{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0001","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-orbit-of-point；資料={\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":2,\"action_table\":[[0,1],[1,0]],\"point\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [0, 1], 2, all_group_elements_enumerated)","explanation_zh":"Wave6第二層核對=(0, [0, 1], 2, all_group_elements_enumerated)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-orbit-of-point","seed":202611081001,"parameters":{"group_table":[[0,1],[1,0]],"identity":0,"point_count":2,"action_table":[[0,1],[1,0]],"point":0,"domain":"Complete_finite_group_action_orbits","operation":"orbit-orbit-of-point","semantic_template":"qf-mod39-orbit-orbit-of-point/{\"action_table\":[[0,1],[1,0]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point\":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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0002","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-orbit-of-point；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]],\"point\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, [0, 1, 2], 3, all_group_elements_enumerated)","explanation_zh":"Wave6第二層核對=(1, [0, 1, 2], 3, all_group_elements_enumerated)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-orbit-of-point","seed":202611081002,"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]],"point":1,"domain":"Complete_finite_group_action_orbits","operation":"orbit-orbit-of-point","semantic_template":"qf-mod39-orbit-orbit-of-point/{\"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\":1,\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0003","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-orbit-of-point；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]],\"point\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [0, 1, 2, 3], 4, all_group_elements_enumerated)","explanation_zh":"Wave6第二層核對=(2, [0, 1, 2, 3], 4, all_group_elements_enumerated)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-orbit-of-point","seed":202611081003,"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]],"point":2,"domain":"Complete_finite_group_action_orbits","operation":"orbit-orbit-of-point","semantic_template":"qf-mod39-orbit-orbit-of-point/{\"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\":2,\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0004","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-orbit-of-point；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]],\"point\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [0, 1, 2, 3, 4], 5, all_group_elements_enumerated)","explanation_zh":"Wave6第二層核對=(3, [0, 1, 2, 3, 4], 5, all_group_elements_enumerated)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-orbit-of-point","seed":202611081004,"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]],"point":3,"domain":"Complete_finite_group_action_orbits","operation":"orbit-orbit-of-point","semantic_template":"qf-mod39-orbit-orbit-of-point/{\"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\":3,\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0005","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-orbit-of-point；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]],\"point\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, [0, 1, 2, 3, 4, 5], 6, all_group_elements_enumerated)","explanation_zh":"Wave6第二層核對=(4, [0, 1, 2, 3, 4, 5], 6, all_group_elements_enumerated)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-orbit-of-point","seed":202611081005,"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]],"point":4,"domain":"Complete_finite_group_action_orbits","operation":"orbit-orbit-of-point","semantic_template":"qf-mod39-orbit-orbit-of-point/{\"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\":4,\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0006","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-orbit-of-point；資料={\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":3,\"action_table\":[[0,1,2],[1,0,2]],\"point\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [2], 2, all_group_elements_enumerated)","explanation_zh":"Wave6第二層核對=(2, [2], 2, all_group_elements_enumerated)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-orbit-of-point","seed":202611081006,"parameters":{"group_table":[[0,1],[1,0]],"identity":0,"point_count":3,"action_table":[[0,1,2],[1,0,2]],"point":2,"domain":"Complete_finite_group_action_orbits","operation":"orbit-orbit-of-point","semantic_template":"qf-mod39-orbit-orbit-of-point/{\"action_table\":[[0,1,2],[1,0,2]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point\":2,\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0007","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-orbit-of-point；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]],\"point\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [0, 1, 2], 3, all_group_elements_enumerated)","explanation_zh":"Wave6第二層核對=(2, [0, 1, 2], 3, all_group_elements_enumerated)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-orbit-of-point","seed":202611081007,"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]],"point":2,"domain":"Complete_finite_group_action_orbits","operation":"orbit-orbit-of-point","semantic_template":"qf-mod39-orbit-orbit-of-point/{\"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\":2,\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0008","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-orbit-of-point；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]],\"point\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [0, 1, 2, 3], 4, all_group_elements_enumerated)","explanation_zh":"Wave6第二層核對=(2, [0, 1, 2, 3], 4, all_group_elements_enumerated)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-orbit-of-point","seed":202611081008,"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]],"point":2,"domain":"Complete_finite_group_action_orbits","operation":"orbit-orbit-of-point","semantic_template":"qf-mod39-orbit-orbit-of-point/{\"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\":2,\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0009","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-orbit-of-point；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]],\"point\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [0, 1, 2, 3, 4], 5, all_group_elements_enumerated)","explanation_zh":"Wave6第二層核對=(2, [0, 1, 2, 3, 4], 5, all_group_elements_enumerated)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-orbit-of-point","seed":202611081009,"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]],"point":2,"domain":"Complete_finite_group_action_orbits","operation":"orbit-orbit-of-point","semantic_template":"qf-mod39-orbit-orbit-of-point/{\"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\":2,\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0010","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-orbit-of-point；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]],\"point\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [0, 1, 2, 3, 4, 5], 6, all_group_elements_enumerated)","explanation_zh":"Wave6第二層核對=(2, [0, 1, 2, 3, 4, 5], 6, all_group_elements_enumerated)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-orbit-of-point","seed":202611081010,"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]],"point":2,"domain":"Complete_finite_group_action_orbits","operation":"orbit-orbit-of-point","semantic_template":"qf-mod39-orbit-orbit-of-point/{\"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\":2,\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0011","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-orbit-of-point；資料={\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":4,\"action_table\":[[0,1,2,3],[1,0,2,3]],\"point\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [2], 2, all_group_elements_enumerated)","explanation_zh":"Wave6第二層核對=(2, [2], 2, all_group_elements_enumerated)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-orbit-of-point","seed":202611081011,"parameters":{"group_table":[[0,1],[1,0]],"identity":0,"point_count":4,"action_table":[[0,1,2,3],[1,0,2,3]],"point":2,"domain":"Complete_finite_group_action_orbits","operation":"orbit-orbit-of-point","semantic_template":"qf-mod39-orbit-orbit-of-point/{\"action_table\":[[0,1,2,3],[1,0,2,3]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point\":2,\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0012","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-orbit-of-point；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]],\"point\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, [0, 1, 2], 3, all_group_elements_enumerated)","explanation_zh":"Wave6第二層核對=(1, [0, 1, 2], 3, all_group_elements_enumerated)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-orbit-of-point","seed":202611081012,"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]],"point":1,"domain":"Complete_finite_group_action_orbits","operation":"orbit-orbit-of-point","semantic_template":"qf-mod39-orbit-orbit-of-point/{\"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\":1,\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0013","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-orbit-of-point；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]],\"point\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [0, 1, 2, 3], 