{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0001","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-qualification；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,6,0,2,4,6],[0,3,6,1,4,7,2,5],[0,4,0,4,0,4,0,4],[0,5,2,7,4,1,6,3],[0,6,4,2,0,6,4,2],[0,7,6,5,4,3,2,1]],\"subset\":[0,1,2,4,6]}。請依 [subset,zero,additive_subgroup,missing_left,missing_right,two_sided_ideal] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 4, 6], 0, 否, [[3, 1, 3], [5, 1, 5], [7, 1, 7]], [[1, 3, 3], [1, 5, 5], [1, 7, 7]], 否)","explanation_zh":"Wave4第二層核對=([0, 1, 2, 4, 6], 0, 否, [[3, 1, 3], [5, 1, 5], [7, 1, 7]], [[1, 3, 3], [1, 5, 5], [1, 7, 7]], 否)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-qualification","seed":202610282001,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,6,0,2,4,6],[0,3,6,1,4,7,2,5],[0,4,0,4,0,4,0,4],[0,5,2,7,4,1,6,3],[0,6,4,2,0,6,4,2],[0,7,6,5,4,3,2,1]],"subset":[0,1,2,4,6],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-qualification","semantic_template":"qf-mod28-ideal-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,6,0,2,4,6],[0,3,6,1,4,7,2,5],[0,4,0,4,0,4,0,4],[0,5,2,7,4,1,6,3],[0,6,4,2,0,6,4,2],[0,7,6,5,4,3,2,1]],\"n\":8,\"subset\":[0,1,2,4,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0002","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-qualification；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,7,0,6],[2,3,4,5,7,6,1,0],[3,4,5,7,6,0,2,1],[4,5,7,6,0,1,3,2],[5,7,6,0,1,2,4,3],[6,0,1,2,3,4,7,5],[7,6,0,1,2,3,5,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,7,0,2,7,4],[0,3,7,1,4,6,5,2],[0,4,0,4,0,4,4,0],[0,5,2,6,4,1,3,7],[0,6,7,5,4,3,1,2],[0,7,4,2,0,7,2,4]],\"subset\":[0,4]}。請依 [subset,zero,additive_subgroup,missing_left,missing_right,two_sided_ideal] 欄序回答。","answer":{"kind":"expression","value":"([0, 4], 0, 是, [], [], 是)","explanation_zh":"Wave4第二層核對=([0, 4], 0, 是, [], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-qualification","seed":202610282002,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,7,0,6],[2,3,4,5,7,6,1,0],[3,4,5,7,6,0,2,1],[4,5,7,6,0,1,3,2],[5,7,6,0,1,2,4,3],[6,0,1,2,3,4,7,5],[7,6,0,1,2,3,5,4]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,7,0,2,7,4],[0,3,7,1,4,6,5,2],[0,4,0,4,0,4,4,0],[0,5,2,6,4,1,3,7],[0,6,7,5,4,3,1,2],[0,7,4,2,0,7,2,4]],"subset":[0,4],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-qualification","semantic_template":"qf-mod28-ideal-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,7,0,6],[2,3,4,5,7,6,1,0],[3,4,5,7,6,0,2,1],[4,5,7,6,0,1,3,2],[5,7,6,0,1,2,4,3],[6,0,1,2,3,4,7,5],[7,6,0,1,2,3,5,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,7,0,2,7,4],[0,3,7,1,4,6,5,2],[0,4,0,4,0,4,4,0],[0,5,2,6,4,1,3,7],[0,6,7,5,4,3,1,2],[0,7,4,2,0,7,2,4]],\"n\":8,\"subset\":[0,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0003","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-qualification；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,6,7,5,0],[2,3,4,6,5,0,7,1],[3,4,6,5,7,1,0,2],[4,6,5,7,0,2,1,3],[5,7,0,1,2,4,3,6],[6,5,7,0,1,3,2,4],[7,0,1,2,3,6,4,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,5,0,4,2,5],[0,3,5,1,4,2,7,6],[0,4,0,4,0,0,4,4],[0,5,4,2,0,4,5,2],[0,6,2,7,4,5,1,3],[0,7,5,6,4,2,3,1]],\"subset\":[0]}。請依 [subset,zero,additive_subgroup,missing_left,missing_right,two_sided_ideal] 欄序回答。","answer":{"kind":"expression","value":"([0], 0, 是, [], [], 是)","explanation_zh":"Wave4第二層核對=([0], 0, 是, [], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-qualification","seed":202610282003,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,4,6,7,5,0],[2,3,4,6,5,0,7,1],[3,4,6,5,7,1,0,2],[4,6,5,7,0,2,1,3],[5,7,0,1,2,4,3,6],[6,5,7,0,1,3,2,4],[7,0,1,2,3,6,4,5]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,5,0,4,2,5],[0,3,5,1,4,2,7,6],[0,4,0,4,0,0,4,4],[0,5,4,2,0,4,5,2],[0,6,2,7,4,5,1,3],[0,7,5,6,4,2,3,1]],"subset":[0],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-qualification","semantic_template":"qf-mod28-ideal-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,6,7,5,0],[2,3,4,6,5,0,7,1],[3,4,6,5,7,1,0,2],[4,6,5,7,0,2,1,3],[5,7,0,1,2,4,3,6],[6,5,7,0,1,3,2,4],[7,0,1,2,3,6,4,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,5,0,4,2,5],[0,3,5,1,4,2,7,6],[0,4,0,4,0,0,4,4],[0,5,4,2,0,4,5,2],[0,6,2,7,4,5,1,3],[0,7,5,6,4,2,3,1]],\"n\":8,\"subset\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0004","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-qualification；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,6,0,7,5],[2,3,4,6,7,1,5,0],[3,4,6,7,5,2,0,1],[4,6,7,5,0,3,1,2],[5,0,1,2,3,7,4,6],[6,7,5,0,1,4,2,3],[7,5,0,1,2,6,3,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,7,0,7,2,4],[0,3,7,1,4,6,5,2],[0,4,0,4,0,4,4,0],[0,5,7,6,4,1,3,2],[0,6,2,5,4,3,1,7],[0,7,4,2,0,2,7,4]],\"subset\":[0,2,4,7]}。請依 [subset,zero,additive_subgroup,missing_left,missing_right,two_sided_ideal] 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 4, 7], 0, 是, [], [], 是)","explanation_zh":"Wave4第二層核對=([0, 2, 4, 7], 0, 是, [], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-qualification","seed":202610282004,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,4,6,0,7,5],[2,3,4,6,7,1,5,0],[3,4,6,7,5,2,0,1],[4,6,7,5,0,3,1,2],[5,0,1,2,3,7,4,6],[6,7,5,0,1,4,2,3],[7,5,0,1,2,6,3,4]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,7,0,7,2,4],[0,3,7,1,4,6,5,2],[0,4,0,4,0,4,4,0],[0,5,7,6,4,1,3,2],[0,6,2,5,4,3,1,7],[0,7,4,2,0,2,7,4]],"subset":[0,2,4,7],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-qualification","semantic_template":"qf-mod28-ideal-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,6,0,7,5],[2,3,4,6,7,1,5,0],[3,4,6,7,5,2,0,1],[4,6,7,5,0,3,1,2],[5,0,1,2,3,7,4,6],[6,7,5,0,1,4,2,3],[7,5,0,1,2,6,3,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,7,0,7,2,4],[0,3,7,1,4,6,5,2],[0,4,0,4,0,4,4,0],[0,5,7,6,4,1,3,2],[0,6,2,5,4,3,1,7],[0,7,4,2,0,2,7,4]],\"n\":8,\"subset\":[0,2,4,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0005","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-qualification；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,7,6,0,5],[2,3,4,7,5,0,1,6],[3,4,7,5,6,1,2,0],[4,7,5,6,0,2,3,1],[5,6,0,1,2,4,7,3],[6,0,1,2,3,7,5,4],[7,5,6,0,1,3,4,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,5,0,4,5,2],[0,3,5,1,4,2,7,6],[0,4,0,4,0,0,4,4],[0,5,4,2,0,4,2,5],[0,6,5,7,4,2,1,3],[0,7,2,6,4,5,3,1]],\"subset\":[0,4]}。請依 [subset,zero,additive_subgroup,missing_left,missing_right,two_sided_ideal] 欄序回答。","answer":{"kind":"expression","value":"([0, 4], 0, 是, [], [], 是)","explanation_zh":"Wave4第二層核對=([0, 4], 0, 是, [], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-qualification","seed":202610282005,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,4,7,6,0,5],[2,3,4,7,5,0,1,6],[3,4,7,5,6,1,2,0],[4,7,5,6,0,2,3,1],[5,6,0,1,2,4,7,3],[6,0,1,2,3,7,5,4],[7,5,6,0,1,3,4,2]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,5,0,4,5,2],[0,3,5,1,4,2,7,6],[0,4,0,4,0,0,4,4],[0,5,4,2,0,4,2,5],[0,6,5,7,4,2,1,3],[0,7,2,6,4,5,3,1]],"subset":[0,4],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-qualification","semantic_template":"qf-mod28-ideal-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,7,6,0,5],[2,3,4,7,5,0,1,6],[3,4,7,5,6,1,2,0],[4,7,5,6,0,2,3,1],[5,6,0,1,2,4,7,3],[6,0,1,2,3,7,5,4],[7,5,6,0,1,3,4,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,5,0,4,5,2],[0,3,5,1,4,2,7,6],[0,4,0,4,0,0,4,4],[0,5,4,2,0,4,2,5],[0,6,5,7,4,2,1,3],[0,7,2,6,4,5,3,1]],\"n\":8,\"subset\":[0,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0006","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-qualification；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,7,0,5,6],[2,3,4,7,6,1,0,5],[3,4,7,6,5,2,1,0],[4,7,6,5,0,3,2,1],[5,0,1,2,3,6,7,4],[6,5,0,1,2,7,4,3],[7,6,5,0,1,4,3,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,6,0,6,4,2],[0,3,6,1,4,7,2,5],[0,4,0,4,0,4,0,4],[0,5,6,7,4,1,2,3],[0,6,4,2,0,2,4,6],[0,7,2,5,4,3,6,1]],\"subset\":[0,1,2,4,6]}。