{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0001","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-counting-identity；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":"(2, 0, [0, 1], 2, [0], 1, 2, 是)","explanation_zh":"Wave6第四層核對=(2, 0, [0, 1], 2, [0], 1, 2, 是)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-counting-identity","seed":202611101001,"parameters":{"group_table":[[0,1],[1,0]],"identity":0,"point_count":2,"action_table":[[0,1],[1,0]],"point":0,"domain":"Complete_finite_action_orbit_stabilizer_bijection","operation":"orbitstab-counting-identity","semantic_template":"qf-mod41-orbitstab-counting-identity/{\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0002","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-counting-identity；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":"(3, 1, [0, 1, 2], 3, [0], 1, 3, 是)","explanation_zh":"Wave6第四層核對=(3, 1, [0, 1, 2], 3, [0], 1, 3, 是)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-counting-identity","seed":202611101002,"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_action_orbit_stabilizer_bijection","operation":"orbitstab-counting-identity","semantic_template":"qf-mod41-orbitstab-counting-identity/{\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0003","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-counting-identity；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":"(4, 2, [0, 1, 2, 3], 4, [0], 1, 4, 是)","explanation_zh":"Wave6第四層核對=(4, 2, [0, 1, 2, 3], 4, [0], 1, 4, 是)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-counting-identity","seed":202611101003,"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_action_orbit_stabilizer_bijection","operation":"orbitstab-counting-identity","semantic_template":"qf-mod41-orbitstab-counting-identity/{\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0004","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-counting-identity；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":"(5, 3, [0, 1, 2, 3, 4], 5, [0], 1, 5, 是)","explanation_zh":"Wave6第四層核對=(5, 3, [0, 1, 2, 3, 4], 5, [0], 1, 5, 是)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-counting-identity","seed":202611101004,"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_action_orbit_stabilizer_bijection","operation":"orbitstab-counting-identity","semantic_template":"qf-mod41-orbitstab-counting-identity/{\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0005","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-counting-identity；資料={\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"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":"(6, 4, [0, 1, 2, 3, 4, 5], 6, [0], 1, 6, 是)","explanation_zh":"Wave6第四層核對=(6, 4, [0, 1, 2, 3, 4, 5], 6, [0], 1, 6, 是)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-counting-identity","seed":202611101005,"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_action_orbit_stabilizer_bijection","operation":"orbitstab-counting-identity","semantic_template":"qf-mod41-orbitstab-counting-identity/{\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0006","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-counting-identity；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], 1, [0, 1], 2, 2, 是)","explanation_zh":"Wave6第四層核對=(2, 2, [2], 1, [0, 1], 2, 2, 是)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-counting-identity","seed":202611101006,"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_action_orbit_stabilizer_bijection","operation":"orbitstab-counting-identity","semantic_template":"qf-mod41-orbitstab-counting-identity/{\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0007","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-counting-identity；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":"(3, 2, [0, 1, 2], 3, [0], 1, 3, 是)","explanation_zh":"Wave6第四層核對=(3, 2, [0, 1, 2], 3, [0], 1, 3, 是)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-counting-identity","seed":202611101007,"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_action_orbit_stabilizer_bijection","operation":"orbitstab-counting-identity","semantic_template":"qf-mod41-orbitstab-counting-identity/{\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0008","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-counting-identity；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":"(4, 2, [0, 1, 2, 3], 4, [0], 1, 4, 是)","explanation_zh":"Wave6第四層核對=(4, 2, [0, 1, 2, 3], 4, [0], 1, 4, 是)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-counting-identity","seed":202611101008,"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_action_orbit_stabilizer_bijection","operation":"orbitstab-counting-identity","semantic_template":"qf-mod41-orbitstab-counting-identity/{\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0009","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-counting-identity；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":"(5, 2, [0, 1, 2, 3, 4], 5, [0], 1, 5, 是)","explanation_zh":"Wave6第四層核對=(5, 2, [0, 1, 2, 3, 4], 5, [0], 1, 5, 是)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-counting-identity","seed":202611101009,"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_action_orbit_stabilizer_bijection","operation":"orbitstab-counting-identity","semantic_template":"qf-mod41-orbitstab-counting-identity/{\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0010","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-counting-identity；資料={\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"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":"(6, 2, [0, 1, 2, 3, 4, 5], 6, [0], 1, 6, 是)","explanation_zh":"Wave6第四層核對=(6, 2, [0, 1, 2, 3, 4, 5], 6, [0], 1, 6, 是)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-counting-identity","seed":202611101010,"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_action_orbit_stabilizer_bijection","operation":"orbitstab-counting-identity","semantic_template":"qf-mod41-orbitstab-counting-identity/{\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0011","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-counting-identity；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], 1, [0, 1], 2, 2, 是)","explanation_zh":"Wave6第四層核對=(2, 2, [2], 1, [0, 1], 2, 2, 是)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-counting-identity","seed":202611101011,"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_action_orbit_stabilizer_bijection","operation":"orbitstab-counting-identity","semantic_template":"qf-mod41-orbitstab-counting-identity/{\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0012","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-counting-identity；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":"(3, 1, [0, 1, 2], 3, [0], 1, 3, 是)","explanation_zh":"Wave6第四層核對=(3, 1, [0, 1, 2], 3, [0], 1, 3, 是)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-counting-identity","seed":202611101012,"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_action_orbit_stabilizer_bijection","operation":"orbitstab-counting-identity","semantic_template":"qf-mod41-orbitstab-counting-identity/{\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0013","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-counting-identity；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":"(4, 0, [0, 1, 2, 3], 4, [0], 1, 4, 是)","explanation_zh":"Wave6第四層核對=(4, 0, [0, 1, 2, 3], 4, [0], 1, 4, 是)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-counting-identity","seed":202611101013,"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_action_orbit_stabilizer_bijection","operation":"orbitstab-counting-identity","semantic_template":"qf-mod41-orbitstab-counting-identity/{\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0014","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-counting-identity；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":"(5, 6, [6], 1, [0, 1, 2, 3, 4], 5, 5, 是)","explanation_zh":"Wave6第四層核對=(5, 6, [6], 1, [0, 1, 2, 3, 4], 5, 5, 是)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-counting-identity","seed":202611101014,"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_action_orbit_stabilizer_bijection","operation":"orbitstab-counting-identity","semantic_template":"qf-mod41-orbitstab-counting-identity/{\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0015","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-counting-identity；資料={\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"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], 1, [0, 1, 2, 3, 4, 5], 6, 6, 是)","explanation_zh":"Wave6第四層核對=(6, 6, [6], 1, [0, 1, 2, 3, 4, 5], 6, 6, 是)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-counting-identity","seed":202611101015,"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_action_orbit_stabilizer_bijection","operation":"orbitstab-counting-identity","semantic_template":"qf-mod41-orbitstab-counting-identity/{\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0016","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-counting-identity；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":"(2, 0, [0, 1], 2, [0], 1, 2, 是)","explanation_zh":"Wave6第四層核對=(2, 0, [0, 1], 2, [0], 1, 2, 是)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-counting-identity","seed":202611101016,"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_action_orbit_stabilizer_bijection","operation":"orbitstab-counting-identity","semantic_template":"qf-mod41-orbitstab-counting-identity/{\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0017","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-counting-identity；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":"(3, 4, [4], 1, [0, 1, 2], 3, 3, 是)","explanation_zh":"Wave6第四層核對=(3, 4, [4], 1, [0, 1, 2], 3, 3, 是)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-counting-identity","seed":202611101017,"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_action_orbit_stabilizer_bijection","operation":"orbitstab-counting-identity","semantic_template":"qf-mod41-orbitstab-counting-identity/{\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0018","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-counting-identity；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":"(4, 3, [0, 1, 2, 3], 4, [0], 1, 4, 是)","explanation_zh":"Wave6第四層核對=(4, 3, [0, 1, 2, 3], 4, [0], 1, 4, 