{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0001","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-stabilizer-of-point；資料={\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":2,\"action_table\":[[0,1],[1,0]],\"point\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [0], 1, 是, all_group_elements_checked)","explanation_zh":"Wave6第三層核對=(0, [0], 1, 是, all_group_elements_checked)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-stabilizer-of-point","seed":202611091001,"parameters":{"group_table":[[0,1],[1,0]],"identity":0,"point_count":2,"action_table":[[0,1],[1,0]],"point":0,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-stabilizer-of-point","semantic_template":"qf-mod40-stabilizer-stabilizer-of-point/{\"action_table\":[[0,1],[1,0]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point\":0,\"point_count\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0002","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-stabilizer-of-point；資料={\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":3,\"action_table\":[[0,1,2],[1,2,0],[2,0,1]],\"point\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, [0], 1, 是, all_group_elements_checked)","explanation_zh":"Wave6第三層核對=(1, [0], 1, 是, all_group_elements_checked)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-stabilizer-of-point","seed":202611091002,"parameters":{"group_table":[[0,1,2],[1,2,0],[2,0,1]],"identity":0,"point_count":3,"action_table":[[0,1,2],[1,2,0],[2,0,1]],"point":1,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-stabilizer-of-point","semantic_template":"qf-mod40-stabilizer-stabilizer-of-point/{\"action_table\":[[0,1,2],[1,2,0],[2,0,1]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point\":1,\"point_count\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0003","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-stabilizer-of-point；資料={\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":4,\"action_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"point\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [0], 1, 是, all_group_elements_checked)","explanation_zh":"Wave6第三層核對=(2, [0], 1, 是, all_group_elements_checked)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-stabilizer-of-point","seed":202611091003,"parameters":{"group_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"identity":0,"point_count":4,"action_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"point":2,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-stabilizer-of-point","semantic_template":"qf-mod40-stabilizer-stabilizer-of-point/{\"action_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point\":2,\"point_count\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0004","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-stabilizer-of-point；資料={\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":5,\"action_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"point\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [0], 1, 是, all_group_elements_checked)","explanation_zh":"Wave6第三層核對=(3, [0], 1, 是, all_group_elements_checked)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-stabilizer-of-point","seed":202611091004,"parameters":{"group_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"identity":0,"point_count":5,"action_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"point":3,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-stabilizer-of-point","semantic_template":"qf-mod40-stabilizer-stabilizer-of-point/{\"action_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point\":3,\"point_count\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0005","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-stabilizer-of-point；資料={\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point_count\":6,\"action_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"point\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, [0], 1, 是, all_group_elements_checked)","explanation_zh":"Wave6第三層核對=(4, [0], 1, 是, all_group_elements_checked)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-stabilizer-of-point","seed":202611091005,"parameters":{"group_table":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],"identity":0,"point_count":6,"action_table":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],"point":4,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-stabilizer-of-point","semantic_template":"qf-mod40-stabilizer-stabilizer-of-point/{\"action_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point\":4,\"point_count\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0006","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-stabilizer-of-point；資料={\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":3,\"action_table\":[[0,1,2],[1,0,2]],\"point\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [0, 1], 2, 是, all_group_elements_checked)","explanation_zh":"Wave6第三層核對=(2, [0, 1], 2, 是, all_group_elements_checked)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-stabilizer-of-point","seed":202611091006,"parameters":{"group_table":[[0,1],[1,0]],"identity":0,"point_count":3,"action_table":[[0,1,2],[1,0,2]],"point":2,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-stabilizer-of-point","semantic_template":"qf-mod40-stabilizer-stabilizer-of-point/{\"action_table\":[[0,1,2],[1,0,2]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point\":2,\"point_count\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0007","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-stabilizer-of-point；資料={\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":4,\"action_table\":[[0,1,2,3],[1,2,0,3],[2,0,1,3]],\"point\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [0], 1, 是, all_group_elements_checked)","explanation_zh":"Wave6第三層核對=(2, [0], 1, 是, all_group_elements_checked)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-stabilizer-of-point","seed":202611091007,"parameters":{"group_table":[[0,1,2],[1,2,0],[2,0,1]],"identity":0,"point_count":4,"action_table":[[0,1,2,3],[1,2,0,3],[2,0,1,3]],"point":2,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-stabilizer-of-point","semantic_template":"qf-mod40-stabilizer-stabilizer-of-point/{\"action_table\":[[0,1,2,3],[1,2,0,3],[2,0,1,3]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point\":2,\"point_count\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0008","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-stabilizer-of-point；資料={\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":5,\"action_table\":[[0,1,2,3,4],[1,2,3,0,4],[2,3,0,1,4],[3,0,1,2,4]],\"point\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [0], 1, 是, all_group_elements_checked)","explanation_zh":"Wave6第三層核對=(2, [0], 1, 是, all_group_elements_checked)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-stabilizer-of-point","seed":202611091008,"parameters":{"group_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"identity":0,"point_count":5,"action_table":[[0,1,2,3,4],[1,2,3,0,4],[2,3,0,1,4],[3,0,1,2,4]],"point":2,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-stabilizer-of-point","semantic_template":"qf-mod40-stabilizer-stabilizer-of-point/{\"action_table\":[[0,1,2,3,4],[1,2,3,0,4],[2,3,0,1,4],[3,0,1,2,4]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point\":2,\"point_count\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0009","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-stabilizer-of-point；資料={\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":6,\"action_table\":[[0,1,2,3,4,5],[1,2,3,4,0,5],[2,3,4,0,1,5],[3,4,0,1,2,5],[4,0,1,2,3,5]],\"point\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [0], 1, 是, all_group_elements_checked)","explanation_zh":"Wave6第三層核對=(2, [0], 1, 是, all_group_elements_checked)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-stabilizer-of-point","seed":202611091009,"parameters":{"group_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"identity":0,"point_count":6,"action_table":[[0,1,2,3,4,5],[1,2,3,4,0,5],[2,3,4,0,1,5],[3,4,0,1,2,5],[4,0,1,2,3,5]],"point":2,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-stabilizer-of-point","semantic_template":"qf-mod40-stabilizer-stabilizer-of-point/{\"action_table\":[[0,1,2,3,4,5],[1,2,3,4,0,5],[2,3,4,0,1,5],[3,4,0,1,2,5],[4,0,1,2,3,5]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point\":2,\"point_count\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0010","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-stabilizer-of-point；資料={\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point_count\":7,\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,4,5,0,6],[2,3,4,5,0,1,6],[3,4,5,0,1,2,6],[4,5,0,1,2,3,6],[5,0,1,2,3,4,6]],\"point\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [0], 1, 是, all_group_elements_checked)","explanation_zh":"Wave6第三層核對=(2, [0], 1, 是, all_group_elements_checked)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-stabilizer-of-point","seed":202611091010,"parameters":{"group_table":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],"identity":0,"point_count":7,"action_table":[[0,1,2,3,4,5,6],[1,2,3,4,5,0,6],[2,3,4,5,0,1,6],[3,4,5,0,1,2,6],[4,5,0,1,2,3,6],[5,0,1,2,3,4,6]],"point":2,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-stabilizer-of-point","semantic_template":"qf-mod40-stabilizer-stabilizer-of-point/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,4,5,0,6],[2,3,4,5,0,1,6],[3,4,5,0,1,2,6],[4,5,0,1,2,3,6],[5,0,1,2,3,4,6]],\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point\":2,\"point_count\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0011","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-stabilizer-of-point；資料={\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":4,\"action_table\":[[0,1,2,3],[1,0,2,3]],\"point\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [0, 1], 2, 是, all_group_elements_checked)","explanation_zh":"Wave6第三層核對=(2, [0, 1], 2, 是, all_group_elements_checked)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-stabilizer-of-point","seed":202611091011,"parameters":{"group_table":[[0,1],[1,0]],"identity":0,"point_count":4,"action_table":[[0,1,2,3],[1,0,2,3]],"point":2,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-stabilizer-of-point","semantic_template":"qf-mod40-stabilizer-stabilizer-of-point/{\"action_table\":[[0,1,2,3],[1,0,2,3]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point\":2,\"point_count\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0012","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-stabilizer-of-point；資料={\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":5,\"action_table\":[[0,1,2,3,4],[1,2,0,3,4],[2,0,1,3,4]],\"point\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, [0], 1, 是, all_group_elements_checked)","explanation_zh":"Wave6第三層核對=(1, [0], 1, 是, all_group_elements_checked)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-stabilizer-of-point","seed":202611091012,"parameters":{"group_table":[[0,1,2],[1,2,0],[2,0,1]],"identity":0,"point_count":5,"action_table":[[0,1,2,3,4],[1,2,0,3,4],[2,0,1,3,4]],"point":1,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-stabilizer-of-point","semantic_template":"qf-mod40-stabilizer-stabilizer-of-point/{\"action_table\":[[0,1,2,3,4],[1,2,0,3,4],[2,0,1,3,4]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point\":1,\"point_count\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0013","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-stabilizer-of-point；資料={\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":6,\"action_table\":[[0,1,2,3,4,5],[1,2,3,0,4,5],[2,3,0,1,4,5],[3,0,1,2,4,5]],\"point\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [0], 