4, all_group_elements_enumerated)","explanation_zh":"Wave6第二層核對=(0, [0, 1, 2, 3], 4, all_group_elements_enumerated)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-orbit-of-point","seed":202611081013,"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]],"point":0,"domain":"Complete_finite_group_action_orbits","operation":"orbit-orbit-of-point","semantic_template":"qf-mod39-orbit-orbit-of-point/{\"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\":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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0014","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-orbit-of-point；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]],\"point\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, [6], 5, all_group_elements_enumerated)","explanation_zh":"Wave6第二層核對=(6, [6], 5, all_group_elements_enumerated)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-orbit-of-point","seed":202611081014,"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]],"point":6,"domain":"Complete_finite_group_action_orbits","operation":"orbit-orbit-of-point","semantic_template":"qf-mod39-orbit-orbit-of-point/{\"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\":6,\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0015","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-orbit-of-point；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]],\"point\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, [6], 6, all_group_elements_enumerated)","explanation_zh":"Wave6第二層核對=(6, [6], 6, all_group_elements_enumerated)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-orbit-of-point","seed":202611081015,"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]],"point":6,"domain":"Complete_finite_group_action_orbits","operation":"orbit-orbit-of-point","semantic_template":"qf-mod39-orbit-orbit-of-point/{\"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\":6,\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0016","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-orbit-of-point；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]],\"point\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [0, 1], 2, all_group_elements_enumerated)","explanation_zh":"Wave6第二層核對=(0, [0, 1], 2, all_group_elements_enumerated)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-orbit-of-point","seed":202611081016,"parameters":{"group_table":[[0,1],[1,0]],"identity":0,"point_count":5,"action_table":[[0,1,2,3,4],[1,0,2,3,4]],"point":0,"domain":"Complete_finite_group_action_orbits","operation":"orbit-orbit-of-point","semantic_template":"qf-mod39-orbit-orbit-of-point/{\"action_table\":[[0,1,2,3,4],[1,0,2,3,4]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point\":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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0017","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-orbit-of-point；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]],\"point\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, [4], 3, all_group_elements_enumerated)","explanation_zh":"Wave6第二層核對=(4, [4], 3, all_group_elements_enumerated)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-orbit-of-point","seed":202611081017,"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]],"point":4,"domain":"Complete_finite_group_action_orbits","operation":"orbit-orbit-of-point","semantic_template":"qf-mod39-orbit-orbit-of-point/{\"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\":4,\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0018","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-orbit-of-point；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]],\"point\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [0, 1, 2, 3], 4, all_group_elements_enumerated)","explanation_zh":"Wave6第二層核對=(3, [0, 1, 2, 3], 4, all_group_elements_enumerated)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-orbit-of-point","seed":202611081018,"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]],"point":3,"domain":"Complete_finite_group_action_orbits","operation":"orbit-orbit-of-point","semantic_template":"qf-mod39-orbit-orbit-of-point/{\"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\":3,\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0019","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-orbit-of-point；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]],\"point\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [0, 1, 2, 3, 4], 5, all_group_elements_enumerated)","explanation_zh":"Wave6第二層核對=(2, [0, 1, 2, 3, 4], 5, all_group_elements_enumerated)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-orbit-of-point","seed":202611081019,"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]],"point":2,"domain":"Complete_finite_group_action_orbits","operation":"orbit-orbit-of-point","semantic_template":"qf-mod39-orbit-orbit-of-point/{\"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\":2,\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0020","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-orbit-of-point；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]],\"point\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, [0, 1, 2, 3, 4, 5], 6, all_group_elements_enumerated)","explanation_zh":"Wave6第二層核對=(1, [0, 1, 2, 3, 4, 5], 6, all_group_elements_enumerated)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-orbit-of-point","seed":202611081020,"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]],"point":1,"domain":"Complete_finite_group_action_orbits","operation":"orbit-orbit-of-point","semantic_template":"qf-mod39-orbit-orbit-of-point/{\"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\":1,\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0021","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-orbit-of-point；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]],\"point\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [2], 2, all_group_elements_enumerated)","explanation_zh":"Wave6第二層核對=(2, [2], 2, all_group_elements_enumerated)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-orbit-of-point","seed":202611081021,"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]],"point":2,"domain":"Complete_finite_group_action_orbits","operation":"orbit-orbit-of-point","semantic_template":"qf-mod39-orbit-orbit-of-point/{\"action_table\":[[0,1,2,3,4,5],[1,0,2,3,4,5]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point\":2,\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0022","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-orbit-of-point；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]],\"point\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [0, 1, 2], 3, all_group_elements_enumerated)","explanation_zh":"Wave6第二層核對=(0, [0, 1, 2], 3, all_group_elements_enumerated)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-orbit-of-point","seed":202611081022,"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]],"point":0,"domain":"Complete_finite_group_action_orbits","operation":"orbit-orbit-of-point","semantic_template":"qf-mod39-orbit-orbit-of-point/{\"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\":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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0023","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-orbit-of-point；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]],\"point\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, [6], 4, all_group_elements_enumerated)","explanation_zh":"Wave6第二層核對=(6, [6], 4, all_group_elements_enumerated)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-orbit-of-point","seed":202611081023,"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]],"point":6,"domain":"Complete_finite_group_action_orbits","operation":"orbit-orbit-of-point","semantic_template":"qf-mod39-orbit-orbit-of-point/{\"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\":6,\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0024","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-orbit-of-point；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]],\"point\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, [5], 5, all_group_elements_enumerated)","explanation_zh":"Wave6第二層核對=(5, [5], 5, all_group_elements_enumerated)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-orbit-of-point","seed":202611081024,"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]],"point":5,"domain":"Complete_finite_group_action_orbits","operation":"orbit-orbit-of-point","semantic_template":"qf-mod39-orbit-orbit-of-point/{\"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\":5,\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0025","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-orbit-of-point；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]],\"point\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, [0, 1, 2, 3, 4, 5], 6, all_group_elements_enumerated)","explanation_zh":"Wave6第二層核對=(4, [0, 1, 2, 3, 4, 5], 6, all_group_elements_enumerated)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-orbit-of-point","seed":202611081025,"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]],"point":4,"domain":"Complete_finite_group_action_orbits","operation":"orbit-orbit-of-point","semantic_template":"qf-mod39-orbit-orbit-of-point/{\"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\":4,\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0026","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-orbit-partition；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, 