template":"qf-mod38-groupaction-compatibility-audit/{\"action_table\":[[0,1,2],[1,2,0],[2,0,1]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0053","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-compatibility-audit；資料={\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":4,\"action_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(64, null, 是, all_g_h_x_checked)","explanation_zh":"Wave6根層核對=(64, null, 是, all_g_h_x_checked)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-compatibility-audit","seed":202611071053,"parameters":{"group_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"identity":0,"point_count":4,"action_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-compatibility-audit","semantic_template":"qf-mod38-groupaction-compatibility-audit/{\"action_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0054","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-compatibility-audit；資料={\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":5,\"action_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(125, null, 是, all_g_h_x_checked)","explanation_zh":"Wave6根層核對=(125, null, 是, all_g_h_x_checked)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-compatibility-audit","seed":202611071054,"parameters":{"group_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"identity":0,"point_count":5,"action_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-compatibility-audit","semantic_template":"qf-mod38-groupaction-compatibility-audit/{\"action_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0055","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-compatibility-audit；資料={\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point_count\":6,\"action_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]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(216, null, 是, all_g_h_x_checked)","explanation_zh":"Wave6根層核對=(216, null, 是, all_g_h_x_checked)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-compatibility-audit","seed":202611071055,"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,"point_count":6,"action_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]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-compatibility-audit","semantic_template":"qf-mod38-groupaction-compatibility-audit/{\"action_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]],\"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,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0056","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-compatibility-audit；資料={\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":3,\"action_table\":[[0,1,2],[1,0,2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, null, 是, all_g_h_x_checked)","explanation_zh":"Wave6根層核對=(12, null, 是, all_g_h_x_checked)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-compatibility-audit","seed":202611071056,"parameters":{"group_table":[[0,1],[1,0]],"identity":0,"point_count":3,"action_table":[[0,1,2],[1,0,2]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-compatibility-audit","semantic_template":"qf-mod38-groupaction-compatibility-audit/{\"action_table\":[[0,1,2],[1,0,2]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0057","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-compatibility-audit；資料={\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":4,\"action_table\":[[0,1,2,3],[1,2,0,3],[2,0,1,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(36, null, 是, all_g_h_x_checked)","explanation_zh":"Wave6根層核對=(36, null, 是, all_g_h_x_checked)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-compatibility-audit","seed":202611071057,"parameters":{"group_table":[[0,1,2],[1,2,0],[2,0,1]],"identity":0,"point_count":4,"action_table":[[0,1,2,3],[1,2,0,3],[2,0,1,3]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-compatibility-audit","semantic_template":"qf-mod38-groupaction-compatibility-audit/{\"action_table\":[[0,1,2,3],[1,2,0,3],[2,0,1,3]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0058","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-compatibility-audit；資料={\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":5,\"action_table\":[[0,1,2,3,4],[1,2,3,0,4],[2,3,0,1,4],[3,0,1,2,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(80, null, 是, all_g_h_x_checked)","explanation_zh":"Wave6根層核對=(80, null, 是, all_g_h_x_checked)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-compatibility-audit","seed":202611071058,"parameters":{"group_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"identity":0,"point_count":5,"action_table":[[0,1,2,3,4],[1,2,3,0,4],[2,3,0,1,4],[3,0,1,2,4]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-compatibility-audit","semantic_template":"qf-mod38-groupaction-compatibility-audit/{\"action_table\":[[0,1,2,3,4],[1,2,3,0,4],[2,3,0,1,4],[3,0,1,2,4]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0059","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-compatibility-audit；資料={\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":6,\"action_table\":[[0,1,2,3,4,5],[1,2,3,4,0,5],[2,3,4,0,1,5],[3,4,0,1,2,5],[4,0,1,2,3,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(150, null, 是, all_g_h_x_checked)","explanation_zh":"Wave6根層核對=(150, null, 是, all_g_h_x_checked)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-compatibility-audit","seed":202611071059,"parameters":{"group_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"identity":0,"point_count":6,"action_table":[[0,1,2,3,4,5],[1,2,3,4,0,5],[2,3,4,0,1,5],[3,4,0,1,2,5],[4,0,1,2,3,5]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-compatibility-audit","semantic_template":"qf-mod38-groupaction-compatibility-audit/{\"action_table\":[[0,1,2,3,4,5],[1,2,3,4,0,5],[2,3,4,0,1,5],[3,4,0,1,2,5],[4,0,1,2,3,5]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0060","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-compatibility-audit；資料={\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point_count\":7,\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,4,5,0,6],[2,3,4,5,0,1,6],[3,4,5,0,1,2,6],[4,5,0,1,2,3,6],[5,0,1,2,3,4,6]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(252, null, 