1, 是, all_group_elements_checked)","explanation_zh":"Wave6第三層核對=(0, [0], 1, 是, all_group_elements_checked)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-stabilizer-of-point","seed":202611091013,"parameters":{"group_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"identity":0,"point_count":6,"action_table":[[0,1,2,3,4,5],[1,2,3,0,4,5],[2,3,0,1,4,5],[3,0,1,2,4,5]],"point":0,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-stabilizer-of-point","semantic_template":"qf-mod40-stabilizer-stabilizer-of-point/{\"action_table\":[[0,1,2,3,4,5],[1,2,3,0,4,5],[2,3,0,1,4,5],[3,0,1,2,4,5]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point\":0,\"point_count\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0014","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-stabilizer-of-point；資料={\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":7,\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,4,0,5,6],[2,3,4,0,1,5,6],[3,4,0,1,2,5,6],[4,0,1,2,3,5,6]],\"point\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, [0, 1, 2, 3, 4], 5, 是, all_group_elements_checked)","explanation_zh":"Wave6第三層核對=(6, [0, 1, 2, 3, 4], 5, 是, all_group_elements_checked)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-stabilizer-of-point","seed":202611091014,"parameters":{"group_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"identity":0,"point_count":7,"action_table":[[0,1,2,3,4,5,6],[1,2,3,4,0,5,6],[2,3,4,0,1,5,6],[3,4,0,1,2,5,6],[4,0,1,2,3,5,6]],"point":6,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-stabilizer-of-point","semantic_template":"qf-mod40-stabilizer-stabilizer-of-point/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,4,0,5,6],[2,3,4,0,1,5,6],[3,4,0,1,2,5,6],[4,0,1,2,3,5,6]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point\":6,\"point_count\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0015","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-stabilizer-of-point；資料={\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point_count\":8,\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,0,6,7],[2,3,4,5,0,1,6,7],[3,4,5,0,1,2,6,7],[4,5,0,1,2,3,6,7],[5,0,1,2,3,4,6,7]],\"point\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, [0, 1, 2, 3, 4, 5], 6, 是, all_group_elements_checked)","explanation_zh":"Wave6第三層核對=(6, [0, 1, 2, 3, 4, 5], 6, 是, all_group_elements_checked)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-stabilizer-of-point","seed":202611091015,"parameters":{"group_table":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],"identity":0,"point_count":8,"action_table":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,0,6,7],[2,3,4,5,0,1,6,7],[3,4,5,0,1,2,6,7],[4,5,0,1,2,3,6,7],[5,0,1,2,3,4,6,7]],"point":6,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-stabilizer-of-point","semantic_template":"qf-mod40-stabilizer-stabilizer-of-point/{\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,0,6,7],[2,3,4,5,0,1,6,7],[3,4,5,0,1,2,6,7],[4,5,0,1,2,3,6,7],[5,0,1,2,3,4,6,7]],\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point\":6,\"point_count\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0016","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-stabilizer-of-point；資料={\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":5,\"action_table\":[[0,1,2,3,4],[1,0,2,3,4]],\"point\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [0], 1, 是, all_group_elements_checked)","explanation_zh":"Wave6第三層核對=(0, [0], 1, 是, all_group_elements_checked)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-stabilizer-of-point","seed":202611091016,"parameters":{"group_table":[[0,1],[1,0]],"identity":0,"point_count":5,"action_table":[[0,1,2,3,4],[1,0,2,3,4]],"point":0,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-stabilizer-of-point","semantic_template":"qf-mod40-stabilizer-stabilizer-of-point/{\"action_table\":[[0,1,2,3,4],[1,0,2,3,4]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point\":0,\"point_count\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0017","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-stabilizer-of-point；資料={\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":6,\"action_table\":[[0,1,2,3,4,5],[1,2,0,3,4,5],[2,0,1,3,4,5]],\"point\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, [0, 1, 2], 3, 是, all_group_elements_checked)","explanation_zh":"Wave6第三層核對=(4, [0, 1, 2], 3, 是, all_group_elements_checked)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-stabilizer-of-point","seed":202611091017,"parameters":{"group_table":[[0,1,2],[1,2,0],[2,0,1]],"identity":0,"point_count":6,"action_table":[[0,1,2,3,4,5],[1,2,0,3,4,5],[2,0,1,3,4,5]],"point":4,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-stabilizer-of-point","semantic_template":"qf-mod40-stabilizer-stabilizer-of-point/{\"action_table\":[[0,1,2,3,4,5],[1,2,0,3,4,5],[2,0,1,3,4,5]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point\":4,\"point_count\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0018","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-stabilizer-of-point；資料={\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":7,\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,0,4,5,6],[2,3,0,1,4,5,6],[3,0,1,2,4,5,6]],\"point\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [0], 1, 是, all_group_elements_checked)","explanation_zh":"Wave6第三層核對=(3, [0], 1, 是, all_group_elements_checked)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-stabilizer-of-point","seed":202611091018,"parameters":{"group_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"identity":0,"point_count":7,"action_table":[[0,1,2,3,4,5,6],[1,2,3,0,4,5,6],[2,3,0,1,4,5,6],[3,0,1,2,4,5,6]],"point":3,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-stabilizer-of-point","semantic_template":"qf-mod40-stabilizer-stabilizer-of-point/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,0,4,5,6],[2,3,0,1,4,5,6],[3,0,1,2,4,5,6]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point\":3,\"point_count\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0019","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-stabilizer-of-point；資料={\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":8,\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,0,5,6,7],[2,3,4,0,1,5,6,7],[3,4,0,1,2,5,6,7],[4,0,1,2,3,5,6,7]],\"point\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [0], 1, 是, all_group_elements_checked)","explanation_zh":"Wave6第三層核對=(2, [0], 1, 是, all_group_elements_checked)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-stabilizer-of-point","seed":202611091019,"parameters":{"group_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"identity":0,"point_count":8,"action_table":[[0,1,2,3,4,5,6,7],[1,2,3,4,0,5,6,7],[2,3,4,0,1,5,6,7],[3,4,0,1,2,5,6,7],[4,0,1,2,3,5,6,7]],"point":2,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-stabilizer-of-point","semantic_template":"qf-mod40-stabilizer-stabilizer-of-point/{\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,0,5,6,7],[2,3,4,0,1,5,6,7],[3,4,0,1,2,5,6,7],[4,0,1,2,3,5,6,7]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point\":2,\"point_count\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0020","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-stabilizer-of-point；資料={\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point_count\":9,\"action_table\":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,5,0,6,7,8],[2,3,4,5,0,1,6,7,8],[3,4,5,0,1,2,6,7,8],[4,5,0,1,2,3,6,7,8],[5,0,1,2,3,4,6,7,8]],\"point\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, [0], 1, 是, all_group_elements_checked)","explanation_zh":"Wave6第三層核對=(1, [0], 1, 是, all_group_elements_checked)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-stabilizer-of-point","seed":202611091020,"parameters":{"group_table":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],"identity":0,"point_count":9,"action_table":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,5,0,6,7,8],[2,3,4,5,0,1,6,7,8],[3,4,5,0,1,2,6,7,8],[4,5,0,1,2,3,6,7,8],[5,0,1,2,3,4,6,7,8]],"point":1,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-stabilizer-of-point","semantic_template":"qf-mod40-stabilizer-stabilizer-of-point/{\"action_table\":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,5,0,6,7,8],[2,3,4,5,0,1,6,7,8],[3,4,5,0,1,2,6,7,8],[4,5,0,1,2,3,6,7,8],[5,0,1,2,3,4,6,7,8]],\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point\":1,\"point_count\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0021","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-stabilizer-of-point；資料={\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":6,\"action_table\":[[0,1,2,3,4,5],[1,0,2,3,4,5]],\"point\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [0, 1], 2, 是, all_group_elements_checked)","explanation_zh":"Wave6第三層核對=(2, [0, 1], 2, 是, all_group_elements_checked)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-stabilizer-of-point","seed":202611091021,"parameters":{"group_table":[[0,1],[1,0]],"identity":0,"point_count":6,"action_table":[[0,1,2,3,4,5],[1,0,2,3,4,5]],"point":2,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-stabilizer-of-point","semantic_template":"qf-mod40-stabilizer-stabilizer-of-point/{\"action_table\":[[0,1,2,3,4,5],[1,0,2,3,4,5]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point\":2,\"point_count\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0022","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-stabilizer-of-point；資料={\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":7,\"action_table\":[[0,1,2,3,4,5,6],[1,2,0,3,4,5,6],[2,0,1,3,4,5,6]],\"point\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [0], 1, 是, all_group_elements_checked)","explanation_zh":"Wave6第三層核對=(0, [0], 1, 是, all_group_elements_checked)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-stabilizer-of-point","seed":202611091022,"parameters":{"group_table":[[0,1,2],[1,2,0],[2,0,1]],"identity":0,"point_count":7,"action_table":[[0,1,2,3,4,5,6],[1,2,0,3,4,5,6],[2,0,1,3,4,5,6]],"point":0,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-stabilizer-of-point","semantic_template":"qf-mod40-stabilizer-stabilizer-of-point/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,0,3,4,5,6],[2,0,1,3,4,5,6]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point\":0,\"point_count\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0023","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-stabilizer-of-point；資料={\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":8,\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,0,4,5,6,7],[2,3,0,1,4,5,6,7],[3,0,1,2,4,5,6,7]],\"point\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, [0, 1, 2, 3], 4, 是, all_group_elements_checked)","explanation_zh":"Wave6第三層核對=(6, [0, 1, 2, 3], 4, 