1]], [0, 1], 是, 是)","explanation_zh":"Wave6第二層核對=([[0, 1]], [0, 1], 是, 是)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-orbit-partition","seed":202611081026,"parameters":{"group_table":[[0,1],[1,0]],"identity":0,"point_count":2,"action_table":[[0,1],[1,0]],"domain":"Complete_finite_group_action_orbits","operation":"orbit-orbit-partition","semantic_template":"qf-mod39-orbit-orbit-partition/{\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0027","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-orbit-partition；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, 1, 2]], [0, 1, 2], 是, 是)","explanation_zh":"Wave6第二層核對=([[0, 1, 2]], [0, 1, 2], 是, 是)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-orbit-partition","seed":202611081027,"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_orbits","operation":"orbit-orbit-partition","semantic_template":"qf-mod39-orbit-orbit-partition/{\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0028","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-orbit-partition；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, 1, 2, 3]], [0, 1, 2, 3], 是, 是)","explanation_zh":"Wave6第二層核對=([[0, 1, 2, 3]], [0, 1, 2, 3], 是, 是)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-orbit-partition","seed":202611081028,"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_orbits","operation":"orbit-orbit-partition","semantic_template":"qf-mod39-orbit-orbit-partition/{\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0029","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-orbit-partition；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, 1, 2, 3, 4]], [0, 1, 2, 3, 4], 是, 是)","explanation_zh":"Wave6第二層核對=([[0, 1, 2, 3, 4]], [0, 1, 2, 3, 4], 是, 是)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-orbit-partition","seed":202611081029,"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_orbits","operation":"orbit-orbit-partition","semantic_template":"qf-mod39-orbit-orbit-partition/{\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0030","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-orbit-partition；資料={\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"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, 1, 2, 3, 4, 5]], [0, 1, 2, 3, 4, 5], 是, 是)","explanation_zh":"Wave6第二層核對=([[0, 1, 2, 3, 4, 5]], [0, 1, 2, 3, 4, 5], 是, 是)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-orbit-partition","seed":202611081030,"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_orbits","operation":"orbit-orbit-partition","semantic_template":"qf-mod39-orbit-orbit-partition/{\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0031","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-orbit-partition；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, 1], [2]], [0, 1, 2], 是, 是)","explanation_zh":"Wave6第二層核對=([[0, 1], [2]], [0, 1, 2], 是, 是)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-orbit-partition","seed":202611081031,"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_orbits","operation":"orbit-orbit-partition","semantic_template":"qf-mod39-orbit-orbit-partition/{\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0032","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-orbit-partition；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, 1, 2], [3]], [0, 1, 2, 3], 是, 是)","explanation_zh":"Wave6第二層核對=([[0, 1, 2], [3]], [0, 1, 2, 3], 是, 是)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-orbit-partition","seed":202611081032,"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_orbits","operation":"orbit-orbit-partition","semantic_template":"qf-mod39-orbit-orbit-partition/{\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0033","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-orbit-partition；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, 1, 2, 3], [4]], [0, 1, 2, 3, 4], 是, 是)","explanation_zh":"Wave6第二層核對=([[0, 1, 2, 3], [4]], [0, 1, 2, 3, 4], 是, 是)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-orbit-partition","seed":202611081033,"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_orbits","operation":"orbit-orbit-partition","semantic_template":"qf-mod39-orbit-orbit-partition/{\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0034","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-orbit-partition；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, 1, 2, 3, 4], [5]], [0, 1, 2, 3, 4, 5], 是, 是)","explanation_zh":"Wave6第二層核對=([[0, 1, 2, 3, 4], [5]], [0, 1, 2, 3, 4, 5], 是, 是)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-orbit-partition","seed":202611081034,"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_orbits","operation":"orbit-orbit-partition","semantic_template":"qf-mod39-orbit-orbit-partition/{\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0035","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-orbit-partition；資料={\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"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, 1, 2, 3, 4, 5], [6]], [0, 1, 2, 3, 4, 5, 6], 是, 是)","explanation_zh":"Wave6第二層核對=([[0, 1, 2, 3, 4, 5], [6]], [0, 1, 2, 3, 4, 5, 6], 是, 是)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-orbit-partition","seed":202611081035,"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_orbits","operation":"orbit-orbit-partition","semantic_template":"qf-mod39-orbit-orbit-partition/{\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0036","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-orbit-partition；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, 1], [2], [3]], [0, 1, 2, 3], 是, 是)","explanation_zh":"Wave6第二層核對=([[0, 1], [2], [3]], [0, 1, 2, 3], 是, 是)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-orbit-partition","seed":202611081036,"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_orbits","operation":"orbit-orbit-partition","semantic_template":"qf-mod39-orbit-orbit-partition/{\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0037","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-orbit-partition；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, 1, 2], [3], [4]], [0, 1, 2, 3, 4], 是, 是)","explanation_zh":"Wave6第二層核對=([[0, 1, 2], [3], [4]], [0, 1, 2, 3, 4], 是, 是)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-orbit-partition","seed":202611081037,"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_orbits","operation":"orbit-orbit-partition","semantic_template":"qf-mod39-orbit-orbit-partition/{\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0038","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-orbit-partition；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, 1, 2, 3], [4], [5]], [0, 1, 2, 3, 4, 5], 是, 是)","explanation_zh":"Wave6第二層核對=([[0, 1, 2, 3], [4], [5]], [0, 1, 2, 3, 4, 5], 是, 是)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-orbit-partition","seed":202611081038,"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_orbits","operation":"orbit-orbit-partition","semantic_template":"qf-mod39-orbit-orbit-partition/{\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0039","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-orbit-partition；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, 1, 2, 3, 4], [5], [6]], [0, 1, 2, 3, 4, 5, 6], 是, 是)","explanation_zh":"Wave6第二層核對=([[0, 1, 2, 3, 4], [5], [6]], [0, 1, 2, 3, 4, 5, 6], 是, 是)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-orbit-partition","seed":202611081039,"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_orbits","operation":"orbit-orbit-partition","semantic_template":"qf-mod39-orbit-orbit-partition/{\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0040","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-orbit-partition；資料={\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"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, 1, 2, 3, 4, 5], [6], [7]], [0, 1, 2, 3, 4, 5, 6, 7], 是, 是)","explanation_zh":"Wave6第二層核對=([[0, 1, 2, 3, 4, 5], [6], [7]], [0, 1, 2, 3, 4, 5, 6, 7], 是, 是)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-orbit-partition","seed":202611081040,"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_orbits","operation":"orbit-orbit-partition","semantic_template":"qf-mod39-orbit-orbit-partition/{\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0041","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-orbit-partition；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, 1], [2], [3], [4]], [0, 1, 2, 3, 4], 是, 是)","explanation_zh":"Wave6第二層核對=([[0, 1], [2], [3], [4]], [0, 1, 2, 3, 4], 是, 是)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-orbit-partition","seed":202611081041,"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_orbits","operation":"orbit-orbit-partition","semantic_template":"qf-mod39-orbit-orbit-partition/{\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0042","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-orbit-partition；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, 1, 2], [3], [4], [5]], [0, 1, 2, 3, 4, 5], 是, 是)","explanation_zh":"Wave6第二層核對=([[0, 1, 2], [3], [4], [5]], [0, 1, 2, 3, 4, 5], 是, 