是, all_g_h_x_checked)","explanation_zh":"Wave6根層核對=(252, null, 是, all_g_h_x_checked)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-compatibility-audit","seed":202611071060,"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,"point_count":7,"action_table":[[0,1,2,3,4,5,6],[1,2,3,4,5,0,6],[2,3,4,5,0,1,6],[3,4,5,0,1,2,6],[4,5,0,1,2,3,6],[5,0,1,2,3,4,6]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-compatibility-audit","semantic_template":"qf-mod38-groupaction-compatibility-audit/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,4,5,0,6],[2,3,4,5,0,1,6],[3,4,5,0,1,2,6],[4,5,0,1,2,3,6],[5,0,1,2,3,4,6]],\"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,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0061","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-compatibility-audit；資料={\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":4,\"action_table\":[[0,1,2,3],[1,0,2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(16, null, 是, all_g_h_x_checked)","explanation_zh":"Wave6根層核對=(16, null, 是, all_g_h_x_checked)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-compatibility-audit","seed":202611071061,"parameters":{"group_table":[[0,1],[1,0]],"identity":0,"point_count":4,"action_table":[[0,1,2,3],[1,0,2,3]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-compatibility-audit","semantic_template":"qf-mod38-groupaction-compatibility-audit/{\"action_table\":[[0,1,2,3],[1,0,2,3]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0062","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-compatibility-audit；資料={\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":5,\"action_table\":[[0,1,2,3,4],[1,2,0,3,4],[2,0,1,3,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(45, null, 是, all_g_h_x_checked)","explanation_zh":"Wave6根層核對=(45, null, 是, all_g_h_x_checked)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-compatibility-audit","seed":202611071062,"parameters":{"group_table":[[0,1,2],[1,2,0],[2,0,1]],"identity":0,"point_count":5,"action_table":[[0,1,2,3,4],[1,2,0,3,4],[2,0,1,3,4]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-compatibility-audit","semantic_template":"qf-mod38-groupaction-compatibility-audit/{\"action_table\":[[0,1,2,3,4],[1,2,0,3,4],[2,0,1,3,4]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0063","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-compatibility-audit；資料={\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":6,\"action_table\":[[0,1,2,3,4,5],[1,2,3,0,4,5],[2,3,0,1,4,5],[3,0,1,2,4,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(96, null, 是, all_g_h_x_checked)","explanation_zh":"Wave6根層核對=(96, null, 是, all_g_h_x_checked)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-compatibility-audit","seed":202611071063,"parameters":{"group_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"identity":0,"point_count":6,"action_table":[[0,1,2,3,4,5],[1,2,3,0,4,5],[2,3,0,1,4,5],[3,0,1,2,4,5]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-compatibility-audit","semantic_template":"qf-mod38-groupaction-compatibility-audit/{\"action_table\":[[0,1,2,3,4,5],[1,2,3,0,4,5],[2,3,0,1,4,5],[3,0,1,2,4,5]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0064","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-compatibility-audit；資料={\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":7,\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,4,0,5,6],[2,3,4,0,1,5,6],[3,4,0,1,2,5,6],[4,0,1,2,3,5,6]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(175, null, 是, all_g_h_x_checked)","explanation_zh":"Wave6根層核對=(175, null, 是, all_g_h_x_checked)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-compatibility-audit","seed":202611071064,"parameters":{"group_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"identity":0,"point_count":7,"action_table":[[0,1,2,3,4,5,6],[1,2,3,4,0,5,6],[2,3,4,0,1,5,6],[3,4,0,1,2,5,6],[4,0,1,2,3,5,6]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-compatibility-audit","semantic_template":"qf-mod38-groupaction-compatibility-audit/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,4,0,5,6],[2,3,4,0,1,5,6],[3,4,0,1,2,5,6],[4,0,1,2,3,5,6]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0065","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-compatibility-audit；資料={\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point_count\":8,\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,0,6,7],[2,3,4,5,0,1,6,7],[3,4,5,0,1,2,6,7],[4,5,0,1,2,3,6,7],[5,0,1,2,3,4,6,7]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(288, null, 是, all_g_h_x_checked)","explanation_zh":"Wave6根層核對=(288, null, 是, all_g_h_x_checked)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-compatibility-audit","seed":202611071065,"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,"point_count":8,"action_table":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,0,6,7],[2,3,4,5,0,1,6,7],[3,4,5,0,1,2,6,7],[4,5,0,1,2,3,6,7],[5,0,1,2,3,4,6,7]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-compatibility-audit","semantic_template":"qf-mod38-groupaction-compatibility-audit/{\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,0,6,7],[2,3,4,5,0,1,6,7],[3,4,5,0,1,2,6,7],[4,5,0,1,2,3,6,7],[5,0,1,2,3,4,6,7]],\"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,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0066","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-compatibility-audit；資料={\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":5,\"action_table\":[[0,1,2,3,4],[1,0,2,3,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(20, null, 