請依 [subset,zero,additive_subgroup,missing_left,missing_right,two_sided_ideal] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 4, 6], 0, 否, [[3, 1, 3], [5, 1, 5], [7, 1, 7]], [[1, 3, 3], [1, 5, 5], [1, 7, 7]], 否)","explanation_zh":"Wave4第二層核對=([0, 1, 2, 4, 6], 0, 否, [[3, 1, 3], [5, 1, 5], [7, 1, 7]], [[1, 3, 3], [1, 5, 5], [1, 7, 7]], 否)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-qualification","seed":202610282006,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,4,7,0,5,6],[2,3,4,7,6,1,0,5],[3,4,7,6,5,2,1,0],[4,7,6,5,0,3,2,1],[5,0,1,2,3,6,7,4],[6,5,0,1,2,7,4,3],[7,6,5,0,1,4,3,2]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,6,0,6,4,2],[0,3,6,1,4,7,2,5],[0,4,0,4,0,4,0,4],[0,5,6,7,4,1,2,3],[0,6,4,2,0,2,4,6],[0,7,2,5,4,3,6,1]],"subset":[0,1,2,4,6],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-qualification","semantic_template":"qf-mod28-ideal-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,7,0,5,6],[2,3,4,7,6,1,0,5],[3,4,7,6,5,2,1,0],[4,7,6,5,0,3,2,1],[5,0,1,2,3,6,7,4],[6,5,0,1,2,7,4,3],[7,6,5,0,1,4,3,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,6,0,6,4,2],[0,3,6,1,4,7,2,5],[0,4,0,4,0,4,0,4],[0,5,6,7,4,1,2,3],[0,6,4,2,0,2,4,6],[0,7,2,5,4,3,6,1]],\"n\":8,\"subset\":[0,1,2,4,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0007","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-qualification；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,6,4,7,0],[2,3,5,4,7,6,0,1],[3,5,4,6,0,7,1,2],[4,6,7,0,2,1,3,5],[5,4,6,7,1,0,2,3],[6,7,0,1,3,2,5,4],[7,0,1,2,5,3,4,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,6,2,0,5,6],[0,3,6,1,7,5,2,4],[0,4,2,7,1,5,6,3],[0,5,0,5,5,0,0,5],[0,6,5,2,6,0,5,2],[0,7,6,4,3,5,2,1]],\"subset\":[0,5]}。請依 [subset,zero,additive_subgroup,missing_left,missing_right,two_sided_ideal] 欄序回答。","answer":{"kind":"expression","value":"([0, 5], 0, 是, [], [], 是)","explanation_zh":"Wave4第二層核對=([0, 5], 0, 是, [], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-qualification","seed":202610282007,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,5,6,4,7,0],[2,3,5,4,7,6,0,1],[3,5,4,6,0,7,1,2],[4,6,7,0,2,1,3,5],[5,4,6,7,1,0,2,3],[6,7,0,1,3,2,5,4],[7,0,1,2,5,3,4,6]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,6,2,0,5,6],[0,3,6,1,7,5,2,4],[0,4,2,7,1,5,6,3],[0,5,0,5,5,0,0,5],[0,6,5,2,6,0,5,2],[0,7,6,4,3,5,2,1]],"subset":[0,5],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-qualification","semantic_template":"qf-mod28-ideal-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,6,4,7,0],[2,3,5,4,7,6,0,1],[3,5,4,6,0,7,1,2],[4,6,7,0,2,1,3,5],[5,4,6,7,1,0,2,3],[6,7,0,1,3,2,5,4],[7,0,1,2,5,3,4,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,6,2,0,5,6],[0,3,6,1,7,5,2,4],[0,4,2,7,1,5,6,3],[0,5,0,5,5,0,0,5],[0,6,5,2,6,0,5,2],[0,7,6,4,3,5,2,1]],\"n\":8,\"subset\":[0,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0008","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-qualification；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,7,4,0,6],[2,3,5,4,6,7,1,0],[3,5,4,7,0,6,2,1],[4,7,6,0,2,1,5,3],[5,4,7,6,1,0,3,2],[6,0,1,2,5,3,7,4],[7,6,0,1,3,2,4,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,7,2,0,7,5],[0,3,7,1,6,5,4,2],[0,4,2,6,1,5,3,7],[0,5,0,5,5,0,5,0],[0,6,7,4,3,5,1,2],[0,7,5,2,7,0,2,5]],\"subset\":[0]}。請依 [subset,zero,additive_subgroup,missing_left,missing_right,two_sided_ideal] 欄序回答。","answer":{"kind":"expression","value":"([0], 0, 是, [], [], 是)","explanation_zh":"Wave4第二層核對=([0], 0, 是, [], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-qualification","seed":202610282008,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,5,7,4,0,6],[2,3,5,4,6,7,1,0],[3,5,4,7,0,6,2,1],[4,7,6,0,2,1,5,3],[5,4,7,6,1,0,3,2],[6,0,1,2,5,3,7,4],[7,6,0,1,3,2,4,5]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,7,2,0,7,5],[0,3,7,1,6,5,4,2],[0,4,2,6,1,5,3,7],[0,5,0,5,5,0,5,0],[0,6,7,4,3,5,1,2],[0,7,5,2,7,0,2,5]],"subset":[0],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-qualification","semantic_template":"qf-mod28-ideal-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,7,4,0,6],[2,3,5,4,6,7,1,0],[3,5,4,7,0,6,2,1],[4,7,6,0,2,1,5,3],[5,4,7,6,1,0,3,2],[6,0,1,2,5,3,7,4],[7,6,0,1,3,2,4,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,7,2,0,7,5],[0,3,7,1,6,5,4,2],[0,4,2,6,1,5,3,7],[0,5,0,5,5,0,5,0],[0,6,7,4,3,5,1,2],[0,7,5,2,7,0,2,5]],\"n\":8,\"subset\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0009","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-qualification；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,7,6,4,0],[2,3,5,6,0,4,7,1],[3,5,6,4,1,7,0,2],[4,7,0,1,5,2,3,6],[5,6,4,7,2,0,1,3],[6,4,7,0,3,1,2,5],[7,0,1,2,6,3,5,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,4,5,0,2,4],[0,3,4,1,2,5,7,6],[0,4,5,2,5,0,4,2],[0,5,0,5,0,0,5,5],[0,6,2,7,4,5,1,3],[0,7,4,6,2,5,3,1]],\"subset\":[0,2,4,5]}。請依 [subset,zero,additive_subgroup,missing_left,missing_right,two_sided_ideal] 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 4, 5], 0, 是, [], [], 是)","explanation_zh":"Wave4第二層核對=([0, 2, 4, 5], 0, 是, [], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-qualification","seed":202610282009,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,5,7,6,4,0],[2,3,5,6,0,4,7,1],[3,5,6,4,1,7,0,2],[4,7,0,1,5,2,3,6],[5,6,4,7,2,0,1,3],[6,4,7,0,3,1,2,5],[7,0,1,2,6,3,5,4]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,4,5,0,2,4],[0,3,4,1,2,5,7,6],[0,4,5,2,5,0,4,2],[0,5,0,5,0,0,5,5],[0,6,2,7,4,5,1,3],[0,7,4,6,2,5,3,1]],"subset":[0,2,4,5],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-qualification","semantic_template":"qf-mod28-ideal-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,7,6,4,0],[2,3,5,6,0,4,7,1],[3,5,6,4,1,7,0,2],[4,7,0,1,5,2,3,6],[5,6,4,7,2,0,1,3],[6,4,7,0,3,1,2,5],[7,0,1,2,6,3,5,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,4,5,0,2,4],[0,3,4,1,2,5,7,6],[0,4,5,2,5,0,4,2],[0,5,0,5,0,0,5,5],[0,6,2,7,4,5,1,3],[0,7,4,6,2,5,3,1]],\"n\":8,\"subset\":[0,2,4,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0010","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-qualification；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,0,6,7,4],[2,3,5,6,1,7,4,0],[3,5,6,7,2,4,0,1],[4,0,1,2,7,3,5,6],[5,6,7,4,3,0,1,2],[6,7,4,0,5,1,2,3],[7,4,0,1,6,2,3,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,7,7,0,2,5],[0,3,7,1,6,5,4,2],[0,4,7,6,1,5,3,2],[0,5,0,5,5,0,5,0],[0,6,2,4,3,5,1,7],[0,7,5,2,2,0,7,5]],\"subset\":[0,5]}。請依 [subset,zero,additive_subgroup,missing_left,missing_right,two_sided_ideal] 欄序回答。","answer":{"kind":"expression","value":"([0, 5], 0, 是, [], [], 是)","explanation_zh":"Wave4第二層核對=([0, 5], 0, 是, [], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-qualification","seed":202610282010,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,5,0,6,7,4],[2,3,5,6,1,7,4,0],[3,5,6,7,2,4,0,1],[4,0,1,2,7,3,5,6],[5,6,7,4,3,0,1,2],[6,7,4,0,5,1,2,3],[7,4,0,1,6,2,3,5]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,7,7,0,2,5],[0,3,7,1,6,5,4,2],[0,4,7,6,1,5,3,2],[0,5,0,5,5,0,5,0],[0,6,2,4,3,5,1,7],[0,7,5,2,2,0,7,5]],"subset":[0,5],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-qualification","semantic_template":"qf-mod28-ideal-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,0,6,7,4],[2,3,5,6,1,7,4,0],[3,5,6,7,2,4,0,1],[4,0,1,2,7,3,5,6],[5,6,7,4,3,0,1,2],[6,7,4,0,5,1,2,3],[7,4,0,1,6,2,3,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,7,7,0,2,5],[0,3,7,1,6,5,4,2],[0,4,7,6,1,5,3,2],[0,5,0,5,5,0,5,0],[0,6,2,4,3,5,1,7],[0,7,5,2,2,0,7,5]],\"n\":8,\"subset\":[0,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0011","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-qualification；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,6,7,0,4],[2,3,5,7,0,4,1,6],[3,5,7,4,1,6,2,0],[4,6,0,1,5,2,7,3],[5,7,4,6,2,0,3,1],[6,0,1,2,7,3,4,5],[7,4,6,0,3,1,5,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,4,5,0,4,2],[0,3,4,1,2,5,7,6],[0,4,5,2,5,0,2,4],[0,5,0,5,0,0,5,5],[0,6,4,7,2,5,1,3],[0,7,2,6,4,5,3,1]],\"subset\":[0,1,2,4,5]}。請依 [subset,zero,additive_subgroup,missing_left,missing_right,two_sided_ideal] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 4, 5], 0, 否, [[3, 1, 3], [6, 1, 6], [7, 1, 7]], [[1, 3, 3], [1, 6, 6], [1, 7, 7]], 否)","explanation_zh":"Wave4第二層核對=([0, 1, 2, 4, 5], 0, 否, [[3, 1, 3], [6, 1, 6], [7, 1, 7]], [[1, 3, 3], [1, 6, 6], [1, 7, 7]], 否)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-qualification","seed":202610282011,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,5,6,7,0,4],[2,3,5,7,0,4,1,6],[3,5,7,4,1,6,2,0],[4,6,0,1,5,2,7,3],[5,7,4,6,2,0,3,1],[6,0,1,2,7,3,4,5],[7,4,6,0,3,1,5,2]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,4,5,0,4,2],[0,3,4,1,2,5,7,6],[0,4,5,2,5,0,2,4],[0,5,0,5,0,0,5,5],[0,6,4,7,2,5,1,3],[0,7,2,6,4,5,3,1]],"subset":[0,1,2,4,5],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-qualification","semantic_template":"qf-mod28-ideal-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,6,7,0,4],[2,3,5,7,0,4,1,6],[3,5,7,4,1,6,2,0],[4,6,0,1,5,2,7,3],[5,7,4,6,2,0,3,1],[6,0,1,2,7,3,4,5],[7,4,6,0,3,1,5,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,4,5,0,4,2],[0,3,4,1,2,5,7,6],[0,4,5,2,5,0,2,4],[0,5,0,5,0,0,5,5],[0,6,4,7,2,5,1,3],[0,7,2,6,4,5,3,1]],\"n\":8,\"subset\":[0,1,2,4,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0012","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-qualification；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,0,7,4,6],[2,3,5,7,1,6,0,4],[3,5,7,6,2,4,1,0],[4,0,1,2,6,3,7,5],[5,7,6,4,3,0,2,1],[6,4,0,1,7,2,5,3],[7,6,4,0,5,1,3,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,6,6,0,5,2],[0,3,6,1,7,5,2,4],[0,4,6,7,1,5,2,3],[0,5,0,5,5,0,0,5],[0,6,5,2,2,0,5,6],[0,7,2,4,3,5,6,1]],\"subset\":[0,5]}。