是)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-orbit-partition","seed":202611081042,"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_orbits","operation":"orbit-orbit-partition","semantic_template":"qf-mod39-orbit-orbit-partition/{\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0043","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-orbit-partition；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, 1, 2, 3], [4], [5], [6]], [0, 1, 2, 3, 4, 5, 6], 是, 是)","explanation_zh":"Wave6第二層核對=([[0, 1, 2, 3], [4], [5], [6]], [0, 1, 2, 3, 4, 5, 6], 是, 是)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-orbit-partition","seed":202611081043,"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_orbits","operation":"orbit-orbit-partition","semantic_template":"qf-mod39-orbit-orbit-partition/{\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0044","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-orbit-partition；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, 1, 2, 3, 4], [5], [6], [7]], [0, 1, 2, 3, 4, 5, 6, 7], 是, 是)","explanation_zh":"Wave6第二層核對=([[0, 1, 2, 3, 4], [5], [6], [7]], [0, 1, 2, 3, 4, 5, 6, 7], 是, 是)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-orbit-partition","seed":202611081044,"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_orbits","operation":"orbit-orbit-partition","semantic_template":"qf-mod39-orbit-orbit-partition/{\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0045","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-orbit-partition；資料={\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"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, 1, 2, 3, 4, 5], [6], [7], [8]], [0, 1, 2, 3, 4, 5, 6, 7, 8], 是, 是)","explanation_zh":"Wave6第二層核對=([[0, 1, 2, 3, 4, 5], [6], [7], [8]], [0, 1, 2, 3, 4, 5, 6, 7, 8], 是, 是)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-orbit-partition","seed":202611081045,"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_orbits","operation":"orbit-orbit-partition","semantic_template":"qf-mod39-orbit-orbit-partition/{\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0046","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-orbit-partition；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, 1], [2], [3], [4], [5]], [0, 1, 2, 3, 4, 5], 是, 是)","explanation_zh":"Wave6第二層核對=([[0, 1], [2], [3], [4], [5]], [0, 1, 2, 3, 4, 5], 是, 是)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-orbit-partition","seed":202611081046,"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_orbits","operation":"orbit-orbit-partition","semantic_template":"qf-mod39-orbit-orbit-partition/{\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0047","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-orbit-partition；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, 1, 2], [3], [4], [5], [6]], [0, 1, 2, 3, 4, 5, 6], 是, 是)","explanation_zh":"Wave6第二層核對=([[0, 1, 2], [3], [4], [5], [6]], [0, 1, 2, 3, 4, 5, 6], 是, 是)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-orbit-partition","seed":202611081047,"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_orbits","operation":"orbit-orbit-partition","semantic_template":"qf-mod39-orbit-orbit-partition/{\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0048","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-orbit-partition；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, 1, 2, 3], [4], [5], [6], [7]], [0, 1, 2, 3, 4, 5, 6, 7], 是, 是)","explanation_zh":"Wave6第二層核對=([[0, 1, 2, 3], [4], [5], [6], [7]], [0, 1, 2, 3, 4, 5, 6, 7], 是, 是)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-orbit-partition","seed":202611081048,"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_orbits","operation":"orbit-orbit-partition","semantic_template":"qf-mod39-orbit-orbit-partition/{\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0049","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-orbit-partition；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, 1, 2, 3, 4], [5], [6], [7], [8]], [0, 1, 2, 3, 4, 5, 6, 7, 8], 是, 是)","explanation_zh":"Wave6第二層核對=([[0, 1, 2, 3, 4], [5], [6], [7], [8]], [0, 1, 2, 3, 4, 5, 6, 7, 8], 是, 是)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-orbit-partition","seed":202611081049,"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_orbits","operation":"orbit-orbit-partition","semantic_template":"qf-mod39-orbit-orbit-partition/{\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0050","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-orbit-partition；資料={\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"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, 1, 2, 3, 4, 5], [6], [7], [8], [9]], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是, 是)","explanation_zh":"Wave6第二層核對=([[0, 1, 2, 3, 4, 5], [6], [7], [8], [9]], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是, 是)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-orbit-partition","seed":202611081050,"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_orbits","operation":"orbit-orbit-partition","semantic_template":"qf-mod39-orbit-orbit-partition/{\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0051","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-equivalence-audit；資料={\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":2,\"action_table\":[[0,1],[1,0]],\"x\":0,\"y\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 1, [1], 是, [0, 1], [0, 1], 是)","explanation_zh":"Wave6第二層核對=(0, 1, [1], 是, [0, 1], [0, 1], 是)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-equivalence-audit","seed":202611081051,"parameters":{"group_table":[[0,1],[1,0]],"identity":0,"point_count":2,"action_table":[[0,1],[1,0]],"x":0,"y":1,"domain":"Complete_finite_group_action_orbits","operation":"orbit-equivalence-audit","semantic_template":"qf-mod39-orbit-equivalence-audit/{\"action_table\":[[0,1],[1,0]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":2,\"x\":0,\"y\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0052","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-equivalence-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]],\"x\":1,\"y\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 2, [1], 是, [0, 1, 2], [0, 1, 2], 是)","explanation_zh":"Wave6第二層核對=(1, 2, [1], 是, [0, 1, 2], [0, 1, 2], 是)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-equivalence-audit","seed":202611081052,"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]],"x":1,"y":2,"domain":"Complete_finite_group_action_orbits","operation":"orbit-equivalence-audit","semantic_template":"qf-mod39-orbit-equivalence-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,\"x\":1,\"y\":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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0053","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-equivalence-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]],\"x\":2,\"y\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 3, [1], 是, [0, 1, 2, 3], [0, 1, 2, 3], 是)","explanation_zh":"Wave6第二層核對=(2, 3, [1], 是, [0, 1, 2, 3], [0, 1, 2, 3], 是)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-equivalence-audit","seed":202611081053,"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]],"x":2,"y":3,"domain":"Complete_finite_group_action_orbits","operation":"orbit-equivalence-audit","semantic_template":"qf-mod39-orbit-equivalence-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,\"x\":2,\"y\":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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0054","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-equivalence-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]],\"x\":3,\"y\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 4, [1], 是, [0, 1, 2, 3, 4], [0, 1, 2, 3, 4], 是)","explanation_zh":"Wave6第二層核對=(3, 4, [1], 是, [0, 1, 2, 3, 4], [0, 1, 2, 3, 4], 是)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-equivalence-audit","seed":202611081054,"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]],"x":3,"y":4,"domain":"Complete_finite_group_action_orbits","operation":"orbit-equivalence-audit","semantic_template":"qf-mod39-orbit-equivalence-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,\"x\":3,\"y\":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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0055","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-equivalence-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]],\"x\":4,\"y\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 5, [1], 是, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是)","explanation_zh":"Wave6第二層核對=(4, 5, [1], 是, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-equivalence-audit","seed":202611081055,"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]],"x":4,"y":5,"domain":"Complete_finite_group_action_orbits","operation":"orbit-equivalence-audit","semantic_template":"qf-mod39-orbit-equivalence-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,\"x\":4,\"y\":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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0056","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-equivalence-audit；資料={\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":3,\"action_table\":[[0,1,2],[1,0,2]],\"x\":2,\"y\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 0, [], 否, [2], [0, 1], 是)","explanation_zh":"Wave6第二層核對=(2, 0, [], 否, [2], [0, 1], 