是, all_g_h_x_checked)","explanation_zh":"Wave6根層核對=(20, null, 是, all_g_h_x_checked)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-compatibility-audit","seed":202611071066,"parameters":{"group_table":[[0,1],[1,0]],"identity":0,"point_count":5,"action_table":[[0,1,2,3,4],[1,0,2,3,4]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-compatibility-audit","semantic_template":"qf-mod38-groupaction-compatibility-audit/{\"action_table\":[[0,1,2,3,4],[1,0,2,3,4]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0067","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-compatibility-audit；資料={\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":6,\"action_table\":[[0,1,2,3,4,5],[1,2,0,3,4,5],[2,0,1,3,4,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(54, null, 是, all_g_h_x_checked)","explanation_zh":"Wave6根層核對=(54, null, 是, all_g_h_x_checked)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-compatibility-audit","seed":202611071067,"parameters":{"group_table":[[0,1,2],[1,2,0],[2,0,1]],"identity":0,"point_count":6,"action_table":[[0,1,2,3,4,5],[1,2,0,3,4,5],[2,0,1,3,4,5]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-compatibility-audit","semantic_template":"qf-mod38-groupaction-compatibility-audit/{\"action_table\":[[0,1,2,3,4,5],[1,2,0,3,4,5],[2,0,1,3,4,5]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0068","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-compatibility-audit；資料={\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":7,\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,0,4,5,6],[2,3,0,1,4,5,6],[3,0,1,2,4,5,6]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(112, null, 是, all_g_h_x_checked)","explanation_zh":"Wave6根層核對=(112, null, 是, all_g_h_x_checked)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-compatibility-audit","seed":202611071068,"parameters":{"group_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"identity":0,"point_count":7,"action_table":[[0,1,2,3,4,5,6],[1,2,3,0,4,5,6],[2,3,0,1,4,5,6],[3,0,1,2,4,5,6]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-compatibility-audit","semantic_template":"qf-mod38-groupaction-compatibility-audit/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,0,4,5,6],[2,3,0,1,4,5,6],[3,0,1,2,4,5,6]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0069","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-compatibility-audit；資料={\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":8,\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,0,5,6,7],[2,3,4,0,1,5,6,7],[3,4,0,1,2,5,6,7],[4,0,1,2,3,5,6,7]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(200, null, 是, all_g_h_x_checked)","explanation_zh":"Wave6根層核對=(200, null, 是, all_g_h_x_checked)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-compatibility-audit","seed":202611071069,"parameters":{"group_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"identity":0,"point_count":8,"action_table":[[0,1,2,3,4,5,6,7],[1,2,3,4,0,5,6,7],[2,3,4,0,1,5,6,7],[3,4,0,1,2,5,6,7],[4,0,1,2,3,5,6,7]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-compatibility-audit","semantic_template":"qf-mod38-groupaction-compatibility-audit/{\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,0,5,6,7],[2,3,4,0,1,5,6,7],[3,4,0,1,2,5,6,7],[4,0,1,2,3,5,6,7]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0070","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-compatibility-audit；資料={\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point_count\":9,\"action_table\":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,5,0,6,7,8],[2,3,4,5,0,1,6,7,8],[3,4,5,0,1,2,6,7,8],[4,5,0,1,2,3,6,7,8],[5,0,1,2,3,4,6,7,8]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(324, null, 是, all_g_h_x_checked)","explanation_zh":"Wave6根層核對=(324, null, 是, all_g_h_x_checked)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-compatibility-audit","seed":202611071070,"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,"point_count":9,"action_table":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,5,0,6,7,8],[2,3,4,5,0,1,6,7,8],[3,4,5,0,1,2,6,7,8],[4,5,0,1,2,3,6,7,8],[5,0,1,2,3,4,6,7,8]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-compatibility-audit","semantic_template":"qf-mod38-groupaction-compatibility-audit/{\"action_table\":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,5,0,6,7,8],[2,3,4,5,0,1,6,7,8],[3,4,5,0,1,2,6,7,8],[4,5,0,1,2,3,6,7,8],[5,0,1,2,3,4,6,7,8]],\"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,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0071","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-compatibility-audit；資料={\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":6,\"action_table\":[[0,1,2,3,4,5],[1,0,2,3,4,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(24, null, 是, all_g_h_x_checked)","explanation_zh":"Wave6根層核對=(24, null, 是, all_g_h_x_checked)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-compatibility-audit","seed":202611071071,"parameters":{"group_table":[[0,1],[1,0]],"identity":0,"point_count":6,"action_table":[[0,1,2,3,4,5],[1,0,2,3,4,5]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-compatibility-audit","semantic_template":"qf-mod38-groupaction-compatibility-audit/{\"action_table\":[[0,1,2,3,4,5],[1,0,2,3,4,5]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0072","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-compatibility-audit；資料={\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":7,\"action_table\":[[0,1,2,3,4,5,6],[1,2,0,3,4,5,6],[2,0,1,3,4,5,6]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(63, null, 是, all_g_h_x_checked)","explanation_zh":"Wave6根層核對=(63, null, 