請依 [subset,zero,additive_subgroup,missing_left,missing_right,two_sided_ideal] 欄序回答。","answer":{"kind":"expression","value":"([0, 5], 0, 是, [], [], 是)","explanation_zh":"Wave4第二層核對=([0, 5], 0, 是, [], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-qualification","seed":202610282012,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,5,0,7,4,6],[2,3,5,7,1,6,0,4],[3,5,7,6,2,4,1,0],[4,0,1,2,6,3,7,5],[5,7,6,4,3,0,2,1],[6,4,0,1,7,2,5,3],[7,6,4,0,5,1,3,2]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,6,6,0,5,2],[0,3,6,1,7,5,2,4],[0,4,6,7,1,5,2,3],[0,5,0,5,5,0,0,5],[0,6,5,2,2,0,5,6],[0,7,2,4,3,5,6,1]],"subset":[0,5],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-qualification","semantic_template":"qf-mod28-ideal-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,0,7,4,6],[2,3,5,7,1,6,0,4],[3,5,7,6,2,4,1,0],[4,0,1,2,6,3,7,5],[5,7,6,4,3,0,2,1],[6,4,0,1,7,2,5,3],[7,6,4,0,5,1,3,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,6,6,0,5,2],[0,3,6,1,7,5,2,4],[0,4,6,7,1,5,2,3],[0,5,0,5,5,0,0,5],[0,6,5,2,2,0,5,6],[0,7,2,4,3,5,6,1]],\"n\":8,\"subset\":[0,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0013","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-qualification；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,5,7,4,0],[2,3,6,4,7,0,5,1],[3,6,4,5,0,1,7,2],[4,5,7,0,2,3,1,6],[5,7,0,1,3,6,2,4],[6,4,5,7,1,2,0,3],[7,0,1,2,6,4,3,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,5,2,6,0,5],[0,3,5,1,7,2,6,4],[0,4,2,7,1,5,6,3],[0,5,6,2,5,6,0,2],[0,6,0,6,6,0,0,6],[0,7,5,4,3,2,6,1]],\"subset\":[0]}。請依 [subset,zero,additive_subgroup,missing_left,missing_right,two_sided_ideal] 欄序回答。","answer":{"kind":"expression","value":"([0], 0, 是, [], [], 是)","explanation_zh":"Wave4第二層核對=([0], 0, 是, [], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-qualification","seed":202610282013,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,6,5,7,4,0],[2,3,6,4,7,0,5,1],[3,6,4,5,0,1,7,2],[4,5,7,0,2,3,1,6],[5,7,0,1,3,6,2,4],[6,4,5,7,1,2,0,3],[7,0,1,2,6,4,3,5]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,5,2,6,0,5],[0,3,5,1,7,2,6,4],[0,4,2,7,1,5,6,3],[0,5,6,2,5,6,0,2],[0,6,0,6,6,0,0,6],[0,7,5,4,3,2,6,1]],"subset":[0],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-qualification","semantic_template":"qf-mod28-ideal-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,5,7,4,0],[2,3,6,4,7,0,5,1],[3,6,4,5,0,1,7,2],[4,5,7,0,2,3,1,6],[5,7,0,1,3,6,2,4],[6,4,5,7,1,2,0,3],[7,0,1,2,6,4,3,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,5,2,6,0,5],[0,3,5,1,7,2,6,4],[0,4,2,7,1,5,6,3],[0,5,6,2,5,6,0,2],[0,6,0,6,6,0,0,6],[0,7,5,4,3,2,6,1]],\"n\":8,\"subset\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0014","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-qualification；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,7,0,4,5],[2,3,6,4,5,1,7,0],[3,6,4,7,0,2,5,1],[4,7,5,0,2,6,1,3],[5,0,1,2,6,7,3,4],[6,4,7,5,1,3,0,2],[7,5,0,1,3,4,2,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,7,2,7,0,6],[0,3,7,1,5,4,6,2],[0,4,2,5,1,3,6,7],[0,5,7,4,3,1,6,2],[0,6,0,6,6,6,0,0],[0,7,6,2,7,2,0,6]],\"subset\":[0,2,6,7]}。請依 [subset,zero,additive_subgroup,missing_left,missing_right,two_sided_ideal] 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 6, 7], 0, 是, [], [], 是)","explanation_zh":"Wave4第二層核對=([0, 2, 6, 7], 0, 是, [], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-qualification","seed":202610282014,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,6,7,0,4,5],[2,3,6,4,5,1,7,0],[3,6,4,7,0,2,5,1],[4,7,5,0,2,6,1,3],[5,0,1,2,6,7,3,4],[6,4,7,5,1,3,0,2],[7,5,0,1,3,4,2,6]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,7,2,7,0,6],[0,3,7,1,5,4,6,2],[0,4,2,5,1,3,6,7],[0,5,7,4,3,1,6,2],[0,6,0,6,6,6,0,0],[0,7,6,2,7,2,0,6]],"subset":[0,2,6,7],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-qualification","semantic_template":"qf-mod28-ideal-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,7,0,4,5],[2,3,6,4,5,1,7,0],[3,6,4,7,0,2,5,1],[4,7,5,0,2,6,1,3],[5,0,1,2,6,7,3,4],[6,4,7,5,1,3,0,2],[7,5,0,1,3,4,2,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,7,2,7,0,6],[0,3,7,1,5,4,6,2],[0,4,2,5,1,3,6,7],[0,5,7,4,3,1,6,2],[0,6,0,6,6,6,0,0],[0,7,6,2,7,2,0,6]],\"n\":8,\"subset\":[0,2,6,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0015","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-qualification；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,7,4,5,0],[2,3,6,5,0,7,4,1],[3,6,5,4,1,0,7,2],[4,7,0,1,6,3,2,5],[5,4,7,0,3,2,1,6],[6,5,4,7,2,1,0,3],[7,0,1,2,5,6,3,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,4,6,2,0,4],[0,3,4,1,2,7,6,5],[0,4,6,2,6,4,0,2],[0,5,2,7,4,1,6,3],[0,6,0,6,0,6,0,6],[0,7,4,5,2,3,6,1]],\"subset\":[0,6]}。請依 [subset,zero,additive_subgroup,missing_left,missing_right,two_sided_ideal] 欄序回答。","answer":{"kind":"expression","value":"([0, 6], 0, 是, [], [], 是)","explanation_zh":"Wave4第二層核對=([0, 6], 0, 是, [], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-qualification","seed":202610282015,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,6,7,4,5,0],[2,3,6,5,0,7,4,1],[3,6,5,4,1,0,7,2],[4,7,0,1,6,3,2,5],[5,4,7,0,3,2,1,6],[6,5,4,7,2,1,0,3],[7,0,1,2,5,6,3,4]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,4,6,2,0,4],[0,3,4,1,2,7,6,5],[0,4,6,2,6,4,0,2],[0,5,2,7,4,1,6,3],[0,6,0,6,0,6,0,6],[0,7,4,5,2,3,6,1]],"subset":[0,6],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-qualification","semantic_template":"qf-mod28-ideal-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,7,4,5,0],[2,3,6,5,0,7,4,1],[3,6,5,4,1,0,7,2],[4,7,0,1,6,3,2,5],[5,4,7,0,3,2,1,6],[6,5,4,7,2,1,0,3],[7,0,1,2,5,6,3,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,4,6,2,0,4],[0,3,4,1,2,7,6,5],[0,4,6,2,6,4,0,2],[0,5,2,7,4,1,6,3],[0,6,0,6,0,6,0,6],[0,7,4,5,2,3,6,1]],\"n\":8,\"subset\":[0,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0016","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-qualification；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,0,7,5,4],[2,3,6,5,1,4,7,0],[3,6,5,7,2,0,4,1],[4,0,1,2,7,6,3,5],[5,7,4,0,6,2,1,3],[6,5,7,4,3,1,0,2],[7,4,0,1,5,3,2,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,7,7,2,0,6],[0,3,7,1,5,4,6,2],[0,4,7,5,1,3,6,2],[0,5,2,4,3,1,6,7],[0,6,0,6,6,6,0,0],[0,7,6,2,2,7,0,6]],\"subset\":[0,1,2,6,7]}。請依 [subset,zero,additive_subgroup,missing_left,missing_right,two_sided_ideal] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 6, 7], 0, 否, [[3, 1, 3], [4, 1, 4], [5, 1, 5]], [[1, 3, 3], [1, 4, 4], [1, 5, 5]], 否)","explanation_zh":"Wave4第二層核對=([0, 1, 2, 6, 7], 0, 否, [[3, 1, 3], [4, 1, 4], [5, 1, 5]], [[1, 3, 3], [1, 4, 4], [1, 5, 5]], 否)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-qualification","seed":202610282016,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,6,0,7,5,4],[2,3,6,5,1,4,7,0],[3,6,5,7,2,0,4,1],[4,0,1,2,7,6,3,5],[5,7,4,0,6,2,1,3],[6,5,7,4,3,1,0,2],[7,4,0,1,5,3,2,6]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,7,7,2,0,6],[0,3,7,1,5,4,6,2],[0,4,7,5,1,3,6,2],[0,5,2,4,3,1,6,7],[0,6,0,6,6,6,0,0],[0,7,6,2,2,7,0,6]],"subset":[0,1,2,6,7],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-qualification","semantic_template":"qf-mod28-ideal-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,0,7,5,4],[2,3,6,5,1,4,7,0],[3,6,5,7,2,0,4,1],[4,0,1,2,7,6,3,5],[5,7,4,0,6,2,1,3],[6,5,7,4,3,1,0,2],[7,4,0,1,5,3,2,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,7,7,2,0,6],[0,3,7,1,5,4,6,2],[0,4,7,5,1,3,6,2],[0,5,2,4,3,1,6,7],[0,6,0,6,6,6,0,0],[0,7,6,2,2,7,0,6]],\"n\":8,\"subset\":[0,1,2,6,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0017","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-qualification；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,5,0,7,4],[2,3,6,7,0,1,4,5],[3,6,7,4,1,2,5,0],[4,5,0,1,6,7,2,3],[5,0,1,2,7,4,3,6],[6,7,4,5,2,3,0,1],[7,4,5,0,3,6,1,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,4,6,4,0,2],[0,3,4,1,2,7,6,5],[0,4,6,2,6,2,0,4],[0,5,4,7,2,1,6,3],[0,6,0,6,0,6,0,6],[0,7,2,5,4,3,6,1]],\"subset\":[0,6]}。