是)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-equivalence-audit","seed":202611081056,"parameters":{"group_table":[[0,1],[1,0]],"identity":0,"point_count":3,"action_table":[[0,1,2],[1,0,2]],"x":2,"y":0,"domain":"Complete_finite_group_action_orbits","operation":"orbit-equivalence-audit","semantic_template":"qf-mod39-orbit-equivalence-audit/{\"action_table\":[[0,1,2],[1,0,2]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":3,\"x\":2,\"y\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0057","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-equivalence-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]],\"x\":2,\"y\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 3, [], 否, [0, 1, 2], [3], 是)","explanation_zh":"Wave6第二層核對=(2, 3, [], 否, [0, 1, 2], [3], 是)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-equivalence-audit","seed":202611081057,"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]],"x":2,"y":3,"domain":"Complete_finite_group_action_orbits","operation":"orbit-equivalence-audit","semantic_template":"qf-mod39-orbit-equivalence-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,\"x\":2,\"y\":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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0058","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-equivalence-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]],\"x\":2,\"y\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 3, [1], 是, [0, 1, 2, 3], [0, 1, 2, 3], 是)","explanation_zh":"Wave6第二層核對=(2, 3, [1], 是, [0, 1, 2, 3], [0, 1, 2, 3], 是)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-equivalence-audit","seed":202611081058,"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]],"x":2,"y":3,"domain":"Complete_finite_group_action_orbits","operation":"orbit-equivalence-audit","semantic_template":"qf-mod39-orbit-equivalence-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,\"x\":2,\"y\":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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0059","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-equivalence-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]],\"x\":2,\"y\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 3, [1], 是, [0, 1, 2, 3, 4], [0, 1, 2, 3, 4], 是)","explanation_zh":"Wave6第二層核對=(2, 3, [1], 是, [0, 1, 2, 3, 4], [0, 1, 2, 3, 4], 是)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-equivalence-audit","seed":202611081059,"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]],"x":2,"y":3,"domain":"Complete_finite_group_action_orbits","operation":"orbit-equivalence-audit","semantic_template":"qf-mod39-orbit-equivalence-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,\"x\":2,\"y\":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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0060","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-equivalence-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]],\"x\":2,\"y\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 3, [1], 是, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是)","explanation_zh":"Wave6第二層核對=(2, 3, [1], 是, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-equivalence-audit","seed":202611081060,"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]],"x":2,"y":3,"domain":"Complete_finite_group_action_orbits","operation":"orbit-equivalence-audit","semantic_template":"qf-mod39-orbit-equivalence-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,\"x\":2,\"y\":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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0061","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-equivalence-audit；資料={\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":4,\"action_table\":[[0,1,2,3],[1,0,2,3]],\"x\":2,\"y\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 3, [], 否, [2], [3], 是)","explanation_zh":"Wave6第二層核對=(2, 3, [], 否, [2], [3], 是)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-equivalence-audit","seed":202611081061,"parameters":{"group_table":[[0,1],[1,0]],"identity":0,"point_count":4,"action_table":[[0,1,2,3],[1,0,2,3]],"x":2,"y":3,"domain":"Complete_finite_group_action_orbits","operation":"orbit-equivalence-audit","semantic_template":"qf-mod39-orbit-equivalence-audit/{\"action_table\":[[0,1,2,3],[1,0,2,3]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":4,\"x\":2,\"y\":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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0062","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-equivalence-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]],\"x\":1,\"y\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 2, [1], 是, [0, 1, 2], [0, 1, 2], 是)","explanation_zh":"Wave6第二層核對=(1, 2, [1], 是, [0, 1, 2], [0, 1, 2], 是)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-equivalence-audit","seed":202611081062,"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]],"x":1,"y":2,"domain":"Complete_finite_group_action_orbits","operation":"orbit-equivalence-audit","semantic_template":"qf-mod39-orbit-equivalence-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,\"x\":1,\"y\":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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0063","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-equivalence-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]],\"x\":0,\"y\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 1, [1], 是, [0, 1, 2, 3], [0, 1, 2, 3], 是)","explanation_zh":"Wave6第二層核對=(0, 1, [1], 是, [0, 1, 2, 3], [0, 1, 2, 3], 是)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-equivalence-audit","seed":202611081063,"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]],"x":0,"y":1,"domain":"Complete_finite_group_action_orbits","operation":"orbit-equivalence-audit","semantic_template":"qf-mod39-orbit-equivalence-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,\"x\":0,\"y\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0064","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-equivalence-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]],\"x\":6,\"y\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 0, [], 否, [6], [0, 1, 2, 3, 4], 是)","explanation_zh":"Wave6第二層核對=(6, 0, [], 否, [6], [0, 1, 2, 3, 4], 是)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-equivalence-audit","seed":202611081064,"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]],"x":6,"y":0,"domain":"Complete_finite_group_action_orbits","operation":"orbit-equivalence-audit","semantic_template":"qf-mod39-orbit-equivalence-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,\"x\":6,\"y\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0065","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-equivalence-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]],\"x\":6,\"y\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 7, [], 否, [6], [7], 是)","explanation_zh":"Wave6第二層核對=(6, 7, [], 否, [6], [7], 是)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-equivalence-audit","seed":202611081065,"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]],"x":6,"y":7,"domain":"Complete_finite_group_action_orbits","operation":"orbit-equivalence-audit","semantic_template":"qf-mod39-orbit-equivalence-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,\"x\":6,\"y\":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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0066","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-equivalence-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]],\"x\":0,\"y\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 1, [1], 是, [0, 1], [0, 1], 是)","explanation_zh":"Wave6第二層核對=(0, 1, [1], 是, [0, 1], [0, 1], 是)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-equivalence-audit","seed":202611081066,"parameters":{"group_table":[[0,1],[1,0]],"identity":0,"point_count":5,"action_table":[[0,1,2,3,4],[1,0,2,3,4]],"x":0,"y":1,"domain":"Complete_finite_group_action_orbits","operation":"orbit-equivalence-audit","semantic_template":"qf-mod39-orbit-equivalence-audit/{\"action_table\":[[0,1,2,3,4],[1,0,2,3,4]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":5,\"x\":0,\"y\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0067","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-equivalence-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]],\"x\":4,\"y\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 5, [], 否, [4], [5], 是)","explanation_zh":"Wave6第二層核對=(4, 5, [], 否, [4], [5], 是)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-equivalence-audit","seed":202611081067,"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]],"x":4,"y":5,"domain":"Complete_finite_group_action_orbits","operation":"orbit-equivalence-audit","semantic_template":"qf-mod39-orbit-equivalence-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,\"x\":4,\"y\":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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0068","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-equivalence-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]],\"x\":3,\"y\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 4, [], 否, [0, 1, 2, 3], [4], 是)","explanation_zh":"Wave6第二層核對=(3, 4, [], 否, [0, 1, 2, 3], [4], 