是, all_g_h_x_checked)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-compatibility-audit","seed":202611071072,"parameters":{"group_table":[[0,1,2],[1,2,0],[2,0,1]],"identity":0,"point_count":7,"action_table":[[0,1,2,3,4,5,6],[1,2,0,3,4,5,6],[2,0,1,3,4,5,6]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-compatibility-audit","semantic_template":"qf-mod38-groupaction-compatibility-audit/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,0,3,4,5,6],[2,0,1,3,4,5,6]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0073","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-compatibility-audit；資料={\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":8,\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,0,4,5,6,7],[2,3,0,1,4,5,6,7],[3,0,1,2,4,5,6,7]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(128, null, 是, all_g_h_x_checked)","explanation_zh":"Wave6根層核對=(128, null, 是, all_g_h_x_checked)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-compatibility-audit","seed":202611071073,"parameters":{"group_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"identity":0,"point_count":8,"action_table":[[0,1,2,3,4,5,6,7],[1,2,3,0,4,5,6,7],[2,3,0,1,4,5,6,7],[3,0,1,2,4,5,6,7]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-compatibility-audit","semantic_template":"qf-mod38-groupaction-compatibility-audit/{\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,0,4,5,6,7],[2,3,0,1,4,5,6,7],[3,0,1,2,4,5,6,7]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0074","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-compatibility-audit；資料={\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":9,\"action_table\":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,0,5,6,7,8],[2,3,4,0,1,5,6,7,8],[3,4,0,1,2,5,6,7,8],[4,0,1,2,3,5,6,7,8]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(225, null, 是, all_g_h_x_checked)","explanation_zh":"Wave6根層核對=(225, null, 是, all_g_h_x_checked)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-compatibility-audit","seed":202611071074,"parameters":{"group_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"identity":0,"point_count":9,"action_table":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,0,5,6,7,8],[2,3,4,0,1,5,6,7,8],[3,4,0,1,2,5,6,7,8],[4,0,1,2,3,5,6,7,8]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-compatibility-audit","semantic_template":"qf-mod38-groupaction-compatibility-audit/{\"action_table\":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,0,5,6,7,8],[2,3,4,0,1,5,6,7,8],[3,4,0,1,2,5,6,7,8],[4,0,1,2,3,5,6,7,8]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0075","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-compatibility-audit；資料={\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point_count\":10,\"action_table\":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,0,6,7,8,9],[2,3,4,5,0,1,6,7,8,9],[3,4,5,0,1,2,6,7,8,9],[4,5,0,1,2,3,6,7,8,9],[5,0,1,2,3,4,6,7,8,9]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(360, null, 是, all_g_h_x_checked)","explanation_zh":"Wave6根層核對=(360, null, 是, all_g_h_x_checked)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-compatibility-audit","seed":202611071075,"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,"point_count":10,"action_table":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,0,6,7,8,9],[2,3,4,5,0,1,6,7,8,9],[3,4,5,0,1,2,6,7,8,9],[4,5,0,1,2,3,6,7,8,9],[5,0,1,2,3,4,6,7,8,9]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-compatibility-audit","semantic_template":"qf-mod38-groupaction-compatibility-audit/{\"action_table\":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,0,6,7,8,9],[2,3,4,5,0,1,6,7,8,9],[3,4,5,0,1,2,6,7,8,9],[4,5,0,1,2,3,6,7,8,9],[5,0,1,2,3,4,6,7,8,9]],\"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,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0076","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-nonproperty-boundary；資料={\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":2,\"action_table\":[[0,1],[1,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 否, 否, 否, valid_action_does_not_automatically_claim_faithful_free_or_transitive)","explanation_zh":"Wave6根層核對=(是, 否, 否, 否, valid_action_does_not_automatically_claim_faithful_free_or_transitive)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-nonproperty-boundary","seed":202611071076,"parameters":{"group_table":[[0,1],[1,0]],"identity":0,"point_count":2,"action_table":[[0,1],[1,0]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-nonproperty-boundary","semantic_template":"qf-mod38-groupaction-nonproperty-boundary/{\"action_table\":[[0,1],[1,0]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0077","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-nonproperty-boundary；資料={\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":3,\"action_table\":[[0,1,2],[1,2,0],[2,0,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 否, 否, 否, valid_action_does_not_automatically_claim_faithful_free_or_transitive)","explanation_zh":"Wave6根層核對=(是, 否, 否, 否, valid_action_does_not_automatically_claim_faithful_free_or_transitive)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-nonproperty-boundary","seed":202611071077,"parameters":{"group_table":[[0,1,2],[1,2,0],[2,0,1]],"identity":0,"point_count":3,"action_table":[[0,1,2],[1,2,0],[2,0,1]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-nonproperty-boundary","semantic_template":"qf-mod38-groupaction-nonproperty-boundary/{\"action_table\":[[0,1,2],[1,2,0],[2,0,1]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0078","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-nonproperty-boundary；資料={\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":4,\"action_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 否, 否, 否, valid_action_does_not_automatically_claim_faithful_free_or_transitive)","explanation_zh":"Wave6根層核對=(是, 否, 否, 