請依 [subset,zero,additive_subgroup,missing_left,missing_right,two_sided_ideal] 欄序回答。","answer":{"kind":"expression","value":"([0, 6], 0, 是, [], [], 是)","explanation_zh":"Wave4第二層核對=([0, 6], 0, 是, [], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-qualification","seed":202610282017,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,6,5,0,7,4],[2,3,6,7,0,1,4,5],[3,6,7,4,1,2,5,0],[4,5,0,1,6,7,2,3],[5,0,1,2,7,4,3,6],[6,7,4,5,2,3,0,1],[7,4,5,0,3,6,1,2]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,4,6,4,0,2],[0,3,4,1,2,7,6,5],[0,4,6,2,6,2,0,4],[0,5,4,7,2,1,6,3],[0,6,0,6,0,6,0,6],[0,7,2,5,4,3,6,1]],"subset":[0,6],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-qualification","semantic_template":"qf-mod28-ideal-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,5,0,7,4],[2,3,6,7,0,1,4,5],[3,6,7,4,1,2,5,0],[4,5,0,1,6,7,2,3],[5,0,1,2,7,4,3,6],[6,7,4,5,2,3,0,1],[7,4,5,0,3,6,1,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,4,6,4,0,2],[0,3,4,1,2,7,6,5],[0,4,6,2,6,2,0,4],[0,5,4,7,2,1,6,3],[0,6,0,6,0,6,0,6],[0,7,2,5,4,3,6,1]],\"n\":8,\"subset\":[0,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0018","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-qualification；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,0,4,7,5],[2,3,6,7,1,0,5,4],[3,6,7,5,2,1,4,0],[4,0,1,2,5,7,3,6],[5,4,0,1,7,6,2,3],[6,7,5,4,3,2,0,1],[7,5,4,0,6,3,1,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,5,5,6,0,2],[0,3,5,1,7,2,6,4],[0,4,5,7,1,2,6,3],[0,5,6,2,2,6,0,5],[0,6,0,6,6,0,0,6],[0,7,2,4,3,5,6,1]],\"subset\":[0]}。請依 [subset,zero,additive_subgroup,missing_left,missing_right,two_sided_ideal] 欄序回答。","answer":{"kind":"expression","value":"([0], 0, 是, [], [], 是)","explanation_zh":"Wave4第二層核對=([0], 0, 是, [], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-qualification","seed":202610282018,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,6,0,4,7,5],[2,3,6,7,1,0,5,4],[3,6,7,5,2,1,4,0],[4,0,1,2,5,7,3,6],[5,4,0,1,7,6,2,3],[6,7,5,4,3,2,0,1],[7,5,4,0,6,3,1,2]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,5,5,6,0,2],[0,3,5,1,7,2,6,4],[0,4,5,7,1,2,6,3],[0,5,6,2,2,6,0,5],[0,6,0,6,6,0,0,6],[0,7,2,4,3,5,6,1]],"subset":[0],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-qualification","semantic_template":"qf-mod28-ideal-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,0,4,7,5],[2,3,6,7,1,0,5,4],[3,6,7,5,2,1,4,0],[4,0,1,2,5,7,3,6],[5,4,0,1,7,6,2,3],[6,7,5,4,3,2,0,1],[7,5,4,0,6,3,1,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,5,5,6,0,2],[0,3,5,1,7,2,6,4],[0,4,5,7,1,2,6,3],[0,5,6,2,2,6,0,5],[0,6,0,6,6,0,0,6],[0,7,2,4,3,5,6,1]],\"n\":8,\"subset\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0019","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-qualification；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,5,6,0,4],[2,3,7,4,6,0,1,5],[3,7,4,5,0,1,2,6],[4,5,6,0,2,3,7,1],[5,6,0,1,3,7,4,2],[6,0,1,2,7,4,5,3],[7,4,5,6,1,2,3,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,5,2,7,5,0],[0,3,5,1,6,2,4,7],[0,4,2,6,1,5,3,7],[0,5,7,2,5,7,2,0],[0,6,5,4,3,2,1,7],[0,7,0,7,7,0,7,0]],\"subset\":[0,2,5,7]}。請依 [subset,zero,additive_subgroup,missing_left,missing_right,two_sided_ideal] 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 5, 7], 0, 是, [], [], 是)","explanation_zh":"Wave4第二層核對=([0, 2, 5, 7], 0, 是, [], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-qualification","seed":202610282019,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,7,5,6,0,4],[2,3,7,4,6,0,1,5],[3,7,4,5,0,1,2,6],[4,5,6,0,2,3,7,1],[5,6,0,1,3,7,4,2],[6,0,1,2,7,4,5,3],[7,4,5,6,1,2,3,0]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,5,2,7,5,0],[0,3,5,1,6,2,4,7],[0,4,2,6,1,5,3,7],[0,5,7,2,5,7,2,0],[0,6,5,4,3,2,1,7],[0,7,0,7,7,0,7,0]],"subset":[0,2,5,7],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-qualification","semantic_template":"qf-mod28-ideal-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,5,6,0,4],[2,3,7,4,6,0,1,5],[3,7,4,5,0,1,2,6],[4,5,6,0,2,3,7,1],[5,6,0,1,3,7,4,2],[6,0,1,2,7,4,5,3],[7,4,5,6,1,2,3,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,5,2,7,5,0],[0,3,5,1,6,2,4,7],[0,4,2,6,1,5,3,7],[0,5,7,2,5,7,2,0],[0,6,5,4,3,2,1,7],[0,7,0,7,7,0,7,0]],\"n\":8,\"subset\":[0,2,5,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0020","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-qualification；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,6,0,5,4],[2,3,7,4,5,1,0,6],[3,7,4,6,0,2,1,5],[4,6,5,0,2,7,3,1],[5,0,1,2,7,6,4,3],[6,5,0,1,3,4,7,2],[7,4,6,5,1,3,2,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,6,2,6,7,0],[0,3,6,1,5,4,2,7],[0,4,2,5,1,3,6,7],[0,5,6,4,3,1,2,7],[0,6,7,2,6,2,7,0],[0,7,0,7,7,7,0,0]],\"subset\":[0,7]}。請依 [subset,zero,additive_subgroup,missing_left,missing_right,two_sided_ideal] 欄序回答。","answer":{"kind":"expression","value":"([0, 7], 0, 是, [], [], 是)","explanation_zh":"Wave4第二層核對=([0, 7], 0, 是, [], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-qualification","seed":202610282020,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,7,6,0,5,4],[2,3,7,4,5,1,0,6],[3,7,4,6,0,2,1,5],[4,6,5,0,2,7,3,1],[5,0,1,2,7,6,4,3],[6,5,0,1,3,4,7,2],[7,4,6,5,1,3,2,0]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,6,2,6,7,0],[0,3,6,1,5,4,2,7],[0,4,2,5,1,3,6,7],[0,5,6,4,3,1,2,7],[0,6,7,2,6,2,7,0],[0,7,0,7,7,7,0,0]],"subset":[0,7],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-qualification","semantic_template":"qf-mod28-ideal-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,6,0,5,4],[2,3,7,4,5,1,0,6],[3,7,4,6,0,2,1,5],[4,6,5,0,2,7,3,1],[5,0,1,2,7,6,4,3],[6,5,0,1,3,4,7,2],[7,4,6,5,1,3,2,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,6,2,6,7,0],[0,3,6,1,5,4,2,7],[0,4,2,5,1,3,6,7],[0,5,6,4,3,1,2,7],[0,6,7,2,6,2,7,0],[0,7,0,7,7,7,0,0]],\"n\":8,\"subset\":[0,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0021","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-qualification；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,6,4,0,5],[2,3,7,5,0,6,1,4],[3,7,5,4,1,0,2,6],[4,6,0,1,7,3,5,2],[5,4,6,0,3,2,7,1],[6,0,1,2,5,7,4,3],[7,5,4,6,2,1,3,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,4,7,2,4,0],[0,3,4,1,2,6,5,7],[0,4,7,2,7,4,2,0],[0,5,2,6,4,1,3,7],[0,6,4,5,2,3,1,7],[0,7,0,7,0,7,7,0]],\"subset\":[0,1,2,4,7]}。請依 [subset,zero,additive_subgroup,missing_left,missing_right,two_sided_ideal] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 4, 7], 0, 否, [[3, 1, 3], [5, 1, 5], [6, 1, 6]], [[1, 3, 3], [1, 5, 5], [1, 6, 6]], 否)","explanation_zh":"Wave4第二層核對=([0, 1, 2, 4, 7], 0, 否, [[3, 1, 3], [5, 1, 5], [6, 1, 6]], [[1, 3, 3], [1, 5, 5], [1, 6, 6]], 否)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-qualification","seed":202610282021,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,7,6,4,0,5],[2,3,7,5,0,6,1,4],[3,7,5,4,1,0,2,6],[4,6,0,1,7,3,5,2],[5,4,6,0,3,2,7,1],[6,0,1,2,5,7,4,3],[7,5,4,6,2,1,3,0]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,4,7,2,4,0],[0,3,4,1,2,6,5,7],[0,4,7,2,7,4,2,0],[0,5,2,6,4,1,3,7],[0,6,4,5,2,3,1,7],[0,7,0,7,0,7,7,0]],"subset":[0,1,2,4,7],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-qualification","semantic_template":"qf-mod28-ideal-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,6,4,0,5],[2,3,7,5,0,6,1,4],[3,7,5,4,1,0,2,6],[4,6,0,1,7,3,5,2],[5,4,6,0,3,2,7,1],[6,0,1,2,5,7,4,3],[7,5,4,6,2,1,3,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,4,7,2,4,0],[0,3,4,1,2,6,5,7],[0,4,7,2,7,4,2,0],[0,5,2,6,4,1,3,7],[0,6,4,5,2,3,1,7],[0,7,0,7,0,7,7,0]],\"n\":8,\"subset\":[0,1,2,4,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0022","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-qualification；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,0,6,4,5],[2,3,7,5,1,4,0,6],[3,7,5,6,2,0,1,4],[4,0,1,2,6,7,5,3],[5,6,4,0,7,2,3,1],[6,4,0,1,5,3,7,2],[7,5,6,4,3,1,2,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,6,6,2,7,0],[0,3,6,1,5,4,2,7],[0,4,6,5,1,3,2,7],[0,5,2,4,3,1,6,7],[0,6,7,2,2,6,7,0],[0,7,0,7,7,7,0,0]],\"subset\":[0,7]}。