是)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-equivalence-audit","seed":202611081068,"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]],"x":3,"y":4,"domain":"Complete_finite_group_action_orbits","operation":"orbit-equivalence-audit","semantic_template":"qf-mod39-orbit-equivalence-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,\"x\":3,\"y\":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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0069","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-equivalence-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]],\"x\":2,\"y\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 3, [1], 是, [0, 1, 2, 3, 4], [0, 1, 2, 3, 4], 是)","explanation_zh":"Wave6第二層核對=(2, 3, [1], 是, [0, 1, 2, 3, 4], [0, 1, 2, 3, 4], 是)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-equivalence-audit","seed":202611081069,"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]],"x":2,"y":3,"domain":"Complete_finite_group_action_orbits","operation":"orbit-equivalence-audit","semantic_template":"qf-mod39-orbit-equivalence-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,\"x\":2,\"y\":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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0070","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-equivalence-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]],\"x\":1,\"y\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 2, [1], 是, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是)","explanation_zh":"Wave6第二層核對=(1, 2, [1], 是, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-equivalence-audit","seed":202611081070,"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]],"x":1,"y":2,"domain":"Complete_finite_group_action_orbits","operation":"orbit-equivalence-audit","semantic_template":"qf-mod39-orbit-equivalence-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,\"x\":1,\"y\":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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0071","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-equivalence-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]],\"x\":2,\"y\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 3, [], 否, [2], [3], 是)","explanation_zh":"Wave6第二層核對=(2, 3, [], 否, [2], [3], 是)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-equivalence-audit","seed":202611081071,"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]],"x":2,"y":3,"domain":"Complete_finite_group_action_orbits","operation":"orbit-equivalence-audit","semantic_template":"qf-mod39-orbit-equivalence-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,\"x\":2,\"y\":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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0072","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-equivalence-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]],\"x\":0,\"y\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 1, [1], 是, [0, 1, 2], [0, 1, 2], 是)","explanation_zh":"Wave6第二層核對=(0, 1, [1], 是, [0, 1, 2], [0, 1, 2], 是)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-equivalence-audit","seed":202611081072,"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]],"x":0,"y":1,"domain":"Complete_finite_group_action_orbits","operation":"orbit-equivalence-audit","semantic_template":"qf-mod39-orbit-equivalence-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,\"x\":0,\"y\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0073","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-equivalence-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]],\"x\":6,\"y\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 7, [], 否, [6], [7], 是)","explanation_zh":"Wave6第二層核對=(6, 7, [], 否, [6], [7], 是)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-equivalence-audit","seed":202611081073,"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]],"x":6,"y":7,"domain":"Complete_finite_group_action_orbits","operation":"orbit-equivalence-audit","semantic_template":"qf-mod39-orbit-equivalence-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,\"x\":6,\"y\":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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0074","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-equivalence-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]],\"x\":5,\"y\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 6, [], 否, [5], [6], 是)","explanation_zh":"Wave6第二層核對=(5, 6, [], 否, [5], [6], 是)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-equivalence-audit","seed":202611081074,"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]],"x":5,"y":6,"domain":"Complete_finite_group_action_orbits","operation":"orbit-equivalence-audit","semantic_template":"qf-mod39-orbit-equivalence-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,\"x\":5,\"y\":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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0075","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-equivalence-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]],\"x\":4,\"y\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 5, [1], 是, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是)","explanation_zh":"Wave6第二層核對=(4, 5, [1], 是, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-equivalence-audit","seed":202611081075,"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]],"x":4,"y":5,"domain":"Complete_finite_group_action_orbits","operation":"orbit-equivalence-audit","semantic_template":"qf-mod39-orbit-equivalence-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,\"x\":4,\"y\":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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0076","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-single-step-boundary；資料={\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":2,\"action_table\":[[0,1],[1,0]],\"point\":0,\"group_element\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 0, 0, [0, 1], 2, 否, single_action_image_not_the_whole_orbit)","explanation_zh":"Wave6第二層核對=(0, 0, 0, [0, 1], 2, 否, single_action_image_not_the_whole_orbit)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-single-step-boundary","seed":202611081076,"parameters":{"group_table":[[0,1],[1,0]],"identity":0,"point_count":2,"action_table":[[0,1],[1,0]],"point":0,"group_element":0,"domain":"Complete_finite_group_action_orbits","operation":"orbit-single-step-boundary","semantic_template":"qf-mod39-orbit-single-step-boundary/{\"action_table\":[[0,1],[1,0]],\"group_element\":0,\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point\":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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0077","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-single-step-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]],\"point\":1,\"group_element\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 1, 2, [0, 1, 2], 3, 否, single_action_image_not_the_whole_orbit)","explanation_zh":"Wave6第二層核對=(1, 1, 2, [0, 1, 2], 3, 否, single_action_image_not_the_whole_orbit)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-single-step-boundary","seed":202611081077,"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]],"point":1,"group_element":1,"domain":"Complete_finite_group_action_orbits","operation":"orbit-single-step-boundary","semantic_template":"qf-mod39-orbit-single-step-boundary/{\"action_table\":[[0,1,2],[1,2,0],[2,0,1]],\"group_element\":1,\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point\":1,\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0078","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-single-step-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]],\"point\":2,\"group_element\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 0, [0, 1, 2, 3], 4, 否, single_action_image_not_the_whole_orbit)","explanation_zh":"Wave6第二層核對=(2, 2, 0, [0, 1, 2, 3], 4, 否, single_action_image_not_the_whole_orbit)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-single-step-boundary","seed":202611081078,"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]],"point":2,"group_element":2,"domain":"Complete_finite_group_action_orbits","operation":"orbit-single-step-boundary","semantic_template":"qf-mod39-orbit-single-step-boundary/{\"action_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"group_element\":2,\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point\":2,\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0079","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-single-step-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]],\"point\":3,\"group_element\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 3, 1, [0, 1, 2, 3, 4], 5, 否, single_action_image_not_the_whole_orbit)","explanation_zh":"Wave6第二層核對=(3, 3, 1, [0, 1, 2, 3, 4], 5, 否, single_action_image_not_the_whole_orbit)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-single-step-boundary","seed":202611081079,"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]],"point":3,"group_element":3,"domain":"Complete_finite_group_action_orbits","operation":"orbit-single-step-boundary","semantic_template":"qf-mod39-orbit-single-step-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_element\":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\":3,\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0080","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-single-step-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]],\"point\":4,\"group_element\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 4, 2, [0, 1, 2, 3, 4, 5], 6, 