否, valid_action_does_not_automatically_claim_faithful_free_or_transitive)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-nonproperty-boundary","seed":202611071078,"parameters":{"group_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"identity":0,"point_count":4,"action_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-nonproperty-boundary","semantic_template":"qf-mod38-groupaction-nonproperty-boundary/{\"action_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0079","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-nonproperty-boundary；資料={\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":5,\"action_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 否, 否, 否, valid_action_does_not_automatically_claim_faithful_free_or_transitive)","explanation_zh":"Wave6根層核對=(是, 否, 否, 否, valid_action_does_not_automatically_claim_faithful_free_or_transitive)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-nonproperty-boundary","seed":202611071079,"parameters":{"group_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"identity":0,"point_count":5,"action_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-nonproperty-boundary","semantic_template":"qf-mod38-groupaction-nonproperty-boundary/{\"action_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0080","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-nonproperty-boundary；資料={\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point_count\":6,\"action_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]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 否, 否, 否, valid_action_does_not_automatically_claim_faithful_free_or_transitive)","explanation_zh":"Wave6根層核對=(是, 否, 否, 否, valid_action_does_not_automatically_claim_faithful_free_or_transitive)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-nonproperty-boundary","seed":202611071080,"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,"point_count":6,"action_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]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-nonproperty-boundary","semantic_template":"qf-mod38-groupaction-nonproperty-boundary/{\"action_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]],\"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,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0081","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-nonproperty-boundary；資料={\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":3,\"action_table\":[[0,1,2],[1,0,2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 否, 否, 否, valid_action_does_not_automatically_claim_faithful_free_or_transitive)","explanation_zh":"Wave6根層核對=(是, 否, 否, 否, valid_action_does_not_automatically_claim_faithful_free_or_transitive)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-nonproperty-boundary","seed":202611071081,"parameters":{"group_table":[[0,1],[1,0]],"identity":0,"point_count":3,"action_table":[[0,1,2],[1,0,2]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-nonproperty-boundary","semantic_template":"qf-mod38-groupaction-nonproperty-boundary/{\"action_table\":[[0,1,2],[1,0,2]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0082","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-nonproperty-boundary；資料={\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":4,\"action_table\":[[0,1,2,3],[1,2,0,3],[2,0,1,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 否, 否, 否, valid_action_does_not_automatically_claim_faithful_free_or_transitive)","explanation_zh":"Wave6根層核對=(是, 否, 否, 否, valid_action_does_not_automatically_claim_faithful_free_or_transitive)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-nonproperty-boundary","seed":202611071082,"parameters":{"group_table":[[0,1,2],[1,2,0],[2,0,1]],"identity":0,"point_count":4,"action_table":[[0,1,2,3],[1,2,0,3],[2,0,1,3]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-nonproperty-boundary","semantic_template":"qf-mod38-groupaction-nonproperty-boundary/{\"action_table\":[[0,1,2,3],[1,2,0,3],[2,0,1,3]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0083","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-nonproperty-boundary；資料={\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":5,\"action_table\":[[0,1,2,3,4],[1,2,3,0,4],[2,3,0,1,4],[3,0,1,2,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 否, 否, 否, valid_action_does_not_automatically_claim_faithful_free_or_transitive)","explanation_zh":"Wave6根層核對=(是, 否, 否, 否, valid_action_does_not_automatically_claim_faithful_free_or_transitive)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-nonproperty-boundary","seed":202611071083,"parameters":{"group_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"identity":0,"point_count":5,"action_table":[[0,1,2,3,4],[1,2,3,0,4],[2,3,0,1,4],[3,0,1,2,4]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-nonproperty-boundary","semantic_template":"qf-mod38-groupaction-nonproperty-boundary/{\"action_table\":[[0,1,2,3,4],[1,2,3,0,4],[2,3,0,1,4],[3,0,1,2,4]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0084","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-nonproperty-boundary；資料={\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":6,\"action_table\":[[0,1,2,3,4,5],[1,2,3,4,0,5],[2,3,4,0,1,5],[3,4,0,1,2,5],[4,0,1,2,3,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 否, 否, 否, valid_action_does_not_automatically_claim_faithful_free_or_transitive)","explanation_zh":"Wave6根層核對=(是, 否, 否, 