請依 [subset,zero,additive_subgroup,missing_left,missing_right,two_sided_ideal] 欄序回答。","answer":{"kind":"expression","value":"([0, 7], 0, 是, [], [], 是)","explanation_zh":"Wave4第二層核對=([0, 7], 0, 是, [], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-qualification","seed":202610282022,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,7,0,6,4,5],[2,3,7,5,1,4,0,6],[3,7,5,6,2,0,1,4],[4,0,1,2,6,7,5,3],[5,6,4,0,7,2,3,1],[6,4,0,1,5,3,7,2],[7,5,6,4,3,1,2,0]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,6,6,2,7,0],[0,3,6,1,5,4,2,7],[0,4,6,5,1,3,2,7],[0,5,2,4,3,1,6,7],[0,6,7,2,2,6,7,0],[0,7,0,7,7,7,0,0]],"subset":[0,7],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-qualification","semantic_template":"qf-mod28-ideal-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,0,6,4,5],[2,3,7,5,1,4,0,6],[3,7,5,6,2,0,1,4],[4,0,1,2,6,7,5,3],[5,6,4,0,7,2,3,1],[6,4,0,1,5,3,7,2],[7,5,6,4,3,1,2,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,6,6,2,7,0],[0,3,6,1,5,4,2,7],[0,4,6,5,1,3,2,7],[0,5,2,4,3,1,6,7],[0,6,7,2,2,6,7,0],[0,7,0,7,7,7,0,0]],\"n\":8,\"subset\":[0,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0023","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-qualification；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,5,0,4,6],[2,3,7,6,0,1,5,4],[3,7,6,4,1,2,0,5],[4,5,0,1,7,6,3,2],[5,0,1,2,6,4,7,3],[6,4,5,0,3,7,2,1],[7,6,4,5,2,3,1,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,4,7,4,2,0],[0,3,4,1,2,6,5,7],[0,4,7,2,7,2,4,0],[0,5,4,6,2,1,3,7],[0,6,2,5,4,3,1,7],[0,7,0,7,0,7,7,0]],\"subset\":[0]}。請依 [subset,zero,additive_subgroup,missing_left,missing_right,two_sided_ideal] 欄序回答。","answer":{"kind":"expression","value":"([0], 0, 是, [], [], 是)","explanation_zh":"Wave4第二層核對=([0], 0, 是, [], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-qualification","seed":202610282023,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,7,5,0,4,6],[2,3,7,6,0,1,5,4],[3,7,6,4,1,2,0,5],[4,5,0,1,7,6,3,2],[5,0,1,2,6,4,7,3],[6,4,5,0,3,7,2,1],[7,6,4,5,2,3,1,0]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,4,7,4,2,0],[0,3,4,1,2,6,5,7],[0,4,7,2,7,2,4,0],[0,5,4,6,2,1,3,7],[0,6,2,5,4,3,1,7],[0,7,0,7,0,7,7,0]],"subset":[0],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-qualification","semantic_template":"qf-mod28-ideal-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,5,0,4,6],[2,3,7,6,0,1,5,4],[3,7,6,4,1,2,0,5],[4,5,0,1,7,6,3,2],[5,0,1,2,6,4,7,3],[6,4,5,0,3,7,2,1],[7,6,4,5,2,3,1,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,4,7,4,2,0],[0,3,4,1,2,6,5,7],[0,4,7,2,7,2,4,0],[0,5,4,6,2,1,3,7],[0,6,2,5,4,3,1,7],[0,7,0,7,0,7,7,0]],\"n\":8,\"subset\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0024","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-qualification；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,0,4,5,6],[2,3,7,6,1,0,4,5],[3,7,6,5,2,1,0,4],[4,0,1,2,5,6,7,3],[5,4,0,1,6,7,3,2],[6,5,4,0,7,3,2,1],[7,6,5,4,3,2,1,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,5,5,7,2,0],[0,3,5,1,6,2,4,7],[0,4,5,6,1,2,3,7],[0,5,7,2,2,7,5,0],[0,6,2,4,3,5,1,7],[0,7,0,7,7,0,7,0]],\"subset\":[0,2,5,7]}。請依 [subset,zero,additive_subgroup,missing_left,missing_right,two_sided_ideal] 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 5, 7], 0, 是, [], [], 是)","explanation_zh":"Wave4第二層核對=([0, 2, 5, 7], 0, 是, [], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-qualification","seed":202610282024,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,7,0,4,5,6],[2,3,7,6,1,0,4,5],[3,7,6,5,2,1,0,4],[4,0,1,2,5,6,7,3],[5,4,0,1,6,7,3,2],[6,5,4,0,7,3,2,1],[7,6,5,4,3,2,1,0]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,5,5,7,2,0],[0,3,5,1,6,2,4,7],[0,4,5,6,1,2,3,7],[0,5,7,2,2,7,5,0],[0,6,2,4,3,5,1,7],[0,7,0,7,7,0,7,0]],"subset":[0,2,5,7],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-qualification","semantic_template":"qf-mod28-ideal-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,0,4,5,6],[2,3,7,6,1,0,4,5],[3,7,6,5,2,1,0,4],[4,0,1,2,5,6,7,3],[5,4,0,1,6,7,3,2],[6,5,4,0,7,3,2,1],[7,6,5,4,3,2,1,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,5,5,7,2,0],[0,3,5,1,6,2,4,7],[0,4,5,6,1,2,3,7],[0,5,7,2,2,7,5,0],[0,6,2,4,3,5,1,7],[0,7,0,7,7,0,7,0]],\"n\":8,\"subset\":[0,2,5,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0025","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-qualification；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,4,5,3,6,7,0],[2,4,3,6,5,7,0,1],[3,5,6,0,7,1,2,4],[4,3,5,7,6,0,1,2],[5,6,7,1,0,2,4,3],[6,7,0,2,1,4,3,5],[7,0,1,4,2,3,5,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,3,0,6,2,3,6],[0,3,0,0,3,3,0,3],[0,4,6,3,1,7,2,5],[0,5,2,3,7,1,6,4],[0,6,3,0,2,6,3,2],[0,7,6,3,5,4,2,1]],\"subset\":[0,3]}。請依 [subset,zero,additive_subgroup,missing_left,missing_right,two_sided_ideal] 欄序回答。","answer":{"kind":"expression","value":"([0, 3], 0, 是, [], [], 是)","explanation_zh":"Wave4第二層核對=([0, 3], 0, 是, [], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-qualification","seed":202610282025,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,4,5,3,6,7,0],[2,4,3,6,5,7,0,1],[3,5,6,0,7,1,2,4],[4,3,5,7,6,0,1,2],[5,6,7,1,0,2,4,3],[6,7,0,2,1,4,3,5],[7,0,1,4,2,3,5,6]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,3,0,6,2,3,6],[0,3,0,0,3,3,0,3],[0,4,6,3,1,7,2,5],[0,5,2,3,7,1,6,4],[0,6,3,0,2,6,3,2],[0,7,6,3,5,4,2,1]],"subset":[0,3],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-qualification","semantic_template":"qf-mod28-ideal-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,4,5,3,6,7,0],[2,4,3,6,5,7,0,1],[3,5,6,0,7,1,2,4],[4,3,5,7,6,0,1,2],[5,6,7,1,0,2,4,3],[6,7,0,2,1,4,3,5],[7,0,1,4,2,3,5,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,3,0,6,2,3,6],[0,3,0,0,3,3,0,3],[0,4,6,3,1,7,2,5],[0,5,2,3,7,1,6,4],[0,6,3,0,2,6,3,2],[0,7,6,3,5,4,2,1]],\"n\":8,\"subset\":[0,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0026","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-additive-subgroup；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,6,0,2,4,6],[0,3,6,1,4,7,2,5],[0,4,0,4,0,4,0,4],[0,5,2,7,4,1,6,3],[0,6,4,2,0,6,4,2],[0,7,6,5,4,3,2,1]],\"subset\":[0,2,4,6]}。請依 [subset,zero,missing_additions,missing_inverses,additive_subgroup] 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 4, 6], 0, [], [], 是)","explanation_zh":"Wave4第二層核對=([0, 2, 4, 6], 0, [], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-additive-subgroup","seed":202610282026,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,6,0,2,4,6],[0,3,6,1,4,7,2,5],[0,4,0,4,0,4,0,4],[0,5,2,7,4,1,6,3],[0,6,4,2,0,6,4,2],[0,7,6,5,4,3,2,1]],"subset":[0,2,4,6],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-additive-subgroup","semantic_template":"qf-mod28-ideal-additive-subgroup/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,6,0,2,4,6],[0,3,6,1,4,7,2,5],[0,4,0,4,0,4,0,4],[0,5,2,7,4,1,6,3],[0,6,4,2,0,6,4,2],[0,7,6,5,4,3,2,1]],\"n\":8,\"subset\":[0,2,4,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0027","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-additive-subgroup；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,7,0,6],[2,3,4,5,7,6,1,0],[3,4,5,7,6,0,2,1],[4,5,7,6,0,1,3,2],[5,7,6,0,1,2,4,3],[6,0,1,2,3,4,7,5],[7,6,0,1,2,3,5,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,7,0,2,7,4],[0,3,7,1,4,6,5,2],[0,4,0,4,0,4,4,0],[0,5,2,6,4,1,3,7],[0,6,7,5,4,3,1,2],[0,7,4,2,0,7,2,4]],\"subset\":[0,4]}。請依 [subset,zero,missing_additions,missing_inverses,additive_subgroup] 欄序回答。","answer":{"kind":"expression","value":"([0, 4], 0, [], [], 是)","explanation_zh":"Wave4第二層核對=([0, 4], 0, [], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-additive-subgroup","seed":202610282027,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,7,0,6],[2,3,4,5,7,6,1,0],[3,4,5,7,6,0,2,1],[4,5,7,6,0,1,3,2],[5,7,6,0,1,2,4,3],[6,0,1,2,3,4,7,5],[7,6,0,1,2,3,5,4]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,7,0,2,7,4],[0,3,7,1,4,6,5,2],[0,4,0,4,0,4,4,0],[0,5,2,6,4,1,3,7],[0,6,7,5,4,3,1,2],[0,7,4,2,0,7,2,4]],"subset":[0,4],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-additive-subgroup","semantic_template":"qf-mod28-ideal-additive-subgroup/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,7,0,6],[2,3,4,5,7,6,1,0],[3,4,5,7,6,0,2,1],[4,5,7,6,0,1,3,2],[5,7,6,0,1,2,4,3],[6,0,1,2,3,4,7,5],[7,6,0,1,2,3,5,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,7,0,2,7,4],[0,3,7,1,4,6,5,2],[0,4,0,4,0,4,4,0],[0,5,2,6,4,1,3,7],[0,6,7,5,4,3,1,2],[0,7,4,2,0,7,2,4]],\"n\":8,\"subset\":[0,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0028","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-additive-subgroup；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,6,7,5,0],[2,3,4,6,5,0,7,1],[3,4,6,5,7,1,0,2],[4,6,5,7,0,2,1,3],[5,7,0,1,2,4,3,6],[6,5,7,0,1,3,2,4],[7,0,1,2,3,6,4,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,5,0,4,2,5],[0,3,5,1,4,2,7,6],[0,4,0,4,0,0,4,4],[0,5,4,2,0,4,5,2],[0,6,2,7,4,5,1,3],[0,7,5,6,4,2,3,1]],\"subset\":[0]}。