是)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-counting-identity","seed":202611101018,"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_action_orbit_stabilizer_bijection","operation":"orbitstab-counting-identity","semantic_template":"qf-mod41-orbitstab-counting-identity/{\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0019","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-counting-identity；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":"(5, 2, [0, 1, 2, 3, 4], 5, [0], 1, 5, 是)","explanation_zh":"Wave6第四層核對=(5, 2, [0, 1, 2, 3, 4], 5, [0], 1, 5, 是)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-counting-identity","seed":202611101019,"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_action_orbit_stabilizer_bijection","operation":"orbitstab-counting-identity","semantic_template":"qf-mod41-orbitstab-counting-identity/{\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0020","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-counting-identity；資料={\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"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":"(6, 1, [0, 1, 2, 3, 4, 5], 6, [0], 1, 6, 是)","explanation_zh":"Wave6第四層核對=(6, 1, [0, 1, 2, 3, 4, 5], 6, [0], 1, 6, 是)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-counting-identity","seed":202611101020,"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_action_orbit_stabilizer_bijection","operation":"orbitstab-counting-identity","semantic_template":"qf-mod41-orbitstab-counting-identity/{\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0021","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-counting-identity；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], 1, [0, 1], 2, 2, 是)","explanation_zh":"Wave6第四層核對=(2, 2, [2], 1, [0, 1], 2, 2, 是)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-counting-identity","seed":202611101021,"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_action_orbit_stabilizer_bijection","operation":"orbitstab-counting-identity","semantic_template":"qf-mod41-orbitstab-counting-identity/{\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0022","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-counting-identity；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":"(3, 0, [0, 1, 2], 3, [0], 1, 3, 是)","explanation_zh":"Wave6第四層核對=(3, 0, [0, 1, 2], 3, [0], 1, 3, 是)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-counting-identity","seed":202611101022,"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_action_orbit_stabilizer_bijection","operation":"orbitstab-counting-identity","semantic_template":"qf-mod41-orbitstab-counting-identity/{\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0023","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-counting-identity；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":"(4, 6, [6], 1, [0, 1, 2, 3], 4, 4, 是)","explanation_zh":"Wave6第四層核對=(4, 6, [6], 1, [0, 1, 2, 3], 4, 4, 是)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-counting-identity","seed":202611101023,"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_action_orbit_stabilizer_bijection","operation":"orbitstab-counting-identity","semantic_template":"qf-mod41-orbitstab-counting-identity/{\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0024","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-counting-identity；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], 1, [0, 1, 2, 3, 4], 5, 5, 是)","explanation_zh":"Wave6第四層核對=(5, 5, [5], 1, [0, 1, 2, 3, 4], 5, 5, 是)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-counting-identity","seed":202611101024,"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_action_orbit_stabilizer_bijection","operation":"orbitstab-counting-identity","semantic_template":"qf-mod41-orbitstab-counting-identity/{\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0025","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-counting-identity；資料={\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"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":"(6, 4, [0, 1, 2, 3, 4, 5], 6, [0], 1, 6, 是)","explanation_zh":"Wave6第四層核對=(6, 4, [0, 1, 2, 3, 4, 5], 6, [0], 1, 6, 是)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-counting-identity","seed":202611101025,"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_action_orbit_stabilizer_bijection","operation":"orbitstab-counting-identity","semantic_template":"qf-mod41-orbitstab-counting-identity/{\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0026","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-action-fibers；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, 1]], [[0, [0]], [1, [1]]], 是, [0, 1], action_map_surjective_onto_orbit)","explanation_zh":"Wave6第四層核對=([[0, 0], [1, 1]], [[0, [0]], [1, [1]]], 是, [0, 1], action_map_surjective_onto_orbit)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-action-fibers","seed":202611101026,"parameters":{"group_table":[[0,1],[1,0]],"identity":0,"point_count":2,"action_table":[[0,1],[1,0]],"point":0,"domain":"Complete_finite_action_orbit_stabilizer_bijection","operation":"orbitstab-action-fibers","semantic_template":"qf-mod41-orbitstab-action-fibers/{\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0027","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-action-fibers；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":"([[0, 1], [1, 2], [2, 0]], [[0, [2]], [1, [0]], [2, [1]]], 是, [0, 1, 2], action_map_surjective_onto_orbit)","explanation_zh":"Wave6第四層核對=([[0, 1], [1, 2], [2, 0]], [[0, [2]], [1, [0]], [2, [1]]], 是, [0, 1, 2], action_map_surjective_onto_orbit)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-action-fibers","seed":202611101027,"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_action_orbit_stabilizer_bijection","operation":"orbitstab-action-fibers","semantic_template":"qf-mod41-orbitstab-action-fibers/{\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0028","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-action-fibers；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":"([[0, 2], [1, 3], [2, 0], [3, 1]], [[0, [2]], [1, [3]], [2, [0]], [3, [1]]], 是, [0, 1, 2, 3], action_map_surjective_onto_orbit)","explanation_zh":"Wave6第四層核對=([[0, 2], [1, 3], [2, 0], [3, 1]], [[0, [2]], [1, [3]], [2, [0]], [3, [1]]], 是, [0, 1, 2, 3], action_map_surjective_onto_orbit)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-action-fibers","seed":202611101028,"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_action_orbit_stabilizer_bijection","operation":"orbitstab-action-fibers","semantic_template":"qf-mod41-orbitstab-action-fibers/{\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0029","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-action-fibers；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":"([[0, 3], [1, 4], [2, 0], [3, 1], [4, 2]], [[0, [2]], [1, [3]], [2, [4]], [3, [0]], [4, [1]]], 是, [0, 1, 2, 3, 4], action_map_surjective_onto_orbit)","explanation_zh":"Wave6第四層核對=([[0, 3], [1, 4], [2, 0], [3, 1], [4, 2]], [[0, [2]], [1, [3]], [2, [4]], [3, [0]], [4, [1]]], 是, [0, 1, 2, 3, 4], action_map_surjective_onto_orbit)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-action-fibers","seed":202611101029,"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_action_orbit_stabilizer_bijection","operation":"orbitstab-action-fibers","semantic_template":"qf-mod41-orbitstab-action-fibers/{\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0030","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-action-fibers；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":"([[0, 4], [1, 5], [2, 0], [3, 1], [4, 2], [5, 3]], [[0, [2]], [1, [3]], [2, [4]], [3, [5]], [4, [0]], [5, [1]]], 是, [0, 1, 2, 3, 4, 5], action_map_surjective_onto_orbit)","explanation_zh":"Wave6第四層核對=([[0, 4], [1, 5], [2, 0], [3, 1], [4, 2], [5, 3]], [[0, [2]], [1, [3]], [2, [4]], [3, [5]], [4, [0]], [5, [1]]], 是, [0, 1, 2, 3, 4, 5], action_map_surjective_onto_orbit)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-action-fibers","seed":202611101030,"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_action_orbit_stabilizer_bijection","operation":"orbitstab-action-fibers","semantic_template":"qf-mod41-orbitstab-action-fibers/{\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0031","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-action-fibers；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":"([[0, 2], [1, 2]], [[2, [0, 1]]], 是, [2], action_map_surjective_onto_orbit)","explanation_zh":"Wave6第四層核對=([[0, 2], [1, 2]], [[2, [0, 1]]], 是, [2], action_map_surjective_onto_orbit)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-action-fibers","seed":202611101031,"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_action_orbit_stabilizer_bijection","operation":"orbitstab-action-fibers","semantic_template":"qf-mod41-orbitstab-action-fibers/{\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0032","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-action-fibers；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":"([[0, 2], [1, 0], [2, 1]], [[0, [1]], [1, [2]], [2, [0]]], 是, [0, 1, 2], action_map_surjective_onto_orbit)","explanation_zh":"Wave6第四層核對=([[0, 2], [1, 0], [2, 1]], [[0, [1]], [1, [2]], [2, [0]]], 是, [0, 1, 2], action_map_surjective_onto_orbit)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-action-fibers","seed":202611101032,"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_action_orbit_stabilizer_bijection","operation":"orbitstab-action-fibers","semantic_template":"qf-mod41-orbitstab-action-fibers/{\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0033","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-action-fibers；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":"([[0, 2], [1, 3], [2, 0], [3, 1]], [[0, [2]], [1, [3]], [2, [0]], [3, [1]]], 是, [0, 1, 2, 3], action_map_surjective_onto_orbit)","explanation_zh":"Wave6第四層核對=([[0, 2], [1, 3], [2, 0], [3, 1]], [[0, [2]], [1, [3]], [2, [0]], [3, [1]]], 是, [0, 1, 2, 3], action_map_surjective_onto_orbit)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-action-fibers","seed":202611101033,"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_action_orbit_stabilizer_bijection","operation":"orbitstab-action-fibers","semantic_template":"qf-mod41-orbitstab-action-fibers/{\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0034","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-action-fibers；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":"([[0, 