否, single_action_image_not_the_whole_orbit)","explanation_zh":"Wave6第二層核對=(4, 4, 2, [0, 1, 2, 3, 4, 5], 6, 否, single_action_image_not_the_whole_orbit)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-single-step-boundary","seed":202611081080,"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]],"point":4,"group_element":4,"domain":"Complete_finite_group_action_orbits","operation":"orbit-single-step-boundary","semantic_template":"qf-mod39-orbit-single-step-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_element\":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\":4,\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0081","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-single-step-boundary；資料={\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":3,\"action_table\":[[0,1,2],[1,0,2]],\"point\":2,\"group_element\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 2, 2, [2], 1, 否, single_action_image_not_the_whole_orbit)","explanation_zh":"Wave6第二層核對=(1, 2, 2, [2], 1, 否, single_action_image_not_the_whole_orbit)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-single-step-boundary","seed":202611081081,"parameters":{"group_table":[[0,1],[1,0]],"identity":0,"point_count":3,"action_table":[[0,1,2],[1,0,2]],"point":2,"group_element":1,"domain":"Complete_finite_group_action_orbits","operation":"orbit-single-step-boundary","semantic_template":"qf-mod39-orbit-single-step-boundary/{\"action_table\":[[0,1,2],[1,0,2]],\"group_element\":1,\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point\":2,\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0082","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-single-step-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]],\"point\":2,\"group_element\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 2, 2, [0, 1, 2], 3, 否, single_action_image_not_the_whole_orbit)","explanation_zh":"Wave6第二層核對=(0, 2, 2, [0, 1, 2], 3, 否, single_action_image_not_the_whole_orbit)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-single-step-boundary","seed":202611081082,"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]],"point":2,"group_element":0,"domain":"Complete_finite_group_action_orbits","operation":"orbit-single-step-boundary","semantic_template":"qf-mod39-orbit-single-step-boundary/{\"action_table\":[[0,1,2,3],[1,2,0,3],[2,0,1,3]],\"group_element\":0,\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point\":2,\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0083","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-single-step-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]],\"point\":2,\"group_element\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 2, 1, [0, 1, 2, 3], 4, 否, single_action_image_not_the_whole_orbit)","explanation_zh":"Wave6第二層核對=(3, 2, 1, [0, 1, 2, 3], 4, 否, single_action_image_not_the_whole_orbit)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-single-step-boundary","seed":202611081083,"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]],"point":2,"group_element":3,"domain":"Complete_finite_group_action_orbits","operation":"orbit-single-step-boundary","semantic_template":"qf-mod39-orbit-single-step-boundary/{\"action_table\":[[0,1,2,3,4],[1,2,3,0,4],[2,3,0,1,4],[3,0,1,2,4]],\"group_element\":3,\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point\":2,\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0084","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-single-step-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]],\"point\":2,\"group_element\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 2, 0, [0, 1, 2, 3, 4], 5, 否, single_action_image_not_the_whole_orbit)","explanation_zh":"Wave6第二層核對=(3, 2, 0, [0, 1, 2, 3, 4], 5, 否, single_action_image_not_the_whole_orbit)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-single-step-boundary","seed":202611081084,"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]],"point":2,"group_element":3,"domain":"Complete_finite_group_action_orbits","operation":"orbit-single-step-boundary","semantic_template":"qf-mod39-orbit-single-step-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_element\":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\":2,\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0085","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-single-step-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]],\"point\":2,\"group_element\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 2, 5, [0, 1, 2, 3, 4, 5], 6, 否, single_action_image_not_the_whole_orbit)","explanation_zh":"Wave6第二層核對=(3, 2, 5, [0, 1, 2, 3, 4, 5], 6, 否, single_action_image_not_the_whole_orbit)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-single-step-boundary","seed":202611081085,"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]],"point":2,"group_element":3,"domain":"Complete_finite_group_action_orbits","operation":"orbit-single-step-boundary","semantic_template":"qf-mod39-orbit-single-step-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_element\":3,\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point\":2,\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0086","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-single-step-boundary；資料={\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":4,\"action_table\":[[0,1,2,3],[1,0,2,3]],\"point\":2,\"group_element\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 2, 2, [2], 1, 否, single_action_image_not_the_whole_orbit)","explanation_zh":"Wave6第二層核對=(0, 2, 2, [2], 1, 否, single_action_image_not_the_whole_orbit)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-single-step-boundary","seed":202611081086,"parameters":{"group_table":[[0,1],[1,0]],"identity":0,"point_count":4,"action_table":[[0,1,2,3],[1,0,2,3]],"point":2,"group_element":0,"domain":"Complete_finite_group_action_orbits","operation":"orbit-single-step-boundary","semantic_template":"qf-mod39-orbit-single-step-boundary/{\"action_table\":[[0,1,2,3],[1,0,2,3]],\"group_element\":0,\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point\":2,\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0087","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-single-step-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]],\"point\":1,\"group_element\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 1, 0, [0, 1, 2], 3, 否, single_action_image_not_the_whole_orbit)","explanation_zh":"Wave6第二層核對=(2, 1, 0, [0, 1, 2], 3, 否, single_action_image_not_the_whole_orbit)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-single-step-boundary","seed":202611081087,"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]],"point":1,"group_element":2,"domain":"Complete_finite_group_action_orbits","operation":"orbit-single-step-boundary","semantic_template":"qf-mod39-orbit-single-step-boundary/{\"action_table\":[[0,1,2,3,4],[1,2,0,3,4],[2,0,1,3,4]],\"group_element\":2,\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point\":1,\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0088","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-single-step-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]],\"point\":0,\"group_element\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 0, 0, [0, 1, 2, 3], 4, 否, single_action_image_not_the_whole_orbit)","explanation_zh":"Wave6第二層核對=(0, 0, 0, [0, 1, 2, 3], 4, 否, single_action_image_not_the_whole_orbit)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-single-step-boundary","seed":202611081088,"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]],"point":0,"group_element":0,"domain":"Complete_finite_group_action_orbits","operation":"orbit-single-step-boundary","semantic_template":"qf-mod39-orbit-single-step-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_element\":0,\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point\":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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0089","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-single-step-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]],\"point\":6,\"group_element\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 6, 6, [6], 1, 否, single_action_image_not_the_whole_orbit)","explanation_zh":"Wave6第二層核對=(3, 6, 6, [6], 1, 否, single_action_image_not_the_whole_orbit)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-single-step-boundary","seed":202611081089,"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]],"point":6,"group_element":3,"domain":"Complete_finite_group_action_orbits","operation":"orbit-single-step-boundary","semantic_template":"qf-mod39-orbit-single-step-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_element\":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\":6,\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0090","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-single-step-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]],\"point\":6,\"group_element\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 6, 6, [6], 1, 否, single_action_image_not_the_whole_orbit)","explanation_zh":"Wave6第二層核對=(2, 6, 6, [6], 1, 