請依 [subset,zero,missing_additions,missing_inverses,additive_subgroup] 欄序回答。","answer":{"kind":"expression","value":"([0], 0, [], [], 是)","explanation_zh":"Wave4第二層核對=([0], 0, [], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-additive-subgroup","seed":202610282028,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,4,6,7,5,0],[2,3,4,6,5,0,7,1],[3,4,6,5,7,1,0,2],[4,6,5,7,0,2,1,3],[5,7,0,1,2,4,3,6],[6,5,7,0,1,3,2,4],[7,0,1,2,3,6,4,5]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,5,0,4,2,5],[0,3,5,1,4,2,7,6],[0,4,0,4,0,0,4,4],[0,5,4,2,0,4,5,2],[0,6,2,7,4,5,1,3],[0,7,5,6,4,2,3,1]],"subset":[0],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-additive-subgroup","semantic_template":"qf-mod28-ideal-additive-subgroup/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,6,7,5,0],[2,3,4,6,5,0,7,1],[3,4,6,5,7,1,0,2],[4,6,5,7,0,2,1,3],[5,7,0,1,2,4,3,6],[6,5,7,0,1,3,2,4],[7,0,1,2,3,6,4,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,5,0,4,2,5],[0,3,5,1,4,2,7,6],[0,4,0,4,0,0,4,4],[0,5,4,2,0,4,5,2],[0,6,2,7,4,5,1,3],[0,7,5,6,4,2,3,1]],\"n\":8,\"subset\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0029","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-additive-subgroup；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,6,0,7,5],[2,3,4,6,7,1,5,0],[3,4,6,7,5,2,0,1],[4,6,7,5,0,3,1,2],[5,0,1,2,3,7,4,6],[6,7,5,0,1,4,2,3],[7,5,0,1,2,6,3,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,7,0,7,2,4],[0,3,7,1,4,6,5,2],[0,4,0,4,0,4,4,0],[0,5,7,6,4,1,3,2],[0,6,2,5,4,3,1,7],[0,7,4,2,0,2,7,4]],\"subset\":[0,2,4,7]}。請依 [subset,zero,missing_additions,missing_inverses,additive_subgroup] 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 4, 7], 0, [], [], 是)","explanation_zh":"Wave4第二層核對=([0, 2, 4, 7], 0, [], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-additive-subgroup","seed":202610282029,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,4,6,0,7,5],[2,3,4,6,7,1,5,0],[3,4,6,7,5,2,0,1],[4,6,7,5,0,3,1,2],[5,0,1,2,3,7,4,6],[6,7,5,0,1,4,2,3],[7,5,0,1,2,6,3,4]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,7,0,7,2,4],[0,3,7,1,4,6,5,2],[0,4,0,4,0,4,4,0],[0,5,7,6,4,1,3,2],[0,6,2,5,4,3,1,7],[0,7,4,2,0,2,7,4]],"subset":[0,2,4,7],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-additive-subgroup","semantic_template":"qf-mod28-ideal-additive-subgroup/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,6,0,7,5],[2,3,4,6,7,1,5,0],[3,4,6,7,5,2,0,1],[4,6,7,5,0,3,1,2],[5,0,1,2,3,7,4,6],[6,7,5,0,1,4,2,3],[7,5,0,1,2,6,3,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,7,0,7,2,4],[0,3,7,1,4,6,5,2],[0,4,0,4,0,4,4,0],[0,5,7,6,4,1,3,2],[0,6,2,5,4,3,1,7],[0,7,4,2,0,2,7,4]],\"n\":8,\"subset\":[0,2,4,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0030","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-additive-subgroup；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,7,6,0,5],[2,3,4,7,5,0,1,6],[3,4,7,5,6,1,2,0],[4,7,5,6,0,2,3,1],[5,6,0,1,2,4,7,3],[6,0,1,2,3,7,5,4],[7,5,6,0,1,3,4,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,5,0,4,5,2],[0,3,5,1,4,2,7,6],[0,4,0,4,0,0,4,4],[0,5,4,2,0,4,2,5],[0,6,5,7,4,2,1,3],[0,7,2,6,4,5,3,1]],\"subset\":[0,4]}。請依 [subset,zero,missing_additions,missing_inverses,additive_subgroup] 欄序回答。","answer":{"kind":"expression","value":"([0, 4], 0, [], [], 是)","explanation_zh":"Wave4第二層核對=([0, 4], 0, [], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-additive-subgroup","seed":202610282030,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,4,7,6,0,5],[2,3,4,7,5,0,1,6],[3,4,7,5,6,1,2,0],[4,7,5,6,0,2,3,1],[5,6,0,1,2,4,7,3],[6,0,1,2,3,7,5,4],[7,5,6,0,1,3,4,2]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,5,0,4,5,2],[0,3,5,1,4,2,7,6],[0,4,0,4,0,0,4,4],[0,5,4,2,0,4,2,5],[0,6,5,7,4,2,1,3],[0,7,2,6,4,5,3,1]],"subset":[0,4],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-additive-subgroup","semantic_template":"qf-mod28-ideal-additive-subgroup/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,7,6,0,5],[2,3,4,7,5,0,1,6],[3,4,7,5,6,1,2,0],[4,7,5,6,0,2,3,1],[5,6,0,1,2,4,7,3],[6,0,1,2,3,7,5,4],[7,5,6,0,1,3,4,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,5,0,4,5,2],[0,3,5,1,4,2,7,6],[0,4,0,4,0,0,4,4],[0,5,4,2,0,4,2,5],[0,6,5,7,4,2,1,3],[0,7,2,6,4,5,3,1]],\"n\":8,\"subset\":[0,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0031","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-additive-subgroup；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,7,0,5,6],[2,3,4,7,6,1,0,5],[3,4,7,6,5,2,1,0],[4,7,6,5,0,3,2,1],[5,0,1,2,3,6,7,4],[6,5,0,1,2,7,4,3],[7,6,5,0,1,4,3,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,6,0,6,4,2],[0,3,6,1,4,7,2,5],[0,4,0,4,0,4,0,4],[0,5,6,7,4,1,2,3],[0,6,4,2,0,2,4,6],[0,7,2,5,4,3,6,1]],\"subset\":[0,2,4,6]}。請依 [subset,zero,missing_additions,missing_inverses,additive_subgroup] 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 4, 6], 0, [], [], 是)","explanation_zh":"Wave4第二層核對=([0, 2, 4, 6], 0, [], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-additive-subgroup","seed":202610282031,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,4,7,0,5,6],[2,3,4,7,6,1,0,5],[3,4,7,6,5,2,1,0],[4,7,6,5,0,3,2,1],[5,0,1,2,3,6,7,4],[6,5,0,1,2,7,4,3],[7,6,5,0,1,4,3,2]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,6,0,6,4,2],[0,3,6,1,4,7,2,5],[0,4,0,4,0,4,0,4],[0,5,6,7,4,1,2,3],[0,6,4,2,0,2,4,6],[0,7,2,5,4,3,6,1]],"subset":[0,2,4,6],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-additive-subgroup","semantic_template":"qf-mod28-ideal-additive-subgroup/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,7,0,5,6],[2,3,4,7,6,1,0,5],[3,4,7,6,5,2,1,0],[4,7,6,5,0,3,2,1],[5,0,1,2,3,6,7,4],[6,5,0,1,2,7,4,3],[7,6,5,0,1,4,3,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,6,0,6,4,2],[0,3,6,1,4,7,2,5],[0,4,0,4,0,4,0,4],[0,5,6,7,4,1,2,3],[0,6,4,2,0,2,4,6],[0,7,2,5,4,3,6,1]],\"n\":8,\"subset\":[0,2,4,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0032","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-additive-subgroup；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,6,4,7,0],[2,3,5,4,7,6,0,1],[3,5,4,6,0,7,1,2],[4,6,7,0,2,1,3,5],[5,4,6,7,1,0,2,3],[6,7,0,1,3,2,5,4],[7,0,1,2,5,3,4,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,6,2,0,5,6],[0,3,6,1,7,5,2,4],[0,4,2,7,1,5,6,3],[0,5,0,5,5,0,0,5],[0,6,5,2,6,0,5,2],[0,7,6,4,3,5,2,1]],\"subset\":[0,5]}。請依 [subset,zero,missing_additions,missing_inverses,additive_subgroup] 欄序回答。","answer":{"kind":"expression","value":"([0, 5], 0, [], [], 是)","explanation_zh":"Wave4第二層核對=([0, 5], 0, [], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-additive-subgroup","seed":202610282032,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,5,6,4,7,0],[2,3,5,4,7,6,0,1],[3,5,4,6,0,7,1,2],[4,6,7,0,2,1,3,5],[5,4,6,7,1,0,2,3],[6,7,0,1,3,2,5,4],[7,0,1,2,5,3,4,6]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,6,2,0,5,6],[0,3,6,1,7,5,2,4],[0,4,2,7,1,5,6,3],[0,5,0,5,5,0,0,5],[0,6,5,2,6,0,5,2],[0,7,6,4,3,5,2,1]],"subset":[0,5],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-additive-subgroup","semantic_template":"qf-mod28-ideal-additive-subgroup/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,6,4,7,0],[2,3,5,4,7,6,0,1],[3,5,4,6,0,7,1,2],[4,6,7,0,2,1,3,5],[5,4,6,7,1,0,2,3],[6,7,0,1,3,2,5,4],[7,0,1,2,5,3,4,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,6,2,0,5,6],[0,3,6,1,7,5,2,4],[0,4,2,7,1,5,6,3],[0,5,0,5,5,0,0,5],[0,6,5,2,6,0,5,2],[0,7,6,4,3,5,2,1]],\"n\":8,\"subset\":[0,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0033","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-additive-subgroup；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,7,4,0,6],[2,3,5,4,6,7,1,0],[3,5,4,7,0,6,2,1],[4,7,6,0,2,1,5,3],[5,4,7,6,1,0,3,2],[6,0,1,2,5,3,7,4],[7,6,0,1,3,2,4,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,7,2,0,7,5],[0,3,7,1,6,5,4,2],[0,4,2,6,1,5,3,7],[0,5,0,5,5,0,5,0],[0,6,7,4,3,5,1,2],[0,7,5,2,7,0,2,5]],\"subset\":[0]}。