2], [1, 3], [2, 4], [3, 0], [4, 1]], [[0, [3]], [1, [4]], [2, [0]], [3, [1]], [4, [2]]], 是, [0, 1, 2, 3, 4], action_map_surjective_onto_orbit)","explanation_zh":"Wave6第四層核對=([[0, 2], [1, 3], [2, 4], [3, 0], [4, 1]], [[0, [3]], [1, [4]], [2, [0]], [3, [1]], [4, [2]]], 是, [0, 1, 2, 3, 4], action_map_surjective_onto_orbit)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-action-fibers","seed":202611101034,"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_action_orbit_stabilizer_bijection","operation":"orbitstab-action-fibers","semantic_template":"qf-mod41-orbitstab-action-fibers/{\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0035","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-action-fibers；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":"([[0, 2], [1, 3], [2, 4], [3, 5], [4, 0], [5, 1]], [[0, [4]], [1, [5]], [2, [0]], [3, [1]], [4, [2]], [5, [3]]], 是, [0, 1, 2, 3, 4, 5], action_map_surjective_onto_orbit)","explanation_zh":"Wave6第四層核對=([[0, 2], [1, 3], [2, 4], [3, 5], [4, 0], [5, 1]], [[0, [4]], [1, [5]], [2, [0]], [3, [1]], [4, [2]], [5, [3]]], 是, [0, 1, 2, 3, 4, 5], action_map_surjective_onto_orbit)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-action-fibers","seed":202611101035,"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_action_orbit_stabilizer_bijection","operation":"orbitstab-action-fibers","semantic_template":"qf-mod41-orbitstab-action-fibers/{\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0036","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-action-fibers；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":"([[0, 2], [1, 2]], [[2, [0, 1]]], 是, [2], action_map_surjective_onto_orbit)","explanation_zh":"Wave6第四層核對=([[0, 2], [1, 2]], [[2, [0, 1]]], 是, [2], action_map_surjective_onto_orbit)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-action-fibers","seed":202611101036,"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_action_orbit_stabilizer_bijection","operation":"orbitstab-action-fibers","semantic_template":"qf-mod41-orbitstab-action-fibers/{\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0037","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-action-fibers；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":"([[0, 1], [1, 2], [2, 0]], [[0, [2]], [1, [0]], [2, [1]]], 是, [0, 1, 2], action_map_surjective_onto_orbit)","explanation_zh":"Wave6第四層核對=([[0, 1], [1, 2], [2, 0]], [[0, [2]], [1, [0]], [2, [1]]], 是, [0, 1, 2], action_map_surjective_onto_orbit)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-action-fibers","seed":202611101037,"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_action_orbit_stabilizer_bijection","operation":"orbitstab-action-fibers","semantic_template":"qf-mod41-orbitstab-action-fibers/{\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0038","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-action-fibers；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, 1], [2, 2], [3, 3]], [[0, [0]], [1, [1]], [2, [2]], [3, [3]]], 是, [0, 1, 2, 3], action_map_surjective_onto_orbit)","explanation_zh":"Wave6第四層核對=([[0, 0], [1, 1], [2, 2], [3, 3]], [[0, [0]], [1, [1]], [2, [2]], [3, [3]]], 是, [0, 1, 2, 3], action_map_surjective_onto_orbit)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-action-fibers","seed":202611101038,"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_action_orbit_stabilizer_bijection","operation":"orbitstab-action-fibers","semantic_template":"qf-mod41-orbitstab-action-fibers/{\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0039","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-action-fibers；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":"([[0, 6], [1, 6], [2, 6], [3, 6], [4, 6]], [[6, [0, 1, 2, 3, 4]]], 是, [6], action_map_surjective_onto_orbit)","explanation_zh":"Wave6第四層核對=([[0, 6], [1, 6], [2, 6], [3, 6], [4, 6]], [[6, [0, 1, 2, 3, 4]]], 是, [6], action_map_surjective_onto_orbit)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-action-fibers","seed":202611101039,"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_action_orbit_stabilizer_bijection","operation":"orbitstab-action-fibers","semantic_template":"qf-mod41-orbitstab-action-fibers/{\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0040","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-action-fibers；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":"([[0, 6], [1, 6], [2, 6], [3, 6], [4, 6], [5, 6]], [[6, [0, 1, 2, 3, 4, 5]]], 是, [6], action_map_surjective_onto_orbit)","explanation_zh":"Wave6第四層核對=([[0, 6], [1, 6], [2, 6], [3, 6], [4, 6], [5, 6]], [[6, [0, 1, 2, 3, 4, 5]]], 是, [6], action_map_surjective_onto_orbit)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-action-fibers","seed":202611101040,"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_action_orbit_stabilizer_bijection","operation":"orbitstab-action-fibers","semantic_template":"qf-mod41-orbitstab-action-fibers/{\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0041","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-action-fibers；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, 1]], [[0, [0]], [1, [1]]], 是, [0, 1], action_map_surjective_onto_orbit)","explanation_zh":"Wave6第四層核對=([[0, 0], [1, 1]], [[0, [0]], [1, [1]]], 是, [0, 1], action_map_surjective_onto_orbit)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-action-fibers","seed":202611101041,"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_action_orbit_stabilizer_bijection","operation":"orbitstab-action-fibers","semantic_template":"qf-mod41-orbitstab-action-fibers/{\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0042","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-action-fibers；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":"([[0, 4], [1, 4], [2, 4]], [[4, [0, 1, 2]]], 是, [4], action_map_surjective_onto_orbit)","explanation_zh":"Wave6第四層核對=([[0, 4], [1, 4], [2, 4]], [[4, [0, 1, 2]]], 是, [4], action_map_surjective_onto_orbit)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-action-fibers","seed":202611101042,"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_action_orbit_stabilizer_bijection","operation":"orbitstab-action-fibers","semantic_template":"qf-mod41-orbitstab-action-fibers/{\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0043","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-action-fibers；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":"([[0, 3], [1, 0], [2, 1], [3, 2]], [[0, [1]], [1, [2]], [2, [3]], [3, [0]]], 是, [0, 1, 2, 3], action_map_surjective_onto_orbit)","explanation_zh":"Wave6第四層核對=([[0, 3], [1, 0], [2, 1], [3, 2]], [[0, [1]], [1, [2]], [2, [3]], [3, [0]]], 是, [0, 1, 2, 3], action_map_surjective_onto_orbit)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-action-fibers","seed":202611101043,"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_action_orbit_stabilizer_bijection","operation":"orbitstab-action-fibers","semantic_template":"qf-mod41-orbitstab-action-fibers/{\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0044","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-action-fibers；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":"([[0, 2], [1, 3], [2, 4], [3, 0], [4, 1]], [[0, [3]], [1, [4]], [2, [0]], [3, [1]], [4, [2]]], 是, [0, 1, 2, 3, 4], action_map_surjective_onto_orbit)","explanation_zh":"Wave6第四層核對=([[0, 2], [1, 3], [2, 4], [3, 0], [4, 1]], [[0, [3]], [1, [4]], [2, [0]], [3, [1]], [4, [2]]], 是, [0, 1, 2, 3, 4], action_map_surjective_onto_orbit)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-action-fibers","seed":202611101044,"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_action_orbit_stabilizer_bijection","operation":"orbitstab-action-fibers","semantic_template":"qf-mod41-orbitstab-action-fibers/{\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0045","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-action-fibers；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":"([[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 0]], [[0, [5]], [1, [0]], [2, [1]], [3, [2]], [4, [3]], [5, [4]]], 是, [0, 1, 2, 3, 4, 5], action_map_surjective_onto_orbit)","explanation_zh":"Wave6第四層核對=([[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 0]], [[0, [5]], [1, [0]], [2, [1]], [3, [2]], [4, [3]], [5, [4]]], 是, [0, 1, 2, 3, 4, 5], action_map_surjective_onto_orbit)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-action-fibers","seed":202611101045,"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_action_orbit_stabilizer_bijection","operation":"orbitstab-action-fibers","semantic_template":"qf-mod41-orbitstab-action-fibers/{\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0046","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-action-fibers；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":"([[0, 2], [1, 2]], [[2, [0, 1]]], 是, [2], action_map_surjective_onto_orbit)","explanation_zh":"Wave6第四層核對=([[0, 2], [1, 2]], [[2, [0, 1]]], 是, [2], action_map_surjective_onto_orbit)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-action-fibers","seed":202611101046,"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_action_orbit_stabilizer_bijection","operation":"orbitstab-action-fibers","semantic_template":"qf-mod41-orbitstab-action-fibers/{\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0047","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-action-fibers；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, 1], [2, 2]], [[0, [0]], [1, [1]], [2, [2]]], 是, [0, 1, 2], action_map_surjective_onto_orbit)","explanation_zh":"Wave6第四層核對=([[0, 0], [1, 1], [2, 2]], [[0, [0]], [1, [1]], [2, [2]]], 是, [0, 1, 2], action_map_surjective_onto_orbit)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-action-fibers","seed":202611101047,"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_action_orbit_stabilizer_bijection","operation":"orbitstab-action-fibers","semantic_template":"qf-mod41-orbitstab-action-fibers/{\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0048","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-action-fibers；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":"([[0, 6], [1, 6], [2, 6], [3, 6]], [[6, [0, 1, 2, 3]]], 是, [6], action_map_surjective_onto_orbit)","explanation_zh":"Wave6第四層核對=([[0, 6], [1, 6], [2, 6], [3, 6]], [[6, [0, 1, 2, 3]]], 