否, single_action_image_not_the_whole_orbit)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-single-step-boundary","seed":202611081090,"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]],"point":6,"group_element":2,"domain":"Complete_finite_group_action_orbits","operation":"orbit-single-step-boundary","semantic_template":"qf-mod39-orbit-single-step-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_element\":2,\"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\":6,\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0091","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-single-step-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]],\"point\":0,\"group_element\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 0, 1, [0, 1], 2, 否, single_action_image_not_the_whole_orbit)","explanation_zh":"Wave6第二層核對=(1, 0, 1, [0, 1], 2, 否, single_action_image_not_the_whole_orbit)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-single-step-boundary","seed":202611081091,"parameters":{"group_table":[[0,1],[1,0]],"identity":0,"point_count":5,"action_table":[[0,1,2,3,4],[1,0,2,3,4]],"point":0,"group_element":1,"domain":"Complete_finite_group_action_orbits","operation":"orbit-single-step-boundary","semantic_template":"qf-mod39-orbit-single-step-boundary/{\"action_table\":[[0,1,2,3,4],[1,0,2,3,4]],\"group_element\":1,\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point\":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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0092","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-single-step-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]],\"point\":4,\"group_element\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 4, 4, [4], 1, 否, single_action_image_not_the_whole_orbit)","explanation_zh":"Wave6第二層核對=(1, 4, 4, [4], 1, 否, single_action_image_not_the_whole_orbit)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-single-step-boundary","seed":202611081092,"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]],"point":4,"group_element":1,"domain":"Complete_finite_group_action_orbits","operation":"orbit-single-step-boundary","semantic_template":"qf-mod39-orbit-single-step-boundary/{\"action_table\":[[0,1,2,3,4,5],[1,2,0,3,4,5],[2,0,1,3,4,5]],\"group_element\":1,\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point\":4,\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0093","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-single-step-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]],\"point\":3,\"group_element\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 3, 0, [0, 1, 2, 3], 4, 否, single_action_image_not_the_whole_orbit)","explanation_zh":"Wave6第二層核對=(1, 3, 0, [0, 1, 2, 3], 4, 否, single_action_image_not_the_whole_orbit)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-single-step-boundary","seed":202611081093,"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]],"point":3,"group_element":1,"domain":"Complete_finite_group_action_orbits","operation":"orbit-single-step-boundary","semantic_template":"qf-mod39-orbit-single-step-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_element\":1,\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point\":3,\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0094","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-single-step-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]],\"point\":2,\"group_element\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 2, 0, [0, 1, 2, 3, 4], 5, 否, single_action_image_not_the_whole_orbit)","explanation_zh":"Wave6第二層核對=(3, 2, 0, [0, 1, 2, 3, 4], 5, 否, single_action_image_not_the_whole_orbit)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-single-step-boundary","seed":202611081094,"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]],"point":2,"group_element":3,"domain":"Complete_finite_group_action_orbits","operation":"orbit-single-step-boundary","semantic_template":"qf-mod39-orbit-single-step-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_element\":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\":2,\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0095","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-single-step-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]],\"point\":1,\"group_element\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 1, 2, [0, 1, 2, 3, 4, 5], 6, 否, single_action_image_not_the_whole_orbit)","explanation_zh":"Wave6第二層核對=(1, 1, 2, [0, 1, 2, 3, 4, 5], 6, 否, single_action_image_not_the_whole_orbit)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-single-step-boundary","seed":202611081095,"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]],"point":1,"group_element":1,"domain":"Complete_finite_group_action_orbits","operation":"orbit-single-step-boundary","semantic_template":"qf-mod39-orbit-single-step-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_element\":1,\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point\":1,\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0096","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-single-step-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]],\"point\":2,\"group_element\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 2, 2, [2], 1, 否, single_action_image_not_the_whole_orbit)","explanation_zh":"Wave6第二層核對=(0, 2, 2, [2], 1, 否, single_action_image_not_the_whole_orbit)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-single-step-boundary","seed":202611081096,"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]],"point":2,"group_element":0,"domain":"Complete_finite_group_action_orbits","operation":"orbit-single-step-boundary","semantic_template":"qf-mod39-orbit-single-step-boundary/{\"action_table\":[[0,1,2,3,4,5],[1,0,2,3,4,5]],\"group_element\":0,\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point\":2,\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0097","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-single-step-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]],\"point\":0,\"group_element\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 0, 0, [0, 1, 2], 3, 否, single_action_image_not_the_whole_orbit)","explanation_zh":"Wave6第二層核對=(0, 0, 0, [0, 1, 2], 3, 否, single_action_image_not_the_whole_orbit)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-single-step-boundary","seed":202611081097,"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]],"point":0,"group_element":0,"domain":"Complete_finite_group_action_orbits","operation":"orbit-single-step-boundary","semantic_template":"qf-mod39-orbit-single-step-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_element\":0,\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point\":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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0098","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-single-step-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]],\"point\":6,\"group_element\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 6, 6, [6], 1, 否, single_action_image_not_the_whole_orbit)","explanation_zh":"Wave6第二層核對=(2, 6, 6, [6], 1, 否, single_action_image_not_the_whole_orbit)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-single-step-boundary","seed":202611081098,"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]],"point":6,"group_element":2,"domain":"Complete_finite_group_action_orbits","operation":"orbit-single-step-boundary","semantic_template":"qf-mod39-orbit-single-step-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_element\":2,\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point\":6,\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0099","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-single-step-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]],\"point\":5,\"group_element\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 5, 5, [5], 1, 否, single_action_image_not_the_whole_orbit)","explanation_zh":"Wave6第二層核對=(3, 5, 5, [5], 1, 否, single_action_image_not_the_whole_orbit)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-single-step-boundary","seed":202611081099,"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]],"point":5,"group_element":3,"domain":"Complete_finite_group_action_orbits","operation":"orbit-single-step-boundary","semantic_template":"qf-mod39-orbit-single-step-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_element\":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\":5,\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r01-finite-group-orbit-0100","batch_id":"batch-20261014-advance-20261108-r01-finite-group-orbit-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-orbit"],"title_zh":"有限群作用的軌道","stem_zh":"完整已認證有限群作用與全部群元素。操作=orbit-single-step-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]],\"point\":4,\"group_element\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 4, 4, [0, 1, 2, 3, 4, 5], 6, 否, single_action_image_not_the_whole_orbit)","explanation_zh":"Wave6第二層核對=(0, 4, 4, [0, 1, 2, 3, 4, 5], 6, 否, single_action_image_not_the_whole_orbit)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的軌道"]},"variant":{"family_id":"qf-mod39-orbit-single-step-boundary","seed":202611081100,"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]],"point":4,"group_element":0,"domain":"Complete_finite_group_action_orbits","operation":"orbit-single-step-boundary","semantic_template":"qf-mod39-orbit-single-step-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_element\":0,\"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\":4,\"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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}