是, all_group_elements_checked)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-stabilizer-of-point","seed":202611091023,"parameters":{"group_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"identity":0,"point_count":8,"action_table":[[0,1,2,3,4,5,6,7],[1,2,3,0,4,5,6,7],[2,3,0,1,4,5,6,7],[3,0,1,2,4,5,6,7]],"point":6,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-stabilizer-of-point","semantic_template":"qf-mod40-stabilizer-stabilizer-of-point/{\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,0,4,5,6,7],[2,3,0,1,4,5,6,7],[3,0,1,2,4,5,6,7]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point\":6,\"point_count\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0024","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-stabilizer-of-point；資料={\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":9,\"action_table\":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,0,5,6,7,8],[2,3,4,0,1,5,6,7,8],[3,4,0,1,2,5,6,7,8],[4,0,1,2,3,5,6,7,8]],\"point\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, [0, 1, 2, 3, 4], 5, 是, all_group_elements_checked)","explanation_zh":"Wave6第三層核對=(5, [0, 1, 2, 3, 4], 5, 是, all_group_elements_checked)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-stabilizer-of-point","seed":202611091024,"parameters":{"group_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"identity":0,"point_count":9,"action_table":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,0,5,6,7,8],[2,3,4,0,1,5,6,7,8],[3,4,0,1,2,5,6,7,8],[4,0,1,2,3,5,6,7,8]],"point":5,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-stabilizer-of-point","semantic_template":"qf-mod40-stabilizer-stabilizer-of-point/{\"action_table\":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,0,5,6,7,8],[2,3,4,0,1,5,6,7,8],[3,4,0,1,2,5,6,7,8],[4,0,1,2,3,5,6,7,8]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point\":5,\"point_count\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0025","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-stabilizer-of-point；資料={\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point_count\":10,\"action_table\":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,0,6,7,8,9],[2,3,4,5,0,1,6,7,8,9],[3,4,5,0,1,2,6,7,8,9],[4,5,0,1,2,3,6,7,8,9],[5,0,1,2,3,4,6,7,8,9]],\"point\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, [0], 1, 是, all_group_elements_checked)","explanation_zh":"Wave6第三層核對=(4, [0], 1, 是, all_group_elements_checked)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-stabilizer-of-point","seed":202611091025,"parameters":{"group_table":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],"identity":0,"point_count":10,"action_table":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,0,6,7,8,9],[2,3,4,5,0,1,6,7,8,9],[3,4,5,0,1,2,6,7,8,9],[4,5,0,1,2,3,6,7,8,9],[5,0,1,2,3,4,6,7,8,9]],"point":4,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-stabilizer-of-point","semantic_template":"qf-mod40-stabilizer-stabilizer-of-point/{\"action_table\":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,0,6,7,8,9],[2,3,4,5,0,1,6,7,8,9],[3,4,5,0,1,2,6,7,8,9],[4,5,0,1,2,3,6,7,8,9],[5,0,1,2,3,4,6,7,8,9]],\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point\":4,\"point_count\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0026","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-subgroup-certificate；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, 是, 是, 是, 是, identity_product_inverse_closure)","explanation_zh":"Wave6第三層核對=([0], 0, 是, 是, 是, 是, identity_product_inverse_closure)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-subgroup-certificate","seed":202611091026,"parameters":{"group_table":[[0,1],[1,0]],"identity":0,"point_count":2,"action_table":[[0,1],[1,0]],"point":0,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-subgroup-certificate","semantic_template":"qf-mod40-stabilizer-subgroup-certificate/{\"action_table\":[[0,1],[1,0]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point\":0,\"point_count\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0027","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-subgroup-certificate；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], 0, 是, 是, 是, 是, identity_product_inverse_closure)","explanation_zh":"Wave6第三層核對=([0], 0, 是, 是, 是, 是, identity_product_inverse_closure)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-subgroup-certificate","seed":202611091027,"parameters":{"group_table":[[0,1,2],[1,2,0],[2,0,1]],"identity":0,"point_count":3,"action_table":[[0,1,2],[1,2,0],[2,0,1]],"point":1,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-subgroup-certificate","semantic_template":"qf-mod40-stabilizer-subgroup-certificate/{\"action_table\":[[0,1,2],[1,2,0],[2,0,1]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point\":1,\"point_count\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0028","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-subgroup-certificate；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], 0, 是, 是, 是, 是, identity_product_inverse_closure)","explanation_zh":"Wave6第三層核對=([0], 0, 是, 是, 是, 是, identity_product_inverse_closure)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-subgroup-certificate","seed":202611091028,"parameters":{"group_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"identity":0,"point_count":4,"action_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"point":2,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-subgroup-certificate","semantic_template":"qf-mod40-stabilizer-subgroup-certificate/{\"action_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point\":2,\"point_count\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0029","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-subgroup-certificate；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], 0, 是, 是, 是, 是, identity_product_inverse_closure)","explanation_zh":"Wave6第三層核對=([0], 0, 是, 是, 是, 是, identity_product_inverse_closure)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-subgroup-certificate","seed":202611091029,"parameters":{"group_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"identity":0,"point_count":5,"action_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"point":3,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-subgroup-certificate","semantic_template":"qf-mod40-stabilizer-subgroup-certificate/{\"action_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point\":3,\"point_count\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0030","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-subgroup-certificate；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], 0, 是, 是, 是, 是, identity_product_inverse_closure)","explanation_zh":"Wave6第三層核對=([0], 0, 是, 是, 是, 是, identity_product_inverse_closure)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-subgroup-certificate","seed":202611091030,"parameters":{"group_table":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],"identity":0,"point_count":6,"action_table":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],"point":4,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-subgroup-certificate","semantic_template":"qf-mod40-stabilizer-subgroup-certificate/{\"action_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point\":4,\"point_count\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0031","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-subgroup-certificate；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, 1], 0, 是, 是, 是, 是, identity_product_inverse_closure)","explanation_zh":"Wave6第三層核對=([0, 1], 0, 是, 是, 是, 是, identity_product_inverse_closure)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-subgroup-certificate","seed":202611091031,"parameters":{"group_table":[[0,1],[1,0]],"identity":0,"point_count":3,"action_table":[[0,1,2],[1,0,2]],"point":2,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-subgroup-certificate","semantic_template":"qf-mod40-stabilizer-subgroup-certificate/{\"action_table\":[[0,1,2],[1,0,2]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point\":2,\"point_count\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0032","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-subgroup-certificate；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], 0, 是, 是, 是, 是, identity_product_inverse_closure)","explanation_zh":"Wave6第三層核對=([0], 0, 是, 是, 是, 是, identity_product_inverse_closure)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-subgroup-certificate","seed":202611091032,"parameters":{"group_table":[[0,1,2],[1,2,0],[2,0,1]],"identity":0,"point_count":4,"action_table":[[0,1,2,3],[1,2,0,3],[2,0,1,3]],"point":2,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-subgroup-certificate","semantic_template":"qf-mod40-stabilizer-subgroup-certificate/{\"action_table\":[[0,1,2,3],[1,2,0,3],[2,0,1,3]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point\":2,\"point_count\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0033","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-subgroup-certificate；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], 0, 是, 是, 是, 是, identity_product_inverse_closure)","explanation_zh":"Wave6第三層核對=([0], 0, 是, 是, 是, 是, identity_product_inverse_closure)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-subgroup-certificate","seed":202611091033,"parameters":{"group_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"identity":0,"point_count":5,"action_table":[[0,1,2,3,4],[1,2,3,0,4],[2,3,0,1,4],[3,0,1,2,4]],"point":2,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-subgroup-certificate","semantic_template":"qf-mod40-stabilizer-subgroup-certificate/{\"action_table\":[[0,1,2,3,4],[1,2,3,0,4],[2,3,0,1,4],[3,0,1,2,4]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point\":2,\"point_count\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0034","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-subgroup-certificate；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], 0, 是, 是, 是, 是, identity_product_inverse_closure)","explanation_zh":"Wave6第三層核對=([0], 0, 是, 是, 是, 是, identity_product_inverse_closure)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-subgroup-certificate","seed":202611091034,"parameters":{"group_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"identity":0,"point_count":6,"action_table":[[0,1,2,3,4,5],[1,2,3,4,0,5],[2,3,4,0,1,5],[3,4,0,1,2,5],[4,0,1,2,3,5]],"point":2,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-subgroup-certificate","semantic_template":"qf-mod40-stabilizer-subgroup-certificate/{\"action_table\":[[0,1,2,3,4,5],[1,2,3,4,0,5],[2,3,4,0,1,5],[3,4,0,1,2,5],[4,0,1,2,3,5]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point\":2,\"point_count\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0035","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-subgroup-certificate；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], 0, 是, 是, 是, 是, identity_product_inverse_closure)","explanation_zh":"Wave6第三層核對=([0], 0, 是, 是, 是, 是, identity_product_inverse_closure)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-subgroup-certificate","seed":202611091035,"parameters":{"group_table":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],"identity":0,"point_count":7,"action_table":[[0,1,2,3,4,5,6],[1,2,3,4,5,0,6],[2,3,4,5,0,1,6],[3,4,5,0,1,2,6],[4,5,0,1,2,3,6],[5,0,1,2,3,4,6]],"point":2,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-subgroup-certificate","semantic_template":"qf-mod40-stabilizer-subgroup-certificate/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,4,5,0,6],[2,3,4,5,0,1,6],[3,4,5,0,1,2,6],[4,5,0,1,2,3,6],[5,0,1,2,3,4,6]],\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point\":2,\"point_count\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0036","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-subgroup-certificate；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, 1], 0, 是, 是, 是, 是, identity_product_inverse_closure)","explanation_zh":"Wave6第三層核對=([0, 1], 0, 是, 是, 是, 