是, [6], action_map_surjective_onto_orbit)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-action-fibers","seed":202611101048,"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_action_orbit_stabilizer_bijection","operation":"orbitstab-action-fibers","semantic_template":"qf-mod41-orbitstab-action-fibers/{\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0049","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-action-fibers；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":"([[0, 5], [1, 5], [2, 5], [3, 5], [4, 5]], [[5, [0, 1, 2, 3, 4]]], 是, [5], action_map_surjective_onto_orbit)","explanation_zh":"Wave6第四層核對=([[0, 5], [1, 5], [2, 5], [3, 5], [4, 5]], [[5, [0, 1, 2, 3, 4]]], 是, [5], action_map_surjective_onto_orbit)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-action-fibers","seed":202611101049,"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_action_orbit_stabilizer_bijection","operation":"orbitstab-action-fibers","semantic_template":"qf-mod41-orbitstab-action-fibers/{\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0050","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-action-fibers；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":"([[0, 4], [1, 5], [2, 0], [3, 1], [4, 2], [5, 3]], [[0, [2]], [1, [3]], [2, [4]], [3, [5]], [4, [0]], [5, [1]]], 是, [0, 1, 2, 3, 4, 5], action_map_surjective_onto_orbit)","explanation_zh":"Wave6第四層核對=([[0, 4], [1, 5], [2, 0], [3, 1], [4, 2], [5, 3]], [[0, [2]], [1, [3]], [2, [4]], [3, [5]], [4, [0]], [5, [1]]], 是, [0, 1, 2, 3, 4, 5], action_map_surjective_onto_orbit)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-action-fibers","seed":202611101050,"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_action_orbit_stabilizer_bijection","operation":"orbitstab-action-fibers","semantic_template":"qf-mod41-orbitstab-action-fibers/{\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0051","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-coset-fiber-bijection；資料={\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":2,\"action_table\":[[0,1],[1,0]],\"point\":0,\"orbit_point\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 1, [1], [1], 是, fiber_equals_left_coset_of_same_point_stabilizer)","explanation_zh":"Wave6第四層核對=(1, 1, [1], [1], 是, fiber_equals_left_coset_of_same_point_stabilizer)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-coset-fiber-bijection","seed":202611101051,"parameters":{"group_table":[[0,1],[1,0]],"identity":0,"point_count":2,"action_table":[[0,1],[1,0]],"point":0,"orbit_point":1,"domain":"Complete_finite_action_orbit_stabilizer_bijection","operation":"orbitstab-coset-fiber-bijection","semantic_template":"qf-mod41-orbitstab-coset-fiber-bijection/{\"action_table\":[[0,1],[1,0]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"orbit_point\":1,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0052","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-coset-fiber-bijection；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,\"orbit_point\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 2, [2], [2], 是, fiber_equals_left_coset_of_same_point_stabilizer)","explanation_zh":"Wave6第四層核對=(0, 2, [2], [2], 是, fiber_equals_left_coset_of_same_point_stabilizer)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-coset-fiber-bijection","seed":202611101052,"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,"orbit_point":0,"domain":"Complete_finite_action_orbit_stabilizer_bijection","operation":"orbitstab-coset-fiber-bijection","semantic_template":"qf-mod41-orbitstab-coset-fiber-bijection/{\"action_table\":[[0,1,2],[1,2,0],[2,0,1]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"orbit_point\":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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0053","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-coset-fiber-bijection；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,\"orbit_point\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 3, [3], [3], 是, fiber_equals_left_coset_of_same_point_stabilizer)","explanation_zh":"Wave6第四層核對=(1, 3, [3], [3], 是, fiber_equals_left_coset_of_same_point_stabilizer)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-coset-fiber-bijection","seed":202611101053,"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,"orbit_point":1,"domain":"Complete_finite_action_orbit_stabilizer_bijection","operation":"orbitstab-coset-fiber-bijection","semantic_template":"qf-mod41-orbitstab-coset-fiber-bijection/{\"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,\"orbit_point\":1,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0054","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-coset-fiber-bijection；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,\"orbit_point\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 4, [4], [4], 是, fiber_equals_left_coset_of_same_point_stabilizer)","explanation_zh":"Wave6第四層核對=(2, 4, [4], [4], 是, fiber_equals_left_coset_of_same_point_stabilizer)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-coset-fiber-bijection","seed":202611101054,"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,"orbit_point":2,"domain":"Complete_finite_action_orbit_stabilizer_bijection","operation":"orbitstab-coset-fiber-bijection","semantic_template":"qf-mod41-orbitstab-coset-fiber-bijection/{\"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,\"orbit_point\":2,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0055","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-coset-fiber-bijection；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,\"orbit_point\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 5, [5], [5], 是, fiber_equals_left_coset_of_same_point_stabilizer)","explanation_zh":"Wave6第四層核對=(3, 5, [5], [5], 是, fiber_equals_left_coset_of_same_point_stabilizer)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-coset-fiber-bijection","seed":202611101055,"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,"orbit_point":3,"domain":"Complete_finite_action_orbit_stabilizer_bijection","operation":"orbitstab-coset-fiber-bijection","semantic_template":"qf-mod41-orbitstab-coset-fiber-bijection/{\"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,\"orbit_point\":3,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0056","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-coset-fiber-bijection；資料={\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":3,\"action_table\":[[0,1,2],[1,0,2]],\"point\":2,\"orbit_point\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 0, [0, 1], [0, 1], 是, fiber_equals_left_coset_of_same_point_stabilizer)","explanation_zh":"Wave6第四層核對=(2, 0, [0, 1], [0, 1], 是, fiber_equals_left_coset_of_same_point_stabilizer)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-coset-fiber-bijection","seed":202611101056,"parameters":{"group_table":[[0,1],[1,0]],"identity":0,"point_count":3,"action_table":[[0,1,2],[1,0,2]],"point":2,"orbit_point":2,"domain":"Complete_finite_action_orbit_stabilizer_bijection","operation":"orbitstab-coset-fiber-bijection","semantic_template":"qf-mod41-orbitstab-coset-fiber-bijection/{\"action_table\":[[0,1,2],[1,0,2]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"orbit_point\":2,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0057","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-coset-fiber-bijection；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,\"orbit_point\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 1, [1], [1], 是, fiber_equals_left_coset_of_same_point_stabilizer)","explanation_zh":"Wave6第四層核對=(0, 1, [1], [1], 是, fiber_equals_left_coset_of_same_point_stabilizer)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-coset-fiber-bijection","seed":202611101057,"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,"orbit_point":0,"domain":"Complete_finite_action_orbit_stabilizer_bijection","operation":"orbitstab-coset-fiber-bijection","semantic_template":"qf-mod41-orbitstab-coset-fiber-bijection/{\"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,\"orbit_point\":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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0058","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-coset-fiber-bijection；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,\"orbit_point\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 0, [0], [0], 是, fiber_equals_left_coset_of_same_point_stabilizer)","explanation_zh":"Wave6第四層核對=(2, 0, [0], [0], 是, fiber_equals_left_coset_of_same_point_stabilizer)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-coset-fiber-bijection","seed":202611101058,"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,"orbit_point":2,"domain":"Complete_finite_action_orbit_stabilizer_bijection","operation":"orbitstab-coset-fiber-bijection","semantic_template":"qf-mod41-orbitstab-coset-fiber-bijection/{\"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,\"orbit_point\":2,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0059","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-coset-fiber-bijection；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,\"orbit_point\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 4, [4], [4], 是, fiber_equals_left_coset_of_same_point_stabilizer)","explanation_zh":"Wave6第四層核對=(1, 4, [4], [4], 是, fiber_equals_left_coset_of_same_point_stabilizer)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-coset-fiber-bijection","seed":202611101059,"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,"orbit_point":1,"domain":"Complete_finite_action_orbit_stabilizer_bijection","operation":"orbitstab-coset-fiber-bijection","semantic_template":"qf-mod41-orbitstab-coset-fiber-bijection/{\"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,\"orbit_point\":1,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0060","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-coset-fiber-bijection；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,\"orbit_point\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 4, [4], [4], 是, fiber_equals_left_coset_of_same_point_stabilizer)","explanation_zh":"Wave6第四層核對=(0, 4, [4], [4], 