否, valid_action_does_not_automatically_claim_faithful_free_or_transitive)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-nonproperty-boundary","seed":202611071084,"parameters":{"group_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"identity":0,"point_count":6,"action_table":[[0,1,2,3,4,5],[1,2,3,4,0,5],[2,3,4,0,1,5],[3,4,0,1,2,5],[4,0,1,2,3,5]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-nonproperty-boundary","semantic_template":"qf-mod38-groupaction-nonproperty-boundary/{\"action_table\":[[0,1,2,3,4,5],[1,2,3,4,0,5],[2,3,4,0,1,5],[3,4,0,1,2,5],[4,0,1,2,3,5]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0085","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-nonproperty-boundary；資料={\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point_count\":7,\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,4,5,0,6],[2,3,4,5,0,1,6],[3,4,5,0,1,2,6],[4,5,0,1,2,3,6],[5,0,1,2,3,4,6]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 否, 否, 否, valid_action_does_not_automatically_claim_faithful_free_or_transitive)","explanation_zh":"Wave6根層核對=(是, 否, 否, 否, valid_action_does_not_automatically_claim_faithful_free_or_transitive)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-nonproperty-boundary","seed":202611071085,"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,"point_count":7,"action_table":[[0,1,2,3,4,5,6],[1,2,3,4,5,0,6],[2,3,4,5,0,1,6],[3,4,5,0,1,2,6],[4,5,0,1,2,3,6],[5,0,1,2,3,4,6]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-nonproperty-boundary","semantic_template":"qf-mod38-groupaction-nonproperty-boundary/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,4,5,0,6],[2,3,4,5,0,1,6],[3,4,5,0,1,2,6],[4,5,0,1,2,3,6],[5,0,1,2,3,4,6]],\"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,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0086","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-nonproperty-boundary；資料={\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":4,\"action_table\":[[0,1,2,3],[1,0,2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 否, 否, 否, valid_action_does_not_automatically_claim_faithful_free_or_transitive)","explanation_zh":"Wave6根層核對=(是, 否, 否, 否, valid_action_does_not_automatically_claim_faithful_free_or_transitive)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-nonproperty-boundary","seed":202611071086,"parameters":{"group_table":[[0,1],[1,0]],"identity":0,"point_count":4,"action_table":[[0,1,2,3],[1,0,2,3]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-nonproperty-boundary","semantic_template":"qf-mod38-groupaction-nonproperty-boundary/{\"action_table\":[[0,1,2,3],[1,0,2,3]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0087","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-nonproperty-boundary；資料={\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":5,\"action_table\":[[0,1,2,3,4],[1,2,0,3,4],[2,0,1,3,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 否, 否, 否, valid_action_does_not_automatically_claim_faithful_free_or_transitive)","explanation_zh":"Wave6根層核對=(是, 否, 否, 否, valid_action_does_not_automatically_claim_faithful_free_or_transitive)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-nonproperty-boundary","seed":202611071087,"parameters":{"group_table":[[0,1,2],[1,2,0],[2,0,1]],"identity":0,"point_count":5,"action_table":[[0,1,2,3,4],[1,2,0,3,4],[2,0,1,3,4]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-nonproperty-boundary","semantic_template":"qf-mod38-groupaction-nonproperty-boundary/{\"action_table\":[[0,1,2,3,4],[1,2,0,3,4],[2,0,1,3,4]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0088","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-nonproperty-boundary；資料={\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":6,\"action_table\":[[0,1,2,3,4,5],[1,2,3,0,4,5],[2,3,0,1,4,5],[3,0,1,2,4,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 否, 否, 否, valid_action_does_not_automatically_claim_faithful_free_or_transitive)","explanation_zh":"Wave6根層核對=(是, 否, 否, 否, valid_action_does_not_automatically_claim_faithful_free_or_transitive)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-nonproperty-boundary","seed":202611071088,"parameters":{"group_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"identity":0,"point_count":6,"action_table":[[0,1,2,3,4,5],[1,2,3,0,4,5],[2,3,0,1,4,5],[3,0,1,2,4,5]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-nonproperty-boundary","semantic_template":"qf-mod38-groupaction-nonproperty-boundary/{\"action_table\":[[0,1,2,3,4,5],[1,2,3,0,4,5],[2,3,0,1,4,5],[3,0,1,2,4,5]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0089","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-nonproperty-boundary；資料={\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":7,\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,4,0,5,6],[2,3,4,0,1,5,6],[3,4,0,1,2,5,6],[4,0,1,2,3,5,6]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 否, 否, 否, valid_action_does_not_automatically_claim_faithful_free_or_transitive)","explanation_zh":"Wave6根層核對=(是, 否, 否, 否, valid_action_does_not_automatically_claim_faithful_free_or_transitive)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-nonproperty-boundary","seed":202611071089,"parameters":{"group_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"identity":0,"point_count":7,"action_table":[[0,1,2,3,4,5,6],[1,2,3,4,0,5,6],[2,3,4,0,1,5,6],[3,4,0,1,2,5,6],[4,0,1,2,3,5,6]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-nonproperty-boundary","semantic_template":"qf-mod38-groupaction-nonproperty-boundary/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,4,0,5,6],[2,3,4,0,1,5,6],[3,4,0,1,2,5,6],[4,0,1,2,3,5,6]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0090","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-nonproperty-boundary；資料={\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point_count\":8,\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,0,6,7],[2,3,4,5,0,1,6,7],[3,4,5,0,1,2,6,7],[4,5,0,1,2,3,6,7],[5,0,1,2,3,4,6,7]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 否, 否, 否, valid_action_does_not_automatically_claim_faithful_free_or_transitive)","explanation_zh":"Wave6根層核對=(是, 否, 否, 