是, identity_product_inverse_closure)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-subgroup-certificate","seed":202611091036,"parameters":{"group_table":[[0,1],[1,0]],"identity":0,"point_count":4,"action_table":[[0,1,2,3],[1,0,2,3]],"point":2,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-subgroup-certificate","semantic_template":"qf-mod40-stabilizer-subgroup-certificate/{\"action_table\":[[0,1,2,3],[1,0,2,3]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point\":2,\"point_count\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0037","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-subgroup-certificate；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], 0, 是, 是, 是, 是, identity_product_inverse_closure)","explanation_zh":"Wave6第三層核對=([0], 0, 是, 是, 是, 是, identity_product_inverse_closure)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-subgroup-certificate","seed":202611091037,"parameters":{"group_table":[[0,1,2],[1,2,0],[2,0,1]],"identity":0,"point_count":5,"action_table":[[0,1,2,3,4],[1,2,0,3,4],[2,0,1,3,4]],"point":1,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-subgroup-certificate","semantic_template":"qf-mod40-stabilizer-subgroup-certificate/{\"action_table\":[[0,1,2,3,4],[1,2,0,3,4],[2,0,1,3,4]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point\":1,\"point_count\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0038","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-subgroup-certificate；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, 是, 是, 是, 是, identity_product_inverse_closure)","explanation_zh":"Wave6第三層核對=([0], 0, 是, 是, 是, 是, identity_product_inverse_closure)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-subgroup-certificate","seed":202611091038,"parameters":{"group_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"identity":0,"point_count":6,"action_table":[[0,1,2,3,4,5],[1,2,3,0,4,5],[2,3,0,1,4,5],[3,0,1,2,4,5]],"point":0,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-subgroup-certificate","semantic_template":"qf-mod40-stabilizer-subgroup-certificate/{\"action_table\":[[0,1,2,3,4,5],[1,2,3,0,4,5],[2,3,0,1,4,5],[3,0,1,2,4,5]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point\":0,\"point_count\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0039","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-subgroup-certificate；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, 1, 2, 3, 4], 0, 是, 是, 是, 是, identity_product_inverse_closure)","explanation_zh":"Wave6第三層核對=([0, 1, 2, 3, 4], 0, 是, 是, 是, 是, identity_product_inverse_closure)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-subgroup-certificate","seed":202611091039,"parameters":{"group_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"identity":0,"point_count":7,"action_table":[[0,1,2,3,4,5,6],[1,2,3,4,0,5,6],[2,3,4,0,1,5,6],[3,4,0,1,2,5,6],[4,0,1,2,3,5,6]],"point":6,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-subgroup-certificate","semantic_template":"qf-mod40-stabilizer-subgroup-certificate/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,4,0,5,6],[2,3,4,0,1,5,6],[3,4,0,1,2,5,6],[4,0,1,2,3,5,6]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point\":6,\"point_count\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0040","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-subgroup-certificate；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, 1, 2, 3, 4, 5], 0, 是, 是, 是, 是, identity_product_inverse_closure)","explanation_zh":"Wave6第三層核對=([0, 1, 2, 3, 4, 5], 0, 是, 是, 是, 是, identity_product_inverse_closure)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-subgroup-certificate","seed":202611091040,"parameters":{"group_table":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],"identity":0,"point_count":8,"action_table":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,0,6,7],[2,3,4,5,0,1,6,7],[3,4,5,0,1,2,6,7],[4,5,0,1,2,3,6,7],[5,0,1,2,3,4,6,7]],"point":6,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-subgroup-certificate","semantic_template":"qf-mod40-stabilizer-subgroup-certificate/{\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,0,6,7],[2,3,4,5,0,1,6,7],[3,4,5,0,1,2,6,7],[4,5,0,1,2,3,6,7],[5,0,1,2,3,4,6,7]],\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point\":6,\"point_count\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0041","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-subgroup-certificate；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, 是, 是, 是, 是, identity_product_inverse_closure)","explanation_zh":"Wave6第三層核對=([0], 0, 是, 是, 是, 是, identity_product_inverse_closure)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-subgroup-certificate","seed":202611091041,"parameters":{"group_table":[[0,1],[1,0]],"identity":0,"point_count":5,"action_table":[[0,1,2,3,4],[1,0,2,3,4]],"point":0,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-subgroup-certificate","semantic_template":"qf-mod40-stabilizer-subgroup-certificate/{\"action_table\":[[0,1,2,3,4],[1,0,2,3,4]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point\":0,\"point_count\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0042","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-subgroup-certificate；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, 1, 2], 0, 是, 是, 是, 是, identity_product_inverse_closure)","explanation_zh":"Wave6第三層核對=([0, 1, 2], 0, 是, 是, 是, 是, identity_product_inverse_closure)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-subgroup-certificate","seed":202611091042,"parameters":{"group_table":[[0,1,2],[1,2,0],[2,0,1]],"identity":0,"point_count":6,"action_table":[[0,1,2,3,4,5],[1,2,0,3,4,5],[2,0,1,3,4,5]],"point":4,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-subgroup-certificate","semantic_template":"qf-mod40-stabilizer-subgroup-certificate/{\"action_table\":[[0,1,2,3,4,5],[1,2,0,3,4,5],[2,0,1,3,4,5]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point\":4,\"point_count\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0043","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-subgroup-certificate；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], 0, 是, 是, 是, 是, identity_product_inverse_closure)","explanation_zh":"Wave6第三層核對=([0], 0, 是, 是, 是, 是, identity_product_inverse_closure)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-subgroup-certificate","seed":202611091043,"parameters":{"group_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"identity":0,"point_count":7,"action_table":[[0,1,2,3,4,5,6],[1,2,3,0,4,5,6],[2,3,0,1,4,5,6],[3,0,1,2,4,5,6]],"point":3,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-subgroup-certificate","semantic_template":"qf-mod40-stabilizer-subgroup-certificate/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,0,4,5,6],[2,3,0,1,4,5,6],[3,0,1,2,4,5,6]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point\":3,\"point_count\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0044","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-subgroup-certificate；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], 0, 是, 是, 是, 是, identity_product_inverse_closure)","explanation_zh":"Wave6第三層核對=([0], 0, 是, 是, 是, 是, identity_product_inverse_closure)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-subgroup-certificate","seed":202611091044,"parameters":{"group_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"identity":0,"point_count":8,"action_table":[[0,1,2,3,4,5,6,7],[1,2,3,4,0,5,6,7],[2,3,4,0,1,5,6,7],[3,4,0,1,2,5,6,7],[4,0,1,2,3,5,6,7]],"point":2,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-subgroup-certificate","semantic_template":"qf-mod40-stabilizer-subgroup-certificate/{\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,0,5,6,7],[2,3,4,0,1,5,6,7],[3,4,0,1,2,5,6,7],[4,0,1,2,3,5,6,7]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point\":2,\"point_count\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0045","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-subgroup-certificate；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], 0, 是, 是, 是, 是, identity_product_inverse_closure)","explanation_zh":"Wave6第三層核對=([0], 0, 是, 是, 是, 是, identity_product_inverse_closure)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-subgroup-certificate","seed":202611091045,"parameters":{"group_table":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],"identity":0,"point_count":9,"action_table":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,5,0,6,7,8],[2,3,4,5,0,1,6,7,8],[3,4,5,0,1,2,6,7,8],[4,5,0,1,2,3,6,7,8],[5,0,1,2,3,4,6,7,8]],"point":1,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-subgroup-certificate","semantic_template":"qf-mod40-stabilizer-subgroup-certificate/{\"action_table\":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,5,0,6,7,8],[2,3,4,5,0,1,6,7,8],[3,4,5,0,1,2,6,7,8],[4,5,0,1,2,3,6,7,8],[5,0,1,2,3,4,6,7,8]],\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point\":1,\"point_count\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0046","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-subgroup-certificate；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, 1], 0, 是, 是, 是, 是, identity_product_inverse_closure)","explanation_zh":"Wave6第三層核對=([0, 1], 0, 是, 是, 是, 是, identity_product_inverse_closure)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-subgroup-certificate","seed":202611091046,"parameters":{"group_table":[[0,1],[1,0]],"identity":0,"point_count":6,"action_table":[[0,1,2,3,4,5],[1,0,2,3,4,5]],"point":2,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-subgroup-certificate","semantic_template":"qf-mod40-stabilizer-subgroup-certificate/{\"action_table\":[[0,1,2,3,4,5],[1,0,2,3,4,5]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point\":2,\"point_count\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0047","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-subgroup-certificate；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, 是, 是, 是, 是, identity_product_inverse_closure)","explanation_zh":"Wave6第三層核對=([0], 0, 是, 是, 是, 是, identity_product_inverse_closure)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-subgroup-certificate","seed":202611091047,"parameters":{"group_table":[[0,1,2],[1,2,0],[2,0,1]],"identity":0,"point_count":7,"action_table":[[0,1,2,3,4,5,6],[1,2,0,3,4,5,6],[2,0,1,3,4,5,6]],"point":0,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-subgroup-certificate","semantic_template":"qf-mod40-stabilizer-subgroup-certificate/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,0,3,4,5,6],[2,0,1,3,4,5,6]],\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point\":0,\"point_count\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0048","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-subgroup-certificate；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, 1, 2, 3], 0, 是, 是, 是, 是, identity_product_inverse_closure)","explanation_zh":"Wave6第三層核對=([0, 1, 2, 3], 0, 是, 是, 是, 是, identity_product_inverse_closure)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-subgroup-certificate","seed":202611091048,"parameters":{"group_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"identity":0,"point_count":8,"action_table":[[0,1,2,3,4,5,6,7],[1,2,3,0,4,5,6,7],[2,3,0,1,4,5,6,7],[3,0,1,2,4,5,6,7]],"point":6,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-subgroup-certificate","semantic_template":"qf-mod40-stabilizer-subgroup-certificate/{\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,0,4,5,6,7],[2,3,0,1,4,5,6,7],[3,0,1,2,4,5,6,7]],\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point\":6,\"point_count\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0049","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-subgroup-certificate；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, 