是, fiber_equals_left_coset_of_same_point_stabilizer)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-coset-fiber-bijection","seed":202611101060,"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,"orbit_point":0,"domain":"Complete_finite_action_orbit_stabilizer_bijection","operation":"orbitstab-coset-fiber-bijection","semantic_template":"qf-mod41-orbitstab-coset-fiber-bijection/{\"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,\"orbit_point\":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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0061","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-coset-fiber-bijection；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,\"orbit_point\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 0, [0, 1], [0, 1], 是, fiber_equals_left_coset_of_same_point_stabilizer)","explanation_zh":"Wave6第四層核對=(2, 0, [0, 1], [0, 1], 是, fiber_equals_left_coset_of_same_point_stabilizer)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-coset-fiber-bijection","seed":202611101061,"parameters":{"group_table":[[0,1],[1,0]],"identity":0,"point_count":4,"action_table":[[0,1,2,3],[1,0,2,3]],"point":2,"orbit_point":2,"domain":"Complete_finite_action_orbit_stabilizer_bijection","operation":"orbitstab-coset-fiber-bijection","semantic_template":"qf-mod41-orbitstab-coset-fiber-bijection/{\"action_table\":[[0,1,2,3],[1,0,2,3]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"orbit_point\":2,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0062","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-coset-fiber-bijection；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,\"orbit_point\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 0, [0], [0], 是, fiber_equals_left_coset_of_same_point_stabilizer)","explanation_zh":"Wave6第四層核對=(1, 0, [0], [0], 是, fiber_equals_left_coset_of_same_point_stabilizer)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-coset-fiber-bijection","seed":202611101062,"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,"orbit_point":1,"domain":"Complete_finite_action_orbit_stabilizer_bijection","operation":"orbitstab-coset-fiber-bijection","semantic_template":"qf-mod41-orbitstab-coset-fiber-bijection/{\"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,\"orbit_point\":1,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0063","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-coset-fiber-bijection；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,\"orbit_point\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 1, [1], [1], 是, fiber_equals_left_coset_of_same_point_stabilizer)","explanation_zh":"Wave6第四層核對=(1, 1, [1], [1], 是, fiber_equals_left_coset_of_same_point_stabilizer)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-coset-fiber-bijection","seed":202611101063,"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,"orbit_point":1,"domain":"Complete_finite_action_orbit_stabilizer_bijection","operation":"orbitstab-coset-fiber-bijection","semantic_template":"qf-mod41-orbitstab-coset-fiber-bijection/{\"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,\"orbit_point\":1,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0064","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-coset-fiber-bijection；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,\"orbit_point\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 0, [0, 1, 2, 3, 4], [0, 1, 2, 3, 4], 是, fiber_equals_left_coset_of_same_point_stabilizer)","explanation_zh":"Wave6第四層核對=(6, 0, [0, 1, 2, 3, 4], [0, 1, 2, 3, 4], 是, fiber_equals_left_coset_of_same_point_stabilizer)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-coset-fiber-bijection","seed":202611101064,"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,"orbit_point":6,"domain":"Complete_finite_action_orbit_stabilizer_bijection","operation":"orbitstab-coset-fiber-bijection","semantic_template":"qf-mod41-orbitstab-coset-fiber-bijection/{\"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,\"orbit_point\":6,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0065","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-coset-fiber-bijection；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,\"orbit_point\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 0, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是, fiber_equals_left_coset_of_same_point_stabilizer)","explanation_zh":"Wave6第四層核對=(6, 0, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是, fiber_equals_left_coset_of_same_point_stabilizer)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-coset-fiber-bijection","seed":202611101065,"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,"orbit_point":6,"domain":"Complete_finite_action_orbit_stabilizer_bijection","operation":"orbitstab-coset-fiber-bijection","semantic_template":"qf-mod41-orbitstab-coset-fiber-bijection/{\"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,\"orbit_point\":6,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0066","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-coset-fiber-bijection；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,\"orbit_point\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 0, [0], [0], 是, fiber_equals_left_coset_of_same_point_stabilizer)","explanation_zh":"Wave6第四層核對=(0, 0, [0], [0], 是, fiber_equals_left_coset_of_same_point_stabilizer)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-coset-fiber-bijection","seed":202611101066,"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,"orbit_point":0,"domain":"Complete_finite_action_orbit_stabilizer_bijection","operation":"orbitstab-coset-fiber-bijection","semantic_template":"qf-mod41-orbitstab-coset-fiber-bijection/{\"action_table\":[[0,1,2,3,4],[1,0,2,3,4]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"orbit_point\":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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0067","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-coset-fiber-bijection；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,\"orbit_point\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 0, [0, 1, 2], [0, 1, 2], 是, fiber_equals_left_coset_of_same_point_stabilizer)","explanation_zh":"Wave6第四層核對=(4, 0, [0, 1, 2], [0, 1, 2], 是, fiber_equals_left_coset_of_same_point_stabilizer)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-coset-fiber-bijection","seed":202611101067,"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,"orbit_point":4,"domain":"Complete_finite_action_orbit_stabilizer_bijection","operation":"orbitstab-coset-fiber-bijection","semantic_template":"qf-mod41-orbitstab-coset-fiber-bijection/{\"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,\"orbit_point\":4,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0068","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-coset-fiber-bijection；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,\"orbit_point\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 2, [2], [2], 是, fiber_equals_left_coset_of_same_point_stabilizer)","explanation_zh":"Wave6第四層核對=(1, 2, [2], [2], 是, fiber_equals_left_coset_of_same_point_stabilizer)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-coset-fiber-bijection","seed":202611101068,"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,"orbit_point":1,"domain":"Complete_finite_action_orbit_stabilizer_bijection","operation":"orbitstab-coset-fiber-bijection","semantic_template":"qf-mod41-orbitstab-coset-fiber-bijection/{\"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,\"orbit_point\":1,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0069","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-coset-fiber-bijection；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,\"orbit_point\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 4, [4], [4], 是, fiber_equals_left_coset_of_same_point_stabilizer)","explanation_zh":"Wave6第四層核對=(1, 4, [4], [4], 是, fiber_equals_left_coset_of_same_point_stabilizer)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-coset-fiber-bijection","seed":202611101069,"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,"orbit_point":1,"domain":"Complete_finite_action_orbit_stabilizer_bijection","operation":"orbitstab-coset-fiber-bijection","semantic_template":"qf-mod41-orbitstab-coset-fiber-bijection/{\"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,\"orbit_point\":1,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0070","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-coset-fiber-bijection；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,\"orbit_point\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 2, [2], [2], 是, fiber_equals_left_coset_of_same_point_stabilizer)","explanation_zh":"Wave6第四層核對=(3, 2, [2], [2], 是, fiber_equals_left_coset_of_same_point_stabilizer)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-coset-fiber-bijection","seed":202611101070,"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,"orbit_point":3,"domain":"Complete_finite_action_orbit_stabilizer_bijection","operation":"orbitstab-coset-fiber-bijection","semantic_template":"qf-mod41-orbitstab-coset-fiber-bijection/{\"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,\"orbit_point\":3,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0071","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-coset-fiber-bijection；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,\"orbit_point\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 0, [0, 1], [0, 1], 是, fiber_equals_left_coset_of_same_point_stabilizer)","explanation_zh":"Wave6第四層核對=(2, 0, [0, 1], [0, 1], 是, fiber_equals_left_coset_of_same_point_stabilizer)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-coset-fiber-bijection","seed":202611101071,"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,"orbit_point":2,"domain":"Complete_finite_action_orbit_stabilizer_bijection","operation":"orbitstab-coset-fiber-bijection","semantic_template":"qf-mod41-orbitstab-coset-fiber-bijection/{\"action_table\":[[0,1,2,3,4,5],[1,0,2,3,4,5]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"orbit_point\":2,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0072","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-coset-fiber-bijection；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,\"orbit_point\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 1, [1], [1], 是, fiber_equals_left_coset_of_same_point_stabilizer)","explanation_zh":"Wave6第四層核對=(1, 1, [1], [1], 