否, valid_action_does_not_automatically_claim_faithful_free_or_transitive)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-nonproperty-boundary","seed":202611071090,"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,"point_count":8,"action_table":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,0,6,7],[2,3,4,5,0,1,6,7],[3,4,5,0,1,2,6,7],[4,5,0,1,2,3,6,7],[5,0,1,2,3,4,6,7]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-nonproperty-boundary","semantic_template":"qf-mod38-groupaction-nonproperty-boundary/{\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,0,6,7],[2,3,4,5,0,1,6,7],[3,4,5,0,1,2,6,7],[4,5,0,1,2,3,6,7],[5,0,1,2,3,4,6,7]],\"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,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0091","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-nonproperty-boundary；資料={\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":5,\"action_table\":[[0,1,2,3,4],[1,0,2,3,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 否, 否, 否, valid_action_does_not_automatically_claim_faithful_free_or_transitive)","explanation_zh":"Wave6根層核對=(是, 否, 否, 否, valid_action_does_not_automatically_claim_faithful_free_or_transitive)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-nonproperty-boundary","seed":202611071091,"parameters":{"group_table":[[0,1],[1,0]],"identity":0,"point_count":5,"action_table":[[0,1,2,3,4],[1,0,2,3,4]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-nonproperty-boundary","semantic_template":"qf-mod38-groupaction-nonproperty-boundary/{\"action_table\":[[0,1,2,3,4],[1,0,2,3,4]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0092","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-nonproperty-boundary；資料={\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":6,\"action_table\":[[0,1,2,3,4,5],[1,2,0,3,4,5],[2,0,1,3,4,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 否, 否, 否, valid_action_does_not_automatically_claim_faithful_free_or_transitive)","explanation_zh":"Wave6根層核對=(是, 否, 否, 否, valid_action_does_not_automatically_claim_faithful_free_or_transitive)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-nonproperty-boundary","seed":202611071092,"parameters":{"group_table":[[0,1,2],[1,2,0],[2,0,1]],"identity":0,"point_count":6,"action_table":[[0,1,2,3,4,5],[1,2,0,3,4,5],[2,0,1,3,4,5]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-nonproperty-boundary","semantic_template":"qf-mod38-groupaction-nonproperty-boundary/{\"action_table\":[[0,1,2,3,4,5],[1,2,0,3,4,5],[2,0,1,3,4,5]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0093","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-nonproperty-boundary；資料={\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":7,\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,0,4,5,6],[2,3,0,1,4,5,6],[3,0,1,2,4,5,6]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 否, 否, 否, valid_action_does_not_automatically_claim_faithful_free_or_transitive)","explanation_zh":"Wave6根層核對=(是, 否, 否, 否, valid_action_does_not_automatically_claim_faithful_free_or_transitive)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-nonproperty-boundary","seed":202611071093,"parameters":{"group_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"identity":0,"point_count":7,"action_table":[[0,1,2,3,4,5,6],[1,2,3,0,4,5,6],[2,3,0,1,4,5,6],[3,0,1,2,4,5,6]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-nonproperty-boundary","semantic_template":"qf-mod38-groupaction-nonproperty-boundary/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,0,4,5,6],[2,3,0,1,4,5,6],[3,0,1,2,4,5,6]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0094","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-nonproperty-boundary；資料={\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":8,\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,0,5,6,7],[2,3,4,0,1,5,6,7],[3,4,0,1,2,5,6,7],[4,0,1,2,3,5,6,7]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 否, 否, 否, valid_action_does_not_automatically_claim_faithful_free_or_transitive)","explanation_zh":"Wave6根層核對=(是, 否, 否, 否, valid_action_does_not_automatically_claim_faithful_free_or_transitive)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-nonproperty-boundary","seed":202611071094,"parameters":{"group_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"identity":0,"point_count":8,"action_table":[[0,1,2,3,4,5,6,7],[1,2,3,4,0,5,6,7],[2,3,4,0,1,5,6,7],[3,4,0,1,2,5,6,7],[4,0,1,2,3,5,6,7]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-nonproperty-boundary","semantic_template":"qf-mod38-groupaction-nonproperty-boundary/{\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,0,5,6,7],[2,3,4,0,1,5,6,7],[3,4,0,1,2,5,6,7],[4,0,1,2,3,5,6,7]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0095","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-nonproperty-boundary；資料={\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point_count\":9,\"action_table\":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,5,0,6,7,8],[2,3,4,5,0,1,6,7,8],[3,4,5,0,1,2,6,7,8],[4,5,0,1,2,3,6,7,8],[5,0,1,2,3,4,6,7,8]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 否, 否, 否, valid_action_does_not_automatically_claim_faithful_free_or_transitive)","explanation_zh":"Wave6根層核對=(是, 否, 否, 否, valid_action_does_not_automatically_claim_faithful_free_or_transitive)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-nonproperty-boundary","seed":202611071095,"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,"point_count":9,"action_table":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,5,0,6,7,8],[2,3,4,5,0,1,6,7,8],[3,4,5,0,1,2,6,7,8],[4,5,0,1,2,3,6,7,8],[5,0,1,2,3,4,6,7,8]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-nonproperty-boundary","semantic_template":"qf-mod38-groupaction-nonproperty-boundary/{\"action_table\":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,5,0,6,7,8],[2,3,4,5,0,1,6,7,8],[3,4,5,0,1,2,6,7,8],[4,5,0,1,2,3,6,7,8],[5,0,1,2,3,4,6,7,8]],\"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,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0096","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-nonproperty-boundary；資料={\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":6,\"action_table\":[[0,1,2,3,4,5],[1,0,2,3,4,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 