1, 2, 3, 4], 0, 是, 是, 是, 是, identity_product_inverse_closure)","explanation_zh":"Wave6第三層核對=([0, 1, 2, 3, 4], 0, 是, 是, 是, 是, identity_product_inverse_closure)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-subgroup-certificate","seed":202611091049,"parameters":{"group_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"identity":0,"point_count":9,"action_table":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,0,5,6,7,8],[2,3,4,0,1,5,6,7,8],[3,4,0,1,2,5,6,7,8],[4,0,1,2,3,5,6,7,8]],"point":5,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-subgroup-certificate","semantic_template":"qf-mod40-stabilizer-subgroup-certificate/{\"action_table\":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,0,5,6,7,8],[2,3,4,0,1,5,6,7,8],[3,4,0,1,2,5,6,7,8],[4,0,1,2,3,5,6,7,8]],\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point\":5,\"point_count\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0050","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-subgroup-certificate；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], 0, 是, 是, 是, 是, identity_product_inverse_closure)","explanation_zh":"Wave6第三層核對=([0], 0, 是, 是, 是, 是, identity_product_inverse_closure)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-subgroup-certificate","seed":202611091050,"parameters":{"group_table":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],"identity":0,"point_count":10,"action_table":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,0,6,7,8,9],[2,3,4,5,0,1,6,7,8,9],[3,4,5,0,1,2,6,7,8,9],[4,5,0,1,2,3,6,7,8,9],[5,0,1,2,3,4,6,7,8,9]],"point":4,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-subgroup-certificate","semantic_template":"qf-mod40-stabilizer-subgroup-certificate/{\"action_table\":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,0,6,7,8,9],[2,3,4,5,0,1,6,7,8,9],[3,4,5,0,1,2,6,7,8,9],[4,5,0,1,2,3,6,7,8,9],[5,0,1,2,3,4,6,7,8,9]],\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point\":4,\"point_count\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0051","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-membership-audit；資料={\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":2,\"action_table\":[[0,1],[1,0]],\"point\":0,\"group_element\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 0, 0, 是, 是)","explanation_zh":"Wave6第三層核對=(0, 0, 0, 是, 是)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-membership-audit","seed":202611091051,"parameters":{"group_table":[[0,1],[1,0]],"identity":0,"point_count":2,"action_table":[[0,1],[1,0]],"point":0,"group_element":0,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-membership-audit","semantic_template":"qf-mod40-stabilizer-membership-audit/{\"action_table\":[[0,1],[1,0]],\"group_element\":0,\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point\":0,\"point_count\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0052","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-membership-audit；資料={\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":3,\"action_table\":[[0,1,2],[1,2,0],[2,0,1]],\"point\":1,\"group_element\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 1, 2, 否, 否)","explanation_zh":"Wave6第三層核對=(1, 1, 2, 否, 否)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-membership-audit","seed":202611091052,"parameters":{"group_table":[[0,1,2],[1,2,0],[2,0,1]],"identity":0,"point_count":3,"action_table":[[0,1,2],[1,2,0],[2,0,1]],"point":1,"group_element":1,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-membership-audit","semantic_template":"qf-mod40-stabilizer-membership-audit/{\"action_table\":[[0,1,2],[1,2,0],[2,0,1]],\"group_element\":1,\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point\":1,\"point_count\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0053","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-membership-audit；資料={\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":4,\"action_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"point\":2,\"group_element\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 0, 否, 否)","explanation_zh":"Wave6第三層核對=(2, 2, 0, 否, 否)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-membership-audit","seed":202611091053,"parameters":{"group_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"identity":0,"point_count":4,"action_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"point":2,"group_element":2,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-membership-audit","semantic_template":"qf-mod40-stabilizer-membership-audit/{\"action_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"group_element\":2,\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point\":2,\"point_count\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0054","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-membership-audit；資料={\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":5,\"action_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"point\":3,\"group_element\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 3, 1, 否, 否)","explanation_zh":"Wave6第三層核對=(3, 3, 1, 否, 否)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-membership-audit","seed":202611091054,"parameters":{"group_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"identity":0,"point_count":5,"action_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"point":3,"group_element":3,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-membership-audit","semantic_template":"qf-mod40-stabilizer-membership-audit/{\"action_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"group_element\":3,\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point\":3,\"point_count\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0055","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-membership-audit；資料={\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point_count\":6,\"action_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"point\":4,\"group_element\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 4, 2, 否, 否)","explanation_zh":"Wave6第三層核對=(4, 4, 2, 否, 否)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-membership-audit","seed":202611091055,"parameters":{"group_table":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],"identity":0,"point_count":6,"action_table":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],"point":4,"group_element":4,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-membership-audit","semantic_template":"qf-mod40-stabilizer-membership-audit/{\"action_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"group_element\":4,\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point\":4,\"point_count\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0056","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-membership-audit；資料={\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":3,\"action_table\":[[0,1,2],[1,0,2]],\"point\":2,\"group_element\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 2, 2, 是, 是)","explanation_zh":"Wave6第三層核對=(1, 2, 2, 是, 是)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-membership-audit","seed":202611091056,"parameters":{"group_table":[[0,1],[1,0]],"identity":0,"point_count":3,"action_table":[[0,1,2],[1,0,2]],"point":2,"group_element":1,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-membership-audit","semantic_template":"qf-mod40-stabilizer-membership-audit/{\"action_table\":[[0,1,2],[1,0,2]],\"group_element\":1,\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point\":2,\"point_count\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0057","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-membership-audit；資料={\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":4,\"action_table\":[[0,1,2,3],[1,2,0,3],[2,0,1,3]],\"point\":2,\"group_element\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 2, 2, 是, 是)","explanation_zh":"Wave6第三層核對=(0, 2, 2, 是, 是)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-membership-audit","seed":202611091057,"parameters":{"group_table":[[0,1,2],[1,2,0],[2,0,1]],"identity":0,"point_count":4,"action_table":[[0,1,2,3],[1,2,0,3],[2,0,1,3]],"point":2,"group_element":0,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-membership-audit","semantic_template":"qf-mod40-stabilizer-membership-audit/{\"action_table\":[[0,1,2,3],[1,2,0,3],[2,0,1,3]],\"group_element\":0,\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point\":2,\"point_count\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0058","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-membership-audit；資料={\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":5,\"action_table\":[[0,1,2,3,4],[1,2,3,0,4],[2,3,0,1,4],[3,0,1,2,4]],\"point\":2,\"group_element\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 2, 1, 否, 否)","explanation_zh":"Wave6第三層核對=(3, 2, 1, 否, 否)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-membership-audit","seed":202611091058,"parameters":{"group_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"identity":0,"point_count":5,"action_table":[[0,1,2,3,4],[1,2,3,0,4],[2,3,0,1,4],[3,0,1,2,4]],"point":2,"group_element":3,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-membership-audit","semantic_template":"qf-mod40-stabilizer-membership-audit/{\"action_table\":[[0,1,2,3,4],[1,2,3,0,4],[2,3,0,1,4],[3,0,1,2,4]],\"group_element\":3,\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point\":2,\"point_count\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0059","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-membership-audit；資料={\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":6,\"action_table\":[[0,1,2,3,4,5],[1,2,3,4,0,5],[2,3,4,0,1,5],[3,4,0,1,2,5],[4,0,1,2,3,5]],\"point\":2,\"group_element\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 2, 0, 否, 否)","explanation_zh":"Wave6第三層核對=(3, 2, 0, 否, 否)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-membership-audit","seed":202611091059,"parameters":{"group_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"identity":0,"point_count":6,"action_table":[[0,1,2,3,4,5],[1,2,3,4,0,5],[2,3,4,0,1,5],[3,4,0,1,2,5],[4,0,1,2,3,5]],"point":2,"group_element":3,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-membership-audit","semantic_template":"qf-mod40-stabilizer-membership-audit/{\"action_table\":[[0,1,2,3,4,5],[1,2,3,4,0,5],[2,3,4,0,1,5],[3,4,0,1,2,5],[4,0,1,2,3,5]],\"group_element\":3,\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point\":2,\"point_count\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0060","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-membership-audit；資料={\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point_count\":7,\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,4,5,0,6],[2,3,4,5,0,1,6],[3,4,5,0,1,2,6],[4,5,0,1,2,3,6],[5,0,1,2,3,4,6]],\"point\":2,\"group_element\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 2, 5, 否, 否)","explanation_zh":"Wave6第三層核對=(3, 2, 5, 否, 