是, fiber_equals_left_coset_of_same_point_stabilizer)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-coset-fiber-bijection","seed":202611101072,"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,"orbit_point":1,"domain":"Complete_finite_action_orbit_stabilizer_bijection","operation":"orbitstab-coset-fiber-bijection","semantic_template":"qf-mod41-orbitstab-coset-fiber-bijection/{\"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,\"orbit_point\":1,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0073","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-coset-fiber-bijection；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,\"orbit_point\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 0, [0, 1, 2, 3], [0, 1, 2, 3], 是, fiber_equals_left_coset_of_same_point_stabilizer)","explanation_zh":"Wave6第四層核對=(6, 0, [0, 1, 2, 3], [0, 1, 2, 3], 是, fiber_equals_left_coset_of_same_point_stabilizer)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-coset-fiber-bijection","seed":202611101073,"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,"orbit_point":6,"domain":"Complete_finite_action_orbit_stabilizer_bijection","operation":"orbitstab-coset-fiber-bijection","semantic_template":"qf-mod41-orbitstab-coset-fiber-bijection/{\"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,\"orbit_point\":6,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0074","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-coset-fiber-bijection；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,\"orbit_point\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 0, [0, 1, 2, 3, 4], [0, 1, 2, 3, 4], 是, fiber_equals_left_coset_of_same_point_stabilizer)","explanation_zh":"Wave6第四層核對=(5, 0, [0, 1, 2, 3, 4], [0, 1, 2, 3, 4], 是, fiber_equals_left_coset_of_same_point_stabilizer)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-coset-fiber-bijection","seed":202611101074,"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,"orbit_point":5,"domain":"Complete_finite_action_orbit_stabilizer_bijection","operation":"orbitstab-coset-fiber-bijection","semantic_template":"qf-mod41-orbitstab-coset-fiber-bijection/{\"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,\"orbit_point\":5,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0075","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-coset-fiber-bijection；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,\"orbit_point\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 1, [1], [1], 是, fiber_equals_left_coset_of_same_point_stabilizer)","explanation_zh":"Wave6第四層核對=(5, 1, [1], [1], 是, fiber_equals_left_coset_of_same_point_stabilizer)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-coset-fiber-bijection","seed":202611101075,"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,"orbit_point":5,"domain":"Complete_finite_action_orbit_stabilizer_bijection","operation":"orbitstab-coset-fiber-bijection","semantic_template":"qf-mod41-orbitstab-coset-fiber-bijection/{\"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,\"orbit_point\":5,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0076","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-binding-boundary；資料={\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":2,\"action_table\":[[0,1],[1,0]],\"point\":0,\"other_point\":1,\"sampled_elements\":[0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], [0], [0, 1], [0], 1, [0], 否, proper_sample_or_other_point_binding_cannot_certify_orbit_stabilizer_identity)","explanation_zh":"Wave6第四層核對=([0], [0], [0, 1], [0], 1, [0], 否, proper_sample_or_other_point_binding_cannot_certify_orbit_stabilizer_identity)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-binding-boundary","seed":202611101076,"parameters":{"group_table":[[0,1],[1,0]],"identity":0,"point_count":2,"action_table":[[0,1],[1,0]],"point":0,"other_point":1,"sampled_elements":[0],"domain":"Complete_finite_action_orbit_stabilizer_bijection","operation":"orbitstab-binding-boundary","semantic_template":"qf-mod41-orbitstab-binding-boundary/{\"action_table\":[[0,1],[1,0]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"other_point\":1,\"point\":0,\"point_count\":2,\"sampled_elements\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0077","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-binding-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,\"other_point\":2,\"sampled_elements\":[0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1], [1, 2], [0, 1, 2], [0], 2, [0], 否, proper_sample_or_other_point_binding_cannot_certify_orbit_stabilizer_identity)","explanation_zh":"Wave6第四層核對=([0, 1], [1, 2], [0, 1, 2], [0], 2, [0], 否, proper_sample_or_other_point_binding_cannot_certify_orbit_stabilizer_identity)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-binding-boundary","seed":202611101077,"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,"other_point":2,"sampled_elements":[0,1],"domain":"Complete_finite_action_orbit_stabilizer_bijection","operation":"orbitstab-binding-boundary","semantic_template":"qf-mod41-orbitstab-binding-boundary/{\"action_table\":[[0,1,2],[1,2,0],[2,0,1]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"other_point\":2,\"point\":1,\"point_count\":3,\"sampled_elements\":[0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0078","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-binding-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,\"other_point\":3,\"sampled_elements\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], [0, 2, 3], [0, 1, 2, 3], [0], 3, [0], 否, proper_sample_or_other_point_binding_cannot_certify_orbit_stabilizer_identity)","explanation_zh":"Wave6第四層核對=([0, 1, 2], [0, 2, 3], [0, 1, 2, 3], [0], 3, [0], 否, proper_sample_or_other_point_binding_cannot_certify_orbit_stabilizer_identity)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-binding-boundary","seed":202611101078,"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,"other_point":3,"sampled_elements":[0,1,2],"domain":"Complete_finite_action_orbit_stabilizer_bijection","operation":"orbitstab-binding-boundary","semantic_template":"qf-mod41-orbitstab-binding-boundary/{\"action_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"other_point\":3,\"point\":2,\"point_count\":4,\"sampled_elements\":[0,1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0079","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-binding-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,\"other_point\":4,\"sampled_elements\":[0,1,2,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4], [0], 4, [0], 否, proper_sample_or_other_point_binding_cannot_certify_orbit_stabilizer_identity)","explanation_zh":"Wave6第四層核對=([0, 1, 2, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4], [0], 4, [0], 否, proper_sample_or_other_point_binding_cannot_certify_orbit_stabilizer_identity)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-binding-boundary","seed":202611101079,"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,"other_point":4,"sampled_elements":[0,1,2,3],"domain":"Complete_finite_action_orbit_stabilizer_bijection","operation":"orbitstab-binding-boundary","semantic_template":"qf-mod41-orbitstab-binding-boundary/{\"action_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"other_point\":4,\"point\":3,\"point_count\":5,\"sampled_elements\":[0,1,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0080","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-binding-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,\"other_point\":5,\"sampled_elements\":[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], [0, 1, 2, 4, 5], [0, 1, 2, 3, 4, 5], [0], 5, [0], 否, proper_sample_or_other_point_binding_cannot_certify_orbit_stabilizer_identity)","explanation_zh":"Wave6第四層核對=([0, 1, 2, 3, 4], [0, 1, 2, 4, 5], [0, 1, 2, 3, 4, 5], [0], 5, [0], 否, proper_sample_or_other_point_binding_cannot_certify_orbit_stabilizer_identity)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-binding-boundary","seed":202611101080,"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,"other_point":5,"sampled_elements":[0,1,2,3,4],"domain":"Complete_finite_action_orbit_stabilizer_bijection","operation":"orbitstab-binding-boundary","semantic_template":"qf-mod41-orbitstab-binding-boundary/{\"action_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"other_point\":5,\"point\":4,\"point_count\":6,\"sampled_elements\":[0,1,2,3,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0081","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-binding-boundary；資料={\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":3,\"action_table\":[[0,1,2],[1,0,2]],\"point\":2,\"other_point\":0,\"sampled_elements\":[0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], [2], [2], [0, 1], 0, [0], 否, proper_sample_or_other_point_binding_cannot_certify_orbit_stabilizer_identity)","explanation_zh":"Wave6第四層核對=([0], [2], [2], [0, 1], 0, [0], 否, proper_sample_or_other_point_binding_cannot_certify_orbit_stabilizer_identity)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-binding-boundary","seed":202611101081,"parameters":{"group_table":[[0,1],[1,0]],"identity":0,"point_count":3,"action_table":[[0,1,2],[1,0,2]],"point":2,"other_point":0,"sampled_elements":[0],"domain":"Complete_finite_action_orbit_stabilizer_bijection","operation":"orbitstab-binding-boundary","semantic_template":"qf-mod41-orbitstab-binding-boundary/{\"action_table\":[[0,1,2],[1,0,2]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"other_point\":0,\"point\":2,\"point_count\":3,\"sampled_elements\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0082","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-binding-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,\"other_point\":3,\"sampled_elements\":[0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], [2], [0, 1, 2], [0], 3, [0, 1, 2], 否, proper_sample_or_other_point_binding_cannot_certify_orbit_stabilizer_identity)","explanation_zh":"Wave6第四層核對=([0], [2], [0, 1, 2], [0], 3, [0, 1, 2], 否, proper_sample_or_other_point_binding_cannot_certify_orbit_stabilizer_identity)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-binding-boundary","seed":202611101082,"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,"other_point":3,"sampled_elements":[0],"domain":"Complete_finite_action_orbit_stabilizer_bijection","operation":"orbitstab-binding-boundary","semantic_template":"qf-mod41-orbitstab-binding-boundary/{\"action_table\":[[0,1,2,3],[1,2,0,3],[2,0,1,3]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"other_point\":3,\"point\":2,\"point_count\":4,\"sampled_elements\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0083","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-binding-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,\"other_point\":3,\"sampled_elements\":[0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1], [2, 