否, 否, 否, valid_action_does_not_automatically_claim_faithful_free_or_transitive)","explanation_zh":"Wave6根層核對=(是, 否, 否, 否, valid_action_does_not_automatically_claim_faithful_free_or_transitive)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-nonproperty-boundary","seed":202611071096,"parameters":{"group_table":[[0,1],[1,0]],"identity":0,"point_count":6,"action_table":[[0,1,2,3,4,5],[1,0,2,3,4,5]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-nonproperty-boundary","semantic_template":"qf-mod38-groupaction-nonproperty-boundary/{\"action_table\":[[0,1,2,3,4,5],[1,0,2,3,4,5]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0097","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-nonproperty-boundary；資料={\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":7,\"action_table\":[[0,1,2,3,4,5,6],[1,2,0,3,4,5,6],[2,0,1,3,4,5,6]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 否, 否, 否, valid_action_does_not_automatically_claim_faithful_free_or_transitive)","explanation_zh":"Wave6根層核對=(是, 否, 否, 否, valid_action_does_not_automatically_claim_faithful_free_or_transitive)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-nonproperty-boundary","seed":202611071097,"parameters":{"group_table":[[0,1,2],[1,2,0],[2,0,1]],"identity":0,"point_count":7,"action_table":[[0,1,2,3,4,5,6],[1,2,0,3,4,5,6],[2,0,1,3,4,5,6]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-nonproperty-boundary","semantic_template":"qf-mod38-groupaction-nonproperty-boundary/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,0,3,4,5,6],[2,0,1,3,4,5,6]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0098","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-nonproperty-boundary；資料={\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":8,\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,0,4,5,6,7],[2,3,0,1,4,5,6,7],[3,0,1,2,4,5,6,7]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 否, 否, 否, valid_action_does_not_automatically_claim_faithful_free_or_transitive)","explanation_zh":"Wave6根層核對=(是, 否, 否, 否, valid_action_does_not_automatically_claim_faithful_free_or_transitive)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-nonproperty-boundary","seed":202611071098,"parameters":{"group_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"identity":0,"point_count":8,"action_table":[[0,1,2,3,4,5,6,7],[1,2,3,0,4,5,6,7],[2,3,0,1,4,5,6,7],[3,0,1,2,4,5,6,7]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-nonproperty-boundary","semantic_template":"qf-mod38-groupaction-nonproperty-boundary/{\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,0,4,5,6,7],[2,3,0,1,4,5,6,7],[3,0,1,2,4,5,6,7]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0099","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-nonproperty-boundary；資料={\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":9,\"action_table\":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,0,5,6,7,8],[2,3,4,0,1,5,6,7,8],[3,4,0,1,2,5,6,7,8],[4,0,1,2,3,5,6,7,8]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 否, 否, 否, valid_action_does_not_automatically_claim_faithful_free_or_transitive)","explanation_zh":"Wave6根層核對=(是, 否, 否, 否, valid_action_does_not_automatically_claim_faithful_free_or_transitive)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-nonproperty-boundary","seed":202611071099,"parameters":{"group_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"identity":0,"point_count":9,"action_table":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,0,5,6,7,8],[2,3,4,0,1,5,6,7,8],[3,4,0,1,2,5,6,7,8],[4,0,1,2,3,5,6,7,8]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-nonproperty-boundary","semantic_template":"qf-mod38-groupaction-nonproperty-boundary/{\"action_table\":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,0,5,6,7,8],[2,3,4,0,1,5,6,7,8],[3,4,0,1,2,5,6,7,8],[4,0,1,2,3,5,6,7,8]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r01-finite-group-action-0100","batch_id":"batch-20261014-advance-20261107-r01-finite-group-action-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-action"],"title_zh":"有限群作用","stem_zh":"完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-nonproperty-boundary；資料={\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point_count\":10,\"action_table\":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,0,6,7,8,9],[2,3,4,5,0,1,6,7,8,9],[3,4,5,0,1,2,6,7,8,9],[4,5,0,1,2,3,6,7,8,9],[5,0,1,2,3,4,6,7,8,9]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 否, 否, 否, valid_action_does_not_automatically_claim_faithful_free_or_transitive)","explanation_zh":"Wave6根層核對=(是, 否, 否, 否, valid_action_does_not_automatically_claim_faithful_free_or_transitive)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用"]},"variant":{"family_id":"qf-mod38-groupaction-nonproperty-boundary","seed":202611071100,"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,"point_count":10,"action_table":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,0,6,7,8,9],[2,3,4,5,0,1,6,7,8,9],[3,4,5,0,1,2,6,7,8,9],[4,5,0,1,2,3,6,7,8,9],[5,0,1,2,3,4,6,7,8,9]],"domain":"Complete_finite_group_action_tables","operation":"groupaction-nonproperty-boundary","semantic_template":"qf-mod38-groupaction-nonproperty-boundary/{\"action_table\":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,0,6,7,8,9],[2,3,4,5,0,1,6,7,8,9],[3,4,5,0,1,2,6,7,8,9],[4,5,0,1,2,3,6,7,8,9],[5,0,1,2,3,4,6,7,8,9]],\"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,\"point_count\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}