否)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-membership-audit","seed":202611091060,"parameters":{"group_table":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],"identity":0,"point_count":7,"action_table":[[0,1,2,3,4,5,6],[1,2,3,4,5,0,6],[2,3,4,5,0,1,6],[3,4,5,0,1,2,6],[4,5,0,1,2,3,6],[5,0,1,2,3,4,6]],"point":2,"group_element":3,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-membership-audit","semantic_template":"qf-mod40-stabilizer-membership-audit/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,4,5,0,6],[2,3,4,5,0,1,6],[3,4,5,0,1,2,6],[4,5,0,1,2,3,6],[5,0,1,2,3,4,6]],\"group_element\":3,\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point\":2,\"point_count\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0061","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-membership-audit；資料={\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":4,\"action_table\":[[0,1,2,3],[1,0,2,3]],\"point\":2,\"group_element\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 2, 2, 是, 是)","explanation_zh":"Wave6第三層核對=(0, 2, 2, 是, 是)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-membership-audit","seed":202611091061,"parameters":{"group_table":[[0,1],[1,0]],"identity":0,"point_count":4,"action_table":[[0,1,2,3],[1,0,2,3]],"point":2,"group_element":0,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-membership-audit","semantic_template":"qf-mod40-stabilizer-membership-audit/{\"action_table\":[[0,1,2,3],[1,0,2,3]],\"group_element\":0,\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point\":2,\"point_count\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0062","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-membership-audit；資料={\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":5,\"action_table\":[[0,1,2,3,4],[1,2,0,3,4],[2,0,1,3,4]],\"point\":1,\"group_element\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 1, 0, 否, 否)","explanation_zh":"Wave6第三層核對=(2, 1, 0, 否, 否)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-membership-audit","seed":202611091062,"parameters":{"group_table":[[0,1,2],[1,2,0],[2,0,1]],"identity":0,"point_count":5,"action_table":[[0,1,2,3,4],[1,2,0,3,4],[2,0,1,3,4]],"point":1,"group_element":2,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-membership-audit","semantic_template":"qf-mod40-stabilizer-membership-audit/{\"action_table\":[[0,1,2,3,4],[1,2,0,3,4],[2,0,1,3,4]],\"group_element\":2,\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point\":1,\"point_count\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0063","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-membership-audit；資料={\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":6,\"action_table\":[[0,1,2,3,4,5],[1,2,3,0,4,5],[2,3,0,1,4,5],[3,0,1,2,4,5]],\"point\":0,\"group_element\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 0, 0, 是, 是)","explanation_zh":"Wave6第三層核對=(0, 0, 0, 是, 是)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-membership-audit","seed":202611091063,"parameters":{"group_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"identity":0,"point_count":6,"action_table":[[0,1,2,3,4,5],[1,2,3,0,4,5],[2,3,0,1,4,5],[3,0,1,2,4,5]],"point":0,"group_element":0,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-membership-audit","semantic_template":"qf-mod40-stabilizer-membership-audit/{\"action_table\":[[0,1,2,3,4,5],[1,2,3,0,4,5],[2,3,0,1,4,5],[3,0,1,2,4,5]],\"group_element\":0,\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point\":0,\"point_count\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0064","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-membership-audit；資料={\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":7,\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,4,0,5,6],[2,3,4,0,1,5,6],[3,4,0,1,2,5,6],[4,0,1,2,3,5,6]],\"point\":6,\"group_element\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 6, 6, 是, 是)","explanation_zh":"Wave6第三層核對=(3, 6, 6, 是, 是)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-membership-audit","seed":202611091064,"parameters":{"group_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"identity":0,"point_count":7,"action_table":[[0,1,2,3,4,5,6],[1,2,3,4,0,5,6],[2,3,4,0,1,5,6],[3,4,0,1,2,5,6],[4,0,1,2,3,5,6]],"point":6,"group_element":3,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-membership-audit","semantic_template":"qf-mod40-stabilizer-membership-audit/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,4,0,5,6],[2,3,4,0,1,5,6],[3,4,0,1,2,5,6],[4,0,1,2,3,5,6]],\"group_element\":3,\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point\":6,\"point_count\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0065","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-membership-audit；資料={\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point_count\":8,\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,0,6,7],[2,3,4,5,0,1,6,7],[3,4,5,0,1,2,6,7],[4,5,0,1,2,3,6,7],[5,0,1,2,3,4,6,7]],\"point\":6,\"group_element\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 6, 6, 是, 是)","explanation_zh":"Wave6第三層核對=(2, 6, 6, 是, 是)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-membership-audit","seed":202611091065,"parameters":{"group_table":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],"identity":0,"point_count":8,"action_table":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,0,6,7],[2,3,4,5,0,1,6,7],[3,4,5,0,1,2,6,7],[4,5,0,1,2,3,6,7],[5,0,1,2,3,4,6,7]],"point":6,"group_element":2,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-membership-audit","semantic_template":"qf-mod40-stabilizer-membership-audit/{\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,0,6,7],[2,3,4,5,0,1,6,7],[3,4,5,0,1,2,6,7],[4,5,0,1,2,3,6,7],[5,0,1,2,3,4,6,7]],\"group_element\":2,\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point\":6,\"point_count\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0066","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-membership-audit；資料={\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":5,\"action_table\":[[0,1,2,3,4],[1,0,2,3,4]],\"point\":0,\"group_element\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 0, 1, 否, 否)","explanation_zh":"Wave6第三層核對=(1, 0, 1, 否, 否)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-membership-audit","seed":202611091066,"parameters":{"group_table":[[0,1],[1,0]],"identity":0,"point_count":5,"action_table":[[0,1,2,3,4],[1,0,2,3,4]],"point":0,"group_element":1,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-membership-audit","semantic_template":"qf-mod40-stabilizer-membership-audit/{\"action_table\":[[0,1,2,3,4],[1,0,2,3,4]],\"group_element\":1,\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point\":0,\"point_count\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0067","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-membership-audit；資料={\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":6,\"action_table\":[[0,1,2,3,4,5],[1,2,0,3,4,5],[2,0,1,3,4,5]],\"point\":4,\"group_element\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 4, 4, 是, 是)","explanation_zh":"Wave6第三層核對=(1, 4, 4, 是, 是)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-membership-audit","seed":202611091067,"parameters":{"group_table":[[0,1,2],[1,2,0],[2,0,1]],"identity":0,"point_count":6,"action_table":[[0,1,2,3,4,5],[1,2,0,3,4,5],[2,0,1,3,4,5]],"point":4,"group_element":1,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-membership-audit","semantic_template":"qf-mod40-stabilizer-membership-audit/{\"action_table\":[[0,1,2,3,4,5],[1,2,0,3,4,5],[2,0,1,3,4,5]],\"group_element\":1,\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point\":4,\"point_count\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0068","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-membership-audit；資料={\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":7,\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,0,4,5,6],[2,3,0,1,4,5,6],[3,0,1,2,4,5,6]],\"point\":3,\"group_element\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 3, 0, 否, 否)","explanation_zh":"Wave6第三層核對=(1, 3, 0, 否, 否)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-membership-audit","seed":202611091068,"parameters":{"group_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"identity":0,"point_count":7,"action_table":[[0,1,2,3,4,5,6],[1,2,3,0,4,5,6],[2,3,0,1,4,5,6],[3,0,1,2,4,5,6]],"point":3,"group_element":1,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-membership-audit","semantic_template":"qf-mod40-stabilizer-membership-audit/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,0,4,5,6],[2,3,0,1,4,5,6],[3,0,1,2,4,5,6]],\"group_element\":1,\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point\":3,\"point_count\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0069","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-membership-audit；資料={\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":8,\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,0,5,6,7],[2,3,4,0,1,5,6,7],[3,4,0,1,2,5,6,7],[4,0,1,2,3,5,6,7]],\"point\":2,\"group_element\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 2, 0, 否, 否)","explanation_zh":"Wave6第三層核對=(3, 2, 0, 否, 否)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-membership-audit","seed":202611091069,"parameters":{"group_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"identity":0,"point_count":8,"action_table":[[0,1,2,3,4,5,6,7],[1,2,3,4,0,5,6,7],[2,3,4,0,1,5,6,7],[3,4,0,1,2,5,6,7],[4,0,1,2,3,5,6,7]],"point":2,"group_element":3,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-membership-audit","semantic_template":"qf-mod40-stabilizer-membership-audit/{\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,0,5,6,7],[2,3,4,0,1,5,6,7],[3,4,0,1,2,5,6,7],[4,0,1,2,3,5,6,7]],\"group_element\":3,\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point\":2,\"point_count\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0070","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-membership-audit；資料={\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point_count\":9,\"action_table\":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,5,0,6,7,8],[2,3,4,5,0,1,6,7,8],[3,4,5,0,1,2,6,7,8],[4,5,0,1,2,3,6,7,8],[5,0,1,2,3,4,6,7,8]],\"point\":1,\"group_element\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 1, 2, 否, 否)","explanation_zh":"Wave6第三層核對=(1, 1, 2, 否, 否)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-membership-audit","seed":202611091070,"parameters":{"group_table":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],"identity":0,"point_count":9,"action_table":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,5,0,6,7,8],[2,3,4,5,0,1,6,7,8],[3,4,5,0,1,2,6,7,8],[4,5,0,1,2,3,6,7,8],[5,0,1,2,3,4,6,7,8]],"point":1,"group_element":1,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-membership-audit","semantic_template":"qf-mod40-stabilizer-membership-audit/{\"action_table\":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,5,0,6,7,8],[2,3,4,5,0,1,6,7,8],[3,4,5,0,1,2,6,7,8],[4,5,0,1,2,3,6,7,8],[5,0,1,2,3,4,6,7,8]],\"group_element\":1,\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point\":1,\"point_count\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0071","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-membership-audit；資料={\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":6,\"action_table\":[[0,1,2,3,4,5],[1,0,2,3,4,5]],\"point\":2,\"group_element\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 2, 2, 是, 是)","explanation_zh":"Wave6第三層核對=(0, 2, 2, 是, 