請依 [subset,zero,missing_additions,missing_inverses,additive_subgroup] 欄序回答。","answer":{"kind":"expression","value":"([0], 0, [], [], 是)","explanation_zh":"Wave4第二層核對=([0], 0, [], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-additive-subgroup","seed":202610282033,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,5,7,4,0,6],[2,3,5,4,6,7,1,0],[3,5,4,7,0,6,2,1],[4,7,6,0,2,1,5,3],[5,4,7,6,1,0,3,2],[6,0,1,2,5,3,7,4],[7,6,0,1,3,2,4,5]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,7,2,0,7,5],[0,3,7,1,6,5,4,2],[0,4,2,6,1,5,3,7],[0,5,0,5,5,0,5,0],[0,6,7,4,3,5,1,2],[0,7,5,2,7,0,2,5]],"subset":[0],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-additive-subgroup","semantic_template":"qf-mod28-ideal-additive-subgroup/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,7,4,0,6],[2,3,5,4,6,7,1,0],[3,5,4,7,0,6,2,1],[4,7,6,0,2,1,5,3],[5,4,7,6,1,0,3,2],[6,0,1,2,5,3,7,4],[7,6,0,1,3,2,4,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,7,2,0,7,5],[0,3,7,1,6,5,4,2],[0,4,2,6,1,5,3,7],[0,5,0,5,5,0,5,0],[0,6,7,4,3,5,1,2],[0,7,5,2,7,0,2,5]],\"n\":8,\"subset\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0034","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-additive-subgroup；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,7,6,4,0],[2,3,5,6,0,4,7,1],[3,5,6,4,1,7,0,2],[4,7,0,1,5,2,3,6],[5,6,4,7,2,0,1,3],[6,4,7,0,3,1,2,5],[7,0,1,2,6,3,5,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,4,5,0,2,4],[0,3,4,1,2,5,7,6],[0,4,5,2,5,0,4,2],[0,5,0,5,0,0,5,5],[0,6,2,7,4,5,1,3],[0,7,4,6,2,5,3,1]],\"subset\":[0,2,4,5]}。請依 [subset,zero,missing_additions,missing_inverses,additive_subgroup] 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 4, 5], 0, [], [], 是)","explanation_zh":"Wave4第二層核對=([0, 2, 4, 5], 0, [], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-additive-subgroup","seed":202610282034,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,5,7,6,4,0],[2,3,5,6,0,4,7,1],[3,5,6,4,1,7,0,2],[4,7,0,1,5,2,3,6],[5,6,4,7,2,0,1,3],[6,4,7,0,3,1,2,5],[7,0,1,2,6,3,5,4]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,4,5,0,2,4],[0,3,4,1,2,5,7,6],[0,4,5,2,5,0,4,2],[0,5,0,5,0,0,5,5],[0,6,2,7,4,5,1,3],[0,7,4,6,2,5,3,1]],"subset":[0,2,4,5],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-additive-subgroup","semantic_template":"qf-mod28-ideal-additive-subgroup/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,7,6,4,0],[2,3,5,6,0,4,7,1],[3,5,6,4,1,7,0,2],[4,7,0,1,5,2,3,6],[5,6,4,7,2,0,1,3],[6,4,7,0,3,1,2,5],[7,0,1,2,6,3,5,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,4,5,0,2,4],[0,3,4,1,2,5,7,6],[0,4,5,2,5,0,4,2],[0,5,0,5,0,0,5,5],[0,6,2,7,4,5,1,3],[0,7,4,6,2,5,3,1]],\"n\":8,\"subset\":[0,2,4,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0035","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-additive-subgroup；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,0,6,7,4],[2,3,5,6,1,7,4,0],[3,5,6,7,2,4,0,1],[4,0,1,2,7,3,5,6],[5,6,7,4,3,0,1,2],[6,7,4,0,5,1,2,3],[7,4,0,1,6,2,3,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,7,7,0,2,5],[0,3,7,1,6,5,4,2],[0,4,7,6,1,5,3,2],[0,5,0,5,5,0,5,0],[0,6,2,4,3,5,1,7],[0,7,5,2,2,0,7,5]],\"subset\":[0,5]}。請依 [subset,zero,missing_additions,missing_inverses,additive_subgroup] 欄序回答。","answer":{"kind":"expression","value":"([0, 5], 0, [], [], 是)","explanation_zh":"Wave4第二層核對=([0, 5], 0, [], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-additive-subgroup","seed":202610282035,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,5,0,6,7,4],[2,3,5,6,1,7,4,0],[3,5,6,7,2,4,0,1],[4,0,1,2,7,3,5,6],[5,6,7,4,3,0,1,2],[6,7,4,0,5,1,2,3],[7,4,0,1,6,2,3,5]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,7,7,0,2,5],[0,3,7,1,6,5,4,2],[0,4,7,6,1,5,3,2],[0,5,0,5,5,0,5,0],[0,6,2,4,3,5,1,7],[0,7,5,2,2,0,7,5]],"subset":[0,5],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-additive-subgroup","semantic_template":"qf-mod28-ideal-additive-subgroup/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,0,6,7,4],[2,3,5,6,1,7,4,0],[3,5,6,7,2,4,0,1],[4,0,1,2,7,3,5,6],[5,6,7,4,3,0,1,2],[6,7,4,0,5,1,2,3],[7,4,0,1,6,2,3,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,7,7,0,2,5],[0,3,7,1,6,5,4,2],[0,4,7,6,1,5,3,2],[0,5,0,5,5,0,5,0],[0,6,2,4,3,5,1,7],[0,7,5,2,2,0,7,5]],\"n\":8,\"subset\":[0,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0036","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-additive-subgroup；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,6,7,0,4],[2,3,5,7,0,4,1,6],[3,5,7,4,1,6,2,0],[4,6,0,1,5,2,7,3],[5,7,4,6,2,0,3,1],[6,0,1,2,7,3,4,5],[7,4,6,0,3,1,5,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,4,5,0,4,2],[0,3,4,1,2,5,7,6],[0,4,5,2,5,0,2,4],[0,5,0,5,0,0,5,5],[0,6,4,7,2,5,1,3],[0,7,2,6,4,5,3,1]],\"subset\":[0,2,4,5]}。請依 [subset,zero,missing_additions,missing_inverses,additive_subgroup] 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 4, 5], 0, [], [], 是)","explanation_zh":"Wave4第二層核對=([0, 2, 4, 5], 0, [], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-additive-subgroup","seed":202610282036,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,5,6,7,0,4],[2,3,5,7,0,4,1,6],[3,5,7,4,1,6,2,0],[4,6,0,1,5,2,7,3],[5,7,4,6,2,0,3,1],[6,0,1,2,7,3,4,5],[7,4,6,0,3,1,5,2]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,4,5,0,4,2],[0,3,4,1,2,5,7,6],[0,4,5,2,5,0,2,4],[0,5,0,5,0,0,5,5],[0,6,4,7,2,5,1,3],[0,7,2,6,4,5,3,1]],"subset":[0,2,4,5],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-additive-subgroup","semantic_template":"qf-mod28-ideal-additive-subgroup/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,6,7,0,4],[2,3,5,7,0,4,1,6],[3,5,7,4,1,6,2,0],[4,6,0,1,5,2,7,3],[5,7,4,6,2,0,3,1],[6,0,1,2,7,3,4,5],[7,4,6,0,3,1,5,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,4,5,0,4,2],[0,3,4,1,2,5,7,6],[0,4,5,2,5,0,2,4],[0,5,0,5,0,0,5,5],[0,6,4,7,2,5,1,3],[0,7,2,6,4,5,3,1]],\"n\":8,\"subset\":[0,2,4,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0037","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-additive-subgroup；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,0,7,4,6],[2,3,5,7,1,6,0,4],[3,5,7,6,2,4,1,0],[4,0,1,2,6,3,7,5],[5,7,6,4,3,0,2,1],[6,4,0,1,7,2,5,3],[7,6,4,0,5,1,3,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,6,6,0,5,2],[0,3,6,1,7,5,2,4],[0,4,6,7,1,5,2,3],[0,5,0,5,5,0,0,5],[0,6,5,2,2,0,5,6],[0,7,2,4,3,5,6,1]],\"subset\":[0,5]}。請依 [subset,zero,missing_additions,missing_inverses,additive_subgroup] 欄序回答。","answer":{"kind":"expression","value":"([0, 5], 0, [], [], 是)","explanation_zh":"Wave4第二層核對=([0, 5], 0, [], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-additive-subgroup","seed":202610282037,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,5,0,7,4,6],[2,3,5,7,1,6,0,4],[3,5,7,6,2,4,1,0],[4,0,1,2,6,3,7,5],[5,7,6,4,3,0,2,1],[6,4,0,1,7,2,5,3],[7,6,4,0,5,1,3,2]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,6,6,0,5,2],[0,3,6,1,7,5,2,4],[0,4,6,7,1,5,2,3],[0,5,0,5,5,0,0,5],[0,6,5,2,2,0,5,6],[0,7,2,4,3,5,6,1]],"subset":[0,5],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-additive-subgroup","semantic_template":"qf-mod28-ideal-additive-subgroup/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,0,7,4,6],[2,3,5,7,1,6,0,4],[3,5,7,6,2,4,1,0],[4,0,1,2,6,3,7,5],[5,7,6,4,3,0,2,1],[6,4,0,1,7,2,5,3],[7,6,4,0,5,1,3,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,6,6,0,5,2],[0,3,6,1,7,5,2,4],[0,4,6,7,1,5,2,3],[0,5,0,5,5,0,0,5],[0,6,5,2,2,0,5,6],[0,7,2,4,3,5,6,1]],\"n\":8,\"subset\":[0,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0038","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-additive-subgroup；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,5,7,4,0],[2,3,6,4,7,0,5,1],[3,6,4,5,0,1,7,2],[4,5,7,0,2,3,1,6],[5,7,0,1,3,6,2,4],[6,4,5,7,1,2,0,3],[7,0,1,2,6,4,3,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,5,2,6,0,5],[0,3,5,1,7,2,6,4],[0,4,2,7,1,5,6,3],[0,5,6,2,5,6,0,2],[0,6,0,6,6,0,0,6],[0,7,5,4,3,2,6,1]],\"subset\":[0]}。