3], [0, 1, 2, 3], [0], 3, [0], 否, proper_sample_or_other_point_binding_cannot_certify_orbit_stabilizer_identity)","explanation_zh":"Wave6第四層核對=([0, 1], [2, 3], [0, 1, 2, 3], [0], 3, [0], 否, proper_sample_or_other_point_binding_cannot_certify_orbit_stabilizer_identity)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-binding-boundary","seed":202611101083,"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,"other_point":3,"sampled_elements":[0,1],"domain":"Complete_finite_action_orbit_stabilizer_bijection","operation":"orbitstab-binding-boundary","semantic_template":"qf-mod41-orbitstab-binding-boundary/{\"action_table\":[[0,1,2,3,4],[1,2,3,0,4],[2,3,0,1,4],[3,0,1,2,4]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"other_point\":3,\"point\":2,\"point_count\":5,\"sampled_elements\":[0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0084","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-binding-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,\"other_point\":3,\"sampled_elements\":[0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], [2], [0, 1, 2, 3, 4], [0], 3, [0], 否, proper_sample_or_other_point_binding_cannot_certify_orbit_stabilizer_identity)","explanation_zh":"Wave6第四層核對=([0], [2], [0, 1, 2, 3, 4], [0], 3, [0], 否, proper_sample_or_other_point_binding_cannot_certify_orbit_stabilizer_identity)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-binding-boundary","seed":202611101084,"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,"other_point":3,"sampled_elements":[0],"domain":"Complete_finite_action_orbit_stabilizer_bijection","operation":"orbitstab-binding-boundary","semantic_template":"qf-mod41-orbitstab-binding-boundary/{\"action_table\":[[0,1,2,3,4,5],[1,2,3,4,0,5],[2,3,4,0,1,5],[3,4,0,1,2,5],[4,0,1,2,3,5]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"other_point\":3,\"point\":2,\"point_count\":6,\"sampled_elements\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0085","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-binding-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,\"other_point\":3,\"sampled_elements\":[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], [0, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], [0], 3, [0], 否, proper_sample_or_other_point_binding_cannot_certify_orbit_stabilizer_identity)","explanation_zh":"Wave6第四層核對=([0, 1, 2, 3, 4], [0, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], [0], 3, [0], 否, proper_sample_or_other_point_binding_cannot_certify_orbit_stabilizer_identity)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-binding-boundary","seed":202611101085,"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,"other_point":3,"sampled_elements":[0,1,2,3,4],"domain":"Complete_finite_action_orbit_stabilizer_bijection","operation":"orbitstab-binding-boundary","semantic_template":"qf-mod41-orbitstab-binding-boundary/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,4,5,0,6],[2,3,4,5,0,1,6],[3,4,5,0,1,2,6],[4,5,0,1,2,3,6],[5,0,1,2,3,4,6]],\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"other_point\":3,\"point\":2,\"point_count\":7,\"sampled_elements\":[0,1,2,3,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0086","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-binding-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,\"other_point\":3,\"sampled_elements\":[0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], [2], [2], [0, 1], 3, [0, 1], 否, proper_sample_or_other_point_binding_cannot_certify_orbit_stabilizer_identity)","explanation_zh":"Wave6第四層核對=([0], [2], [2], [0, 1], 3, [0, 1], 否, proper_sample_or_other_point_binding_cannot_certify_orbit_stabilizer_identity)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-binding-boundary","seed":202611101086,"parameters":{"group_table":[[0,1],[1,0]],"identity":0,"point_count":4,"action_table":[[0,1,2,3],[1,0,2,3]],"point":2,"other_point":3,"sampled_elements":[0],"domain":"Complete_finite_action_orbit_stabilizer_bijection","operation":"orbitstab-binding-boundary","semantic_template":"qf-mod41-orbitstab-binding-boundary/{\"action_table\":[[0,1,2,3],[1,0,2,3]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"other_point\":3,\"point\":2,\"point_count\":4,\"sampled_elements\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0087","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-binding-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,\"other_point\":2,\"sampled_elements\":[0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1], [1, 2], [0, 1, 2], [0], 2, [0], 否, proper_sample_or_other_point_binding_cannot_certify_orbit_stabilizer_identity)","explanation_zh":"Wave6第四層核對=([0, 1], [1, 2], [0, 1, 2], [0], 2, [0], 否, proper_sample_or_other_point_binding_cannot_certify_orbit_stabilizer_identity)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-binding-boundary","seed":202611101087,"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,"other_point":2,"sampled_elements":[0,1],"domain":"Complete_finite_action_orbit_stabilizer_bijection","operation":"orbitstab-binding-boundary","semantic_template":"qf-mod41-orbitstab-binding-boundary/{\"action_table\":[[0,1,2,3,4],[1,2,0,3,4],[2,0,1,3,4]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"other_point\":2,\"point\":1,\"point_count\":5,\"sampled_elements\":[0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0088","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-binding-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,\"other_point\":1,\"sampled_elements\":[0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], [0], [0, 1, 2, 3], [0], 1, [0], 否, proper_sample_or_other_point_binding_cannot_certify_orbit_stabilizer_identity)","explanation_zh":"Wave6第四層核對=([0], [0], [0, 1, 2, 3], [0], 1, [0], 否, proper_sample_or_other_point_binding_cannot_certify_orbit_stabilizer_identity)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-binding-boundary","seed":202611101088,"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,"other_point":1,"sampled_elements":[0],"domain":"Complete_finite_action_orbit_stabilizer_bijection","operation":"orbitstab-binding-boundary","semantic_template":"qf-mod41-orbitstab-binding-boundary/{\"action_table\":[[0,1,2,3,4,5],[1,2,3,0,4,5],[2,3,0,1,4,5],[3,0,1,2,4,5]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"other_point\":1,\"point\":0,\"point_count\":6,\"sampled_elements\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0089","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-binding-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,\"other_point\":0,\"sampled_elements\":[0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1], [6], [6], [0, 1, 2, 3, 4], 0, [0], 否, proper_sample_or_other_point_binding_cannot_certify_orbit_stabilizer_identity)","explanation_zh":"Wave6第四層核對=([0, 1], [6], [6], [0, 1, 2, 3, 4], 0, [0], 否, proper_sample_or_other_point_binding_cannot_certify_orbit_stabilizer_identity)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-binding-boundary","seed":202611101089,"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,"other_point":0,"sampled_elements":[0,1],"domain":"Complete_finite_action_orbit_stabilizer_bijection","operation":"orbitstab-binding-boundary","semantic_template":"qf-mod41-orbitstab-binding-boundary/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,4,0,5,6],[2,3,4,0,1,5,6],[3,4,0,1,2,5,6],[4,0,1,2,3,5,6]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"other_point\":0,\"point\":6,\"point_count\":7,\"sampled_elements\":[0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0090","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-binding-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,\"other_point\":7,\"sampled_elements\":[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], [6], [6], [0, 1, 2, 3, 4, 5], 7, [0, 1, 2, 3, 4, 5], 否, proper_sample_or_other_point_binding_cannot_certify_orbit_stabilizer_identity)","explanation_zh":"Wave6第四層核對=([0, 1, 2, 3, 4], [6], [6], [0, 1, 2, 3, 4, 5], 7, [0, 1, 2, 3, 4, 5], 否, proper_sample_or_other_point_binding_cannot_certify_orbit_stabilizer_identity)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-binding-boundary","seed":202611101090,"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,"other_point":7,"sampled_elements":[0,1,2,3,4],"domain":"Complete_finite_action_orbit_stabilizer_bijection","operation":"orbitstab-binding-boundary","semantic_template":"qf-mod41-orbitstab-binding-boundary/{\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,0,6,7],[2,3,4,5,0,1,6,7],[3,4,5,0,1,2,6,7],[4,5,0,1,2,3,6,7],[5,0,1,2,3,4,6,7]],\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"other_point\":7,\"point\":6,\"point_count\":8,\"sampled_elements\":[0,1,2,3,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0091","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-binding-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,\"other_point\":1,\"sampled_elements\":[0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], [0], [0, 1], [0], 1, [0], 否, proper_sample_or_other_point_binding_cannot_certify_orbit_stabilizer_identity)","explanation_zh":"Wave6第四層核對=([0], [0], [0, 1], [0], 1, [0], 否, proper_sample_or_other_point_binding_cannot_certify_orbit_stabilizer_identity)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-binding-boundary","seed":202611101091,"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,"other_point":1,"sampled_elements":[0],"domain":"Complete_finite_action_orbit_stabilizer_bijection","operation":"orbitstab-binding-boundary","semantic_template":"qf-mod41-orbitstab-binding-boundary/{\"action_table\":[[0,1,2,3,4],[1,0,2,3,4]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"other_point\":1,\"point\":0,\"point_count\":5,\"sampled_elements\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0092","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-binding-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,\"other_point\":5,\"sampled_elements\":[0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], [4], [4], [0, 