是)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-membership-audit","seed":202611091071,"parameters":{"group_table":[[0,1],[1,0]],"identity":0,"point_count":6,"action_table":[[0,1,2,3,4,5],[1,0,2,3,4,5]],"point":2,"group_element":0,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-membership-audit","semantic_template":"qf-mod40-stabilizer-membership-audit/{\"action_table\":[[0,1,2,3,4,5],[1,0,2,3,4,5]],\"group_element\":0,\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point\":2,\"point_count\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0072","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-membership-audit；資料={\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point_count\":7,\"action_table\":[[0,1,2,3,4,5,6],[1,2,0,3,4,5,6],[2,0,1,3,4,5,6]],\"point\":0,\"group_element\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 0, 0, 是, 是)","explanation_zh":"Wave6第三層核對=(0, 0, 0, 是, 是)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-membership-audit","seed":202611091072,"parameters":{"group_table":[[0,1,2],[1,2,0],[2,0,1]],"identity":0,"point_count":7,"action_table":[[0,1,2,3,4,5,6],[1,2,0,3,4,5,6],[2,0,1,3,4,5,6]],"point":0,"group_element":0,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-membership-audit","semantic_template":"qf-mod40-stabilizer-membership-audit/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,0,3,4,5,6],[2,0,1,3,4,5,6]],\"group_element\":0,\"group_table\":[[0,1,2],[1,2,0],[2,0,1]],\"identity\":0,\"point\":0,\"point_count\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0073","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-membership-audit；資料={\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point_count\":8,\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,0,4,5,6,7],[2,3,0,1,4,5,6,7],[3,0,1,2,4,5,6,7]],\"point\":6,\"group_element\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 6, 6, 是, 是)","explanation_zh":"Wave6第三層核對=(2, 6, 6, 是, 是)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-membership-audit","seed":202611091073,"parameters":{"group_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"identity":0,"point_count":8,"action_table":[[0,1,2,3,4,5,6,7],[1,2,3,0,4,5,6,7],[2,3,0,1,4,5,6,7],[3,0,1,2,4,5,6,7]],"point":6,"group_element":2,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-membership-audit","semantic_template":"qf-mod40-stabilizer-membership-audit/{\"action_table\":[[0,1,2,3,4,5,6,7],[1,2,3,0,4,5,6,7],[2,3,0,1,4,5,6,7],[3,0,1,2,4,5,6,7]],\"group_element\":2,\"group_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"identity\":0,\"point\":6,\"point_count\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0074","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-membership-audit；資料={\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point_count\":9,\"action_table\":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,0,5,6,7,8],[2,3,4,0,1,5,6,7,8],[3,4,0,1,2,5,6,7,8],[4,0,1,2,3,5,6,7,8]],\"point\":5,\"group_element\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 5, 5, 是, 是)","explanation_zh":"Wave6第三層核對=(3, 5, 5, 是, 是)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-membership-audit","seed":202611091074,"parameters":{"group_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"identity":0,"point_count":9,"action_table":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,0,5,6,7,8],[2,3,4,0,1,5,6,7,8],[3,4,0,1,2,5,6,7,8],[4,0,1,2,3,5,6,7,8]],"point":5,"group_element":3,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-membership-audit","semantic_template":"qf-mod40-stabilizer-membership-audit/{\"action_table\":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,0,5,6,7,8],[2,3,4,0,1,5,6,7,8],[3,4,0,1,2,5,6,7,8],[4,0,1,2,3,5,6,7,8]],\"group_element\":3,\"group_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"identity\":0,\"point\":5,\"point_count\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0075","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-membership-audit；資料={\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point_count\":10,\"action_table\":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,0,6,7,8,9],[2,3,4,5,0,1,6,7,8,9],[3,4,5,0,1,2,6,7,8,9],[4,5,0,1,2,3,6,7,8,9],[5,0,1,2,3,4,6,7,8,9]],\"point\":4,\"group_element\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 4, 4, 是, 是)","explanation_zh":"Wave6第三層核對=(0, 4, 4, 是, 是)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-membership-audit","seed":202611091075,"parameters":{"group_table":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],"identity":0,"point_count":10,"action_table":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,0,6,7,8,9],[2,3,4,5,0,1,6,7,8,9],[3,4,5,0,1,2,6,7,8,9],[4,5,0,1,2,3,6,7,8,9],[5,0,1,2,3,4,6,7,8,9]],"point":4,"group_element":0,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-membership-audit","semantic_template":"qf-mod40-stabilizer-membership-audit/{\"action_table\":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,0,6,7,8,9],[2,3,4,5,0,1,6,7,8,9],[3,4,5,0,1,2,6,7,8,9],[4,5,0,1,2,3,6,7,8,9],[5,0,1,2,3,4,6,7,8,9]],\"group_element\":0,\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point\":4,\"point_count\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0076","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-point-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}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [0], 1, [0], 是, 否, stabilizer_not_orbit_or_action_kernel)","explanation_zh":"Wave6第三層核對=(0, [0], 1, [0], 是, 否, stabilizer_not_orbit_or_action_kernel)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-point-boundary","seed":202611091076,"parameters":{"group_table":[[0,1],[1,0]],"identity":0,"point_count":2,"action_table":[[0,1],[1,0]],"point":0,"other_point":1,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-point-boundary","semantic_template":"qf-mod40-stabilizer-point-boundary/{\"action_table\":[[0,1],[1,0]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"other_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、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0077","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-point-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}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, [0], 2, [0], 是, 否, stabilizer_not_orbit_or_action_kernel)","explanation_zh":"Wave6第三層核對=(1, [0], 2, [0], 是, 否, stabilizer_not_orbit_or_action_kernel)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-point-boundary","seed":202611091077,"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,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-point-boundary","semantic_template":"qf-mod40-stabilizer-point-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}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0078","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-point-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}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [0], 3, [0], 是, 否, stabilizer_not_orbit_or_action_kernel)","explanation_zh":"Wave6第三層核對=(2, [0], 3, [0], 是, 否, stabilizer_not_orbit_or_action_kernel)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-point-boundary","seed":202611091078,"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,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-point-boundary","semantic_template":"qf-mod40-stabilizer-point-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}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0079","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-point-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}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [0], 4, [0], 是, 否, stabilizer_not_orbit_or_action_kernel)","explanation_zh":"Wave6第三層核對=(3, [0], 4, [0], 是, 否, stabilizer_not_orbit_or_action_kernel)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-point-boundary","seed":202611091079,"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,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-point-boundary","semantic_template":"qf-mod40-stabilizer-point-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}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0080","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-point-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}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, [0], 5, [0], 是, 否, stabilizer_not_orbit_or_action_kernel)","explanation_zh":"Wave6第三層核對=(4, [0], 5, [0], 是, 否, stabilizer_not_orbit_or_action_kernel)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-point-boundary","seed":202611091080,"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,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-point-boundary","semantic_template":"qf-mod40-stabilizer-point-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}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0081","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-point-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}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [0, 1], 0, [0], 否, 否, stabilizer_not_orbit_or_action_kernel)","explanation_zh":"Wave6第三層核對=(2, [0, 1], 0, [0], 否, 否, stabilizer_not_orbit_or_action_kernel)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-point-boundary","seed":202611091081,"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,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-point-boundary","semantic_template":"qf-mod40-stabilizer-point-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}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0082","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-point-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}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [0], 3, [0, 1, 2], 否, 否, stabilizer_not_orbit_or_action_kernel)","explanation_zh":"Wave6第三層核對=(2, [0], 3, [0, 1, 2], 否, 否, stabilizer_not_orbit_or_action_kernel)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-point-boundary","seed":202611091082,"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,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-point-boundary","semantic_template":"qf-mod40-stabilizer-point-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}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0083","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-point-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}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [0], 3, [0], 是, 否, stabilizer_not_orbit_or_action_kernel)","explanation_zh":"Wave6第三層核對=(2, [0], 3, [0], 是, 否, stabilizer_not_orbit_or_action_kernel)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-point-boundary","seed":202611091083,"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,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-point-boundary","semantic_template":"qf-mod40-stabilizer-point-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}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0084","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-point-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}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [0], 3, [0], 是, 否, stabilizer_not_orbit_or_action_kernel)","explanation_zh":"Wave6第三層核對=(2, [0], 3, [0], 是, 否, stabilizer_not_orbit_or_action_kernel)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-point-boundary","seed":202611091084,"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,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-point-boundary","semantic_template":"qf-mod40-stabilizer-point-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}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0085","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-point-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}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [0], 3, [0], 是, 否, stabilizer_not_orbit_or_action_kernel)","explanation_zh":"Wave6第三層核對=(2, [0], 3, [0], 是, 否, stabilizer_not_orbit_or_action_kernel)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-point-boundary","seed":202611091085,"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,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-point-boundary","semantic_template":"qf-mod40-stabilizer-point-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}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0086","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-point-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}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [0, 1], 3, [0, 1], 是, 否, stabilizer_not_orbit_or_action_kernel)","explanation_zh":"Wave6第三層核對=(2, [0, 1], 3, [0, 1], 是, 否, stabilizer_not_orbit_or_action_kernel)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-point-boundary","seed":202611091086,"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,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-point-boundary","semantic_template":"qf-mod40-stabilizer-point-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}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0087","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-point-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}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, [0], 2, [0], 是, 否, stabilizer_not_orbit_or_action_kernel)","explanation_zh":"Wave6第三層核對=(1, [0], 2, [0], 是, 否, stabilizer_not_orbit_or_action_kernel)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-point-boundary","seed":202611091087,"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,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-point-boundary","semantic_template":"qf-mod40-stabilizer-point-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}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0088","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-point-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}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [0], 1, [0], 是, 否, stabilizer_not_orbit_or_action_kernel)","explanation_zh":"Wave6第三層核對=(0, [0], 1, [0], 是, 否, stabilizer_not_orbit_or_action_kernel)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-point-boundary","seed":202611091088,"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,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-point-boundary","semantic_template":"qf-mod40-stabilizer-point-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}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0089","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-point-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}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, [0, 1, 2, 3, 4], 0, [0], 否, 否, stabilizer_not_orbit_or_action_kernel)","explanation_zh":"Wave6第三層核對=(6, [0, 1, 2, 3, 4], 0, [0], 否, 否, stabilizer_not_orbit_or_action_kernel)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-point-boundary","seed":202611091089,"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,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-point-boundary","semantic_template":"qf-mod40-stabilizer-point-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}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0090","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-point-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}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, [0, 1, 2, 3, 4, 5], 7, [0, 1, 2, 3, 4, 5], 是, 否, stabilizer_not_orbit_or_action_kernel)","explanation_zh":"Wave6第三層核對=(6, [0, 1, 2, 3, 4, 5], 7, [0, 1, 2, 3, 4, 5], 是, 否, stabilizer_not_orbit_or_action_kernel)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-point-boundary","seed":202611091090,"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,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-point-boundary","semantic_template":"qf-mod40-stabilizer-point-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}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0091","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-point-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}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [0], 1, [0], 是, 否, stabilizer_not_orbit_or_action_kernel)","explanation_zh":"Wave6第三層核對=(0, [0], 1, [0], 是, 否, stabilizer_not_orbit_or_action_kernel)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-point-boundary","seed":202611091091,"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,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-point-boundary","semantic_template":"qf-mod40-stabilizer-point-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}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0092","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-point-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}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, [0, 1, 2], 5, [0, 1, 2], 是, 否, stabilizer_not_orbit_or_action_kernel)","explanation_zh":"Wave6第三層核對=(4, [0, 1, 2], 5, [0, 1, 2], 是, 否, stabilizer_not_orbit_or_action_kernel)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-point-boundary","seed":202611091092,"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,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-point-boundary","semantic_template":"qf-mod40-stabilizer-point-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}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0093","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-point-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}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [0], 4, [0, 1, 2, 3], 否, 否, stabilizer_not_orbit_or_action_kernel)","explanation_zh":"Wave6第三層核對=(3, [0], 4, [0, 1, 2, 3], 否, 否, stabilizer_not_orbit_or_action_kernel)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-point-boundary","seed":202611091093,"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,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-point-boundary","semantic_template":"qf-mod40-stabilizer-point-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}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0094","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-point-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}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [0], 3, [0], 是, 否, stabilizer_not_orbit_or_action_kernel)","explanation_zh":"Wave6第三層核對=(2, [0], 3, [0], 是, 否, stabilizer_not_orbit_or_action_kernel)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-point-boundary","seed":202611091094,"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,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-point-boundary","semantic_template":"qf-mod40-stabilizer-point-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}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0095","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-point-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}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, [0], 2, [0], 是, 否, stabilizer_not_orbit_or_action_kernel)","explanation_zh":"Wave6第三層核對=(1, [0], 2, [0], 是, 否, stabilizer_not_orbit_or_action_kernel)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-point-boundary","seed":202611091095,"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,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-point-boundary","semantic_template":"qf-mod40-stabilizer-point-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}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0096","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-point-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}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [0, 1], 3, [0, 1], 是, 否, stabilizer_not_orbit_or_action_kernel)","explanation_zh":"Wave6第三層核對=(2, [0, 1], 3, [0, 1], 是, 否, stabilizer_not_orbit_or_action_kernel)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-point-boundary","seed":202611091096,"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,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-point-boundary","semantic_template":"qf-mod40-stabilizer-point-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}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0097","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-point-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}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [0], 1, [0], 是, 否, stabilizer_not_orbit_or_action_kernel)","explanation_zh":"Wave6第三層核對=(0, [0], 1, [0], 是, 否, stabilizer_not_orbit_or_action_kernel)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-point-boundary","seed":202611091097,"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,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-point-boundary","semantic_template":"qf-mod40-stabilizer-point-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}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0098","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-point-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}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, [0, 1, 2, 3], 7, [0, 1, 2, 3], 是, 否, stabilizer_not_orbit_or_action_kernel)","explanation_zh":"Wave6第三層核對=(6, [0, 1, 2, 3], 7, [0, 1, 2, 3], 是, 否, stabilizer_not_orbit_or_action_kernel)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-point-boundary","seed":202611091098,"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,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-point-boundary","semantic_template":"qf-mod40-stabilizer-point-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}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0099","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-point-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}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, [0, 1, 2, 3, 4], 6, [0, 1, 2, 3, 4], 是, 否, stabilizer_not_orbit_or_action_kernel)","explanation_zh":"Wave6第三層核對=(5, [0, 1, 2, 3, 4], 6, [0, 1, 2, 3, 4], 是, 否, stabilizer_not_orbit_or_action_kernel)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-point-boundary","seed":202611091099,"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,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-point-boundary","semantic_template":"qf-mod40-stabilizer-point-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}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r01-finite-group-stabilizer-0100","batch_id":"batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-stabilizer"],"title_zh":"有限群作用的穩定子群","stem_zh":"完整已認證有限群作用與指定point。操作=stabilizer-point-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}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, [0], 5, [0], 是, 否, stabilizer_not_orbit_or_action_kernel)","explanation_zh":"Wave6第三層核對=(4, [0], 5, [0], 是, 否, stabilizer_not_orbit_or_action_kernel)；依完整明示資料獨立重算；stabilizer遍歷全部g並完整驗identity、product與inverse closure；不得把single fixed element、action kernel或orbit混同stabilizer。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群作用的穩定子群"]},"variant":{"family_id":"qf-mod40-stabilizer-point-boundary","seed":202611091100,"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,"domain":"Complete_finite_group_action_stabilizer","operation":"stabilizer-point-boundary","semantic_template":"qf-mod40-stabilizer-point-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}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}