1, 2], 5, [0, 1, 2], 否, proper_sample_or_other_point_binding_cannot_certify_orbit_stabilizer_identity)","explanation_zh":"Wave6第四層核對=([0], [4], [4], [0, 1, 2], 5, [0, 1, 2], 否, proper_sample_or_other_point_binding_cannot_certify_orbit_stabilizer_identity)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-binding-boundary","seed":202611101092,"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,"other_point":5,"sampled_elements":[0],"domain":"Complete_finite_action_orbit_stabilizer_bijection","operation":"orbitstab-binding-boundary","semantic_template":"qf-mod41-orbitstab-binding-boundary/{\"action_table\":[[0,1,2,3,4,5],[1,2,0,3,4,5],[2,0,1,3,4,5]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"other_point\":5,\"point\":4,\"point_count\":6,\"sampled_elements\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0093","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-binding-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,\"other_point\":4,\"sampled_elements\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], [0, 1, 3], [0, 1, 2, 3], [0], 4, [0, 1, 2, 3], 否, proper_sample_or_other_point_binding_cannot_certify_orbit_stabilizer_identity)","explanation_zh":"Wave6第四層核對=([0, 1, 2], [0, 1, 3], [0, 1, 2, 3], [0], 4, [0, 1, 2, 3], 否, proper_sample_or_other_point_binding_cannot_certify_orbit_stabilizer_identity)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-binding-boundary","seed":202611101093,"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,"other_point":4,"sampled_elements":[0,1,2],"domain":"Complete_finite_action_orbit_stabilizer_bijection","operation":"orbitstab-binding-boundary","semantic_template":"qf-mod41-orbitstab-binding-boundary/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,0,4,5,6],[2,3,0,1,4,5,6],[3,0,1,2,4,5,6]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"other_point\":4,\"point\":3,\"point_count\":7,\"sampled_elements\":[0,1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0094","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-binding-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,\"other_point\":3,\"sampled_elements\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], [2, 3, 4], [0, 1, 2, 3, 4], [0], 3, [0], 否, proper_sample_or_other_point_binding_cannot_certify_orbit_stabilizer_identity)","explanation_zh":"Wave6第四層核對=([0, 1, 2], [2, 3, 4], [0, 1, 2, 3, 4], [0], 3, [0], 否, proper_sample_or_other_point_binding_cannot_certify_orbit_stabilizer_identity)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-binding-boundary","seed":202611101094,"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,"other_point":3,"sampled_elements":[0,1,2],"domain":"Complete_finite_action_orbit_stabilizer_bijection","operation":"orbitstab-binding-boundary","semantic_template":"qf-mod41-orbitstab-binding-boundary/{\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,0,5,6,7],[2,3,4,0,1,5,6,7],[3,4,0,1,2,5,6,7],[4,0,1,2,3,5,6,7]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"other_point\":3,\"point\":2,\"point_count\":8,\"sampled_elements\":[0,1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0095","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-binding-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,\"other_point\":2,\"sampled_elements\":[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], [1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], [0], 2, [0], 否, proper_sample_or_other_point_binding_cannot_certify_orbit_stabilizer_identity)","explanation_zh":"Wave6第四層核對=([0, 1, 2, 3, 4], [1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], [0], 2, [0], 否, proper_sample_or_other_point_binding_cannot_certify_orbit_stabilizer_identity)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-binding-boundary","seed":202611101095,"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,"other_point":2,"sampled_elements":[0,1,2,3,4],"domain":"Complete_finite_action_orbit_stabilizer_bijection","operation":"orbitstab-binding-boundary","semantic_template":"qf-mod41-orbitstab-binding-boundary/{\"action_table\":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,5,0,6,7,8],[2,3,4,5,0,1,6,7,8],[3,4,5,0,1,2,6,7,8],[4,5,0,1,2,3,6,7,8],[5,0,1,2,3,4,6,7,8]],\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"other_point\":2,\"point\":1,\"point_count\":9,\"sampled_elements\":[0,1,2,3,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0096","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-binding-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,\"other_point\":3,\"sampled_elements\":[0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], [2], [2], [0, 1], 3, [0, 1], 否, proper_sample_or_other_point_binding_cannot_certify_orbit_stabilizer_identity)","explanation_zh":"Wave6第四層核對=([0], [2], [2], [0, 1], 3, [0, 1], 否, proper_sample_or_other_point_binding_cannot_certify_orbit_stabilizer_identity)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-binding-boundary","seed":202611101096,"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,"other_point":3,"sampled_elements":[0],"domain":"Complete_finite_action_orbit_stabilizer_bijection","operation":"orbitstab-binding-boundary","semantic_template":"qf-mod41-orbitstab-binding-boundary/{\"action_table\":[[0,1,2,3,4,5],[1,0,2,3,4,5]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"other_point\":3,\"point\":2,\"point_count\":6,\"sampled_elements\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0097","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-binding-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,\"other_point\":1,\"sampled_elements\":[0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1], [0, 1], [0, 1, 2], [0], 1, [0], 否, proper_sample_or_other_point_binding_cannot_certify_orbit_stabilizer_identity)","explanation_zh":"Wave6第四層核對=([0, 1], [0, 1], [0, 1, 2], [0], 1, [0], 否, proper_sample_or_other_point_binding_cannot_certify_orbit_stabilizer_identity)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-binding-boundary","seed":202611101097,"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,"other_point":1,"sampled_elements":[0,1],"domain":"Complete_finite_action_orbit_stabilizer_bijection","operation":"orbitstab-binding-boundary","semantic_template":"qf-mod41-orbitstab-binding-boundary/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,0,3,4,5,6],[2,0,1,3,4,5,6]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"other_point\":1,\"point\":0,\"point_count\":7,\"sampled_elements\":[0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0098","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-binding-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,\"other_point\":7,\"sampled_elements\":[0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1], [6], [6], [0, 1, 2, 3], 7, [0, 1, 2, 3], 否, proper_sample_or_other_point_binding_cannot_certify_orbit_stabilizer_identity)","explanation_zh":"Wave6第四層核對=([0, 1], [6], [6], [0, 1, 2, 3], 7, [0, 1, 2, 3], 否, proper_sample_or_other_point_binding_cannot_certify_orbit_stabilizer_identity)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-binding-boundary","seed":202611101098,"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,"other_point":7,"sampled_elements":[0,1],"domain":"Complete_finite_action_orbit_stabilizer_bijection","operation":"orbitstab-binding-boundary","semantic_template":"qf-mod41-orbitstab-binding-boundary/{\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,0,4,5,6,7],[2,3,0,1,4,5,6,7],[3,0,1,2,4,5,6,7]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"other_point\":7,\"point\":6,\"point_count\":8,\"sampled_elements\":[0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0099","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-binding-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,\"other_point\":6,\"sampled_elements\":[0,1,2,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3], [5], [5], [0, 1, 2, 3, 4], 6, [0, 1, 2, 3, 4], 否, proper_sample_or_other_point_binding_cannot_certify_orbit_stabilizer_identity)","explanation_zh":"Wave6第四層核對=([0, 1, 2, 3], [5], [5], [0, 1, 2, 3, 4], 6, [0, 1, 2, 3, 4], 否, proper_sample_or_other_point_binding_cannot_certify_orbit_stabilizer_identity)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-binding-boundary","seed":202611101099,"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,"other_point":6,"sampled_elements":[0,1,2,3],"domain":"Complete_finite_action_orbit_stabilizer_bijection","operation":"orbitstab-binding-boundary","semantic_template":"qf-mod41-orbitstab-binding-boundary/{\"action_table\":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,0,5,6,7,8],[2,3,4,0,1,5,6,7,8],[3,4,0,1,2,5,6,7,8],[4,0,1,2,3,5,6,7,8]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"other_point\":6,\"point\":5,\"point_count\":9,\"sampled_elements\":[0,1,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0100","batch_id":"batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-stabilizer-theorem"],"title_zh":"軌道－穩定子定理","stem_zh":"同一完整已認證有限群作用與同一指定point。操作=orbitstab-binding-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,\"other_point\":5,\"sampled_elements\":[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], [0, 1, 2, 4, 5], [0, 1, 2, 3, 4, 5], [0], 5, [0], 否, proper_sample_or_other_point_binding_cannot_certify_orbit_stabilizer_identity)","explanation_zh":"Wave6第四層核對=([0, 1, 2, 3, 4], [0, 1, 2, 4, 5], [0, 1, 2, 3, 4, 5], [0], 5, [0], 否, proper_sample_or_other_point_binding_cannot_certify_orbit_stabilizer_identity)；依完整明示資料獨立重算；action map對orbit滿射且每個fiber須核對為同一point stabilizer的左陪集；不要求free、faithful或transitive；proper sample、不同point或只有數值乘積皆不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["軌道－穩定子定理"]},"variant":{"family_id":"qf-mod41-orbitstab-binding-boundary","seed":202611101100,"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,"other_point":5,"sampled_elements":[0,1,2,3,4],"domain":"Complete_finite_action_orbit_stabilizer_bijection","operation":"orbitstab-binding-boundary","semantic_template":"qf-mod41-orbitstab-binding-boundary/{\"action_table\":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,0,6,7,8,9],[2,3,4,5,0,1,6,7,8,9],[3,4,5,0,1,2,6,7,8,9],[4,5,0,1,2,3,6,7,8,9],[5,0,1,2,3,4,6,7,8,9]],\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"other_point\":5,\"point\":4,\"point_count\":10,\"sampled_elements\":[0,1,2,3,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}