請依 [subset,zero,missing_additions,missing_inverses,additive_subgroup] 欄序回答。","answer":{"kind":"expression","value":"([0], 0, [], [], 是)","explanation_zh":"Wave4第二層核對=([0], 0, [], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-additive-subgroup","seed":202610282038,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,6,5,7,4,0],[2,3,6,4,7,0,5,1],[3,6,4,5,0,1,7,2],[4,5,7,0,2,3,1,6],[5,7,0,1,3,6,2,4],[6,4,5,7,1,2,0,3],[7,0,1,2,6,4,3,5]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,5,2,6,0,5],[0,3,5,1,7,2,6,4],[0,4,2,7,1,5,6,3],[0,5,6,2,5,6,0,2],[0,6,0,6,6,0,0,6],[0,7,5,4,3,2,6,1]],"subset":[0],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-additive-subgroup","semantic_template":"qf-mod28-ideal-additive-subgroup/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,5,7,4,0],[2,3,6,4,7,0,5,1],[3,6,4,5,0,1,7,2],[4,5,7,0,2,3,1,6],[5,7,0,1,3,6,2,4],[6,4,5,7,1,2,0,3],[7,0,1,2,6,4,3,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,5,2,6,0,5],[0,3,5,1,7,2,6,4],[0,4,2,7,1,5,6,3],[0,5,6,2,5,6,0,2],[0,6,0,6,6,0,0,6],[0,7,5,4,3,2,6,1]],\"n\":8,\"subset\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0039","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-additive-subgroup；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,7,0,4,5],[2,3,6,4,5,1,7,0],[3,6,4,7,0,2,5,1],[4,7,5,0,2,6,1,3],[5,0,1,2,6,7,3,4],[6,4,7,5,1,3,0,2],[7,5,0,1,3,4,2,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,7,2,7,0,6],[0,3,7,1,5,4,6,2],[0,4,2,5,1,3,6,7],[0,5,7,4,3,1,6,2],[0,6,0,6,6,6,0,0],[0,7,6,2,7,2,0,6]],\"subset\":[0,2,6,7]}。請依 [subset,zero,missing_additions,missing_inverses,additive_subgroup] 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 6, 7], 0, [], [], 是)","explanation_zh":"Wave4第二層核對=([0, 2, 6, 7], 0, [], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-additive-subgroup","seed":202610282039,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,6,7,0,4,5],[2,3,6,4,5,1,7,0],[3,6,4,7,0,2,5,1],[4,7,5,0,2,6,1,3],[5,0,1,2,6,7,3,4],[6,4,7,5,1,3,0,2],[7,5,0,1,3,4,2,6]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,7,2,7,0,6],[0,3,7,1,5,4,6,2],[0,4,2,5,1,3,6,7],[0,5,7,4,3,1,6,2],[0,6,0,6,6,6,0,0],[0,7,6,2,7,2,0,6]],"subset":[0,2,6,7],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-additive-subgroup","semantic_template":"qf-mod28-ideal-additive-subgroup/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,7,0,4,5],[2,3,6,4,5,1,7,0],[3,6,4,7,0,2,5,1],[4,7,5,0,2,6,1,3],[5,0,1,2,6,7,3,4],[6,4,7,5,1,3,0,2],[7,5,0,1,3,4,2,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,7,2,7,0,6],[0,3,7,1,5,4,6,2],[0,4,2,5,1,3,6,7],[0,5,7,4,3,1,6,2],[0,6,0,6,6,6,0,0],[0,7,6,2,7,2,0,6]],\"n\":8,\"subset\":[0,2,6,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0040","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-additive-subgroup；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,7,4,5,0],[2,3,6,5,0,7,4,1],[3,6,5,4,1,0,7,2],[4,7,0,1,6,3,2,5],[5,4,7,0,3,2,1,6],[6,5,4,7,2,1,0,3],[7,0,1,2,5,6,3,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,4,6,2,0,4],[0,3,4,1,2,7,6,5],[0,4,6,2,6,4,0,2],[0,5,2,7,4,1,6,3],[0,6,0,6,0,6,0,6],[0,7,4,5,2,3,6,1]],\"subset\":[0,6]}。請依 [subset,zero,missing_additions,missing_inverses,additive_subgroup] 欄序回答。","answer":{"kind":"expression","value":"([0, 6], 0, [], [], 是)","explanation_zh":"Wave4第二層核對=([0, 6], 0, [], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-additive-subgroup","seed":202610282040,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,6,7,4,5,0],[2,3,6,5,0,7,4,1],[3,6,5,4,1,0,7,2],[4,7,0,1,6,3,2,5],[5,4,7,0,3,2,1,6],[6,5,4,7,2,1,0,3],[7,0,1,2,5,6,3,4]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,4,6,2,0,4],[0,3,4,1,2,7,6,5],[0,4,6,2,6,4,0,2],[0,5,2,7,4,1,6,3],[0,6,0,6,0,6,0,6],[0,7,4,5,2,3,6,1]],"subset":[0,6],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-additive-subgroup","semantic_template":"qf-mod28-ideal-additive-subgroup/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,7,4,5,0],[2,3,6,5,0,7,4,1],[3,6,5,4,1,0,7,2],[4,7,0,1,6,3,2,5],[5,4,7,0,3,2,1,6],[6,5,4,7,2,1,0,3],[7,0,1,2,5,6,3,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,4,6,2,0,4],[0,3,4,1,2,7,6,5],[0,4,6,2,6,4,0,2],[0,5,2,7,4,1,6,3],[0,6,0,6,0,6,0,6],[0,7,4,5,2,3,6,1]],\"n\":8,\"subset\":[0,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0041","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-additive-subgroup；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,0,7,5,4],[2,3,6,5,1,4,7,0],[3,6,5,7,2,0,4,1],[4,0,1,2,7,6,3,5],[5,7,4,0,6,2,1,3],[6,5,7,4,3,1,0,2],[7,4,0,1,5,3,2,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,7,7,2,0,6],[0,3,7,1,5,4,6,2],[0,4,7,5,1,3,6,2],[0,5,2,4,3,1,6,7],[0,6,0,6,6,6,0,0],[0,7,6,2,2,7,0,6]],\"subset\":[0,2,6,7]}。請依 [subset,zero,missing_additions,missing_inverses,additive_subgroup] 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 6, 7], 0, [], [], 是)","explanation_zh":"Wave4第二層核對=([0, 2, 6, 7], 0, [], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-additive-subgroup","seed":202610282041,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,6,0,7,5,4],[2,3,6,5,1,4,7,0],[3,6,5,7,2,0,4,1],[4,0,1,2,7,6,3,5],[5,7,4,0,6,2,1,3],[6,5,7,4,3,1,0,2],[7,4,0,1,5,3,2,6]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,7,7,2,0,6],[0,3,7,1,5,4,6,2],[0,4,7,5,1,3,6,2],[0,5,2,4,3,1,6,7],[0,6,0,6,6,6,0,0],[0,7,6,2,2,7,0,6]],"subset":[0,2,6,7],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-additive-subgroup","semantic_template":"qf-mod28-ideal-additive-subgroup/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,0,7,5,4],[2,3,6,5,1,4,7,0],[3,6,5,7,2,0,4,1],[4,0,1,2,7,6,3,5],[5,7,4,0,6,2,1,3],[6,5,7,4,3,1,0,2],[7,4,0,1,5,3,2,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,7,7,2,0,6],[0,3,7,1,5,4,6,2],[0,4,7,5,1,3,6,2],[0,5,2,4,3,1,6,7],[0,6,0,6,6,6,0,0],[0,7,6,2,2,7,0,6]],\"n\":8,\"subset\":[0,2,6,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0042","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-additive-subgroup；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,5,0,7,4],[2,3,6,7,0,1,4,5],[3,6,7,4,1,2,5,0],[4,5,0,1,6,7,2,3],[5,0,1,2,7,4,3,6],[6,7,4,5,2,3,0,1],[7,4,5,0,3,6,1,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,4,6,4,0,2],[0,3,4,1,2,7,6,5],[0,4,6,2,6,2,0,4],[0,5,4,7,2,1,6,3],[0,6,0,6,0,6,0,6],[0,7,2,5,4,3,6,1]],\"subset\":[0,6]}。請依 [subset,zero,missing_additions,missing_inverses,additive_subgroup] 欄序回答。","answer":{"kind":"expression","value":"([0, 6], 0, [], [], 是)","explanation_zh":"Wave4第二層核對=([0, 6], 0, [], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-additive-subgroup","seed":202610282042,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,6,5,0,7,4],[2,3,6,7,0,1,4,5],[3,6,7,4,1,2,5,0],[4,5,0,1,6,7,2,3],[5,0,1,2,7,4,3,6],[6,7,4,5,2,3,0,1],[7,4,5,0,3,6,1,2]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,4,6,4,0,2],[0,3,4,1,2,7,6,5],[0,4,6,2,6,2,0,4],[0,5,4,7,2,1,6,3],[0,6,0,6,0,6,0,6],[0,7,2,5,4,3,6,1]],"subset":[0,6],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-additive-subgroup","semantic_template":"qf-mod28-ideal-additive-subgroup/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,5,0,7,4],[2,3,6,7,0,1,4,5],[3,6,7,4,1,2,5,0],[4,5,0,1,6,7,2,3],[5,0,1,2,7,4,3,6],[6,7,4,5,2,3,0,1],[7,4,5,0,3,6,1,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,4,6,4,0,2],[0,3,4,1,2,7,6,5],[0,4,6,2,6,2,0,4],[0,5,4,7,2,1,6,3],[0,6,0,6,0,6,0,6],[0,7,2,5,4,3,6,1]],\"n\":8,\"subset\":[0,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0043","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-additive-subgroup；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,0,4,7,5],[2,3,6,7,1,0,5,4],[3,6,7,5,2,1,4,0],[4,0,1,2,5,7,3,6],[5,4,0,1,7,6,2,3],[6,7,5,4,3,2,0,1],[7,5,4,0,6,3,1,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,5,5,6,0,2],[0,3,5,1,7,2,6,4],[0,4,5,7,1,2,6,3],[0,5,6,2,2,6,0,5],[0,6,0,6,6,0,0,6],[0,7,2,4,3,5,6,1]],\"subset\":[0]}。請依 [subset,zero,missing_additions,missing_inverses,additive_subgroup] 欄序回答。","answer":{"kind":"expression","value":"([0], 0, [], [], 是)","explanation_zh":"Wave4第二層核對=([0], 0, [], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-additive-subgroup","seed":202610282043,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,6,0,4,7,5],[2,3,6,7,1,0,5,4],[3,6,7,5,2,1,4,0],[4,0,1,2,5,7,3,6],[5,4,0,1,7,6,2,3],[6,7,5,4,3,2,0,1],[7,5,4,0,6,3,1,2]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,5,5,6,0,2],[0,3,5,1,7,2,6,4],[0,4,5,7,1,2,6,3],[0,5,6,2,2,6,0,5],[0,6,0,6,6,0,0,6],[0,7,2,4,3,5,6,1]],"subset":[0],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-additive-subgroup","semantic_template":"qf-mod28-ideal-additive-subgroup/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,0,4,7,5],[2,3,6,7,1,0,5,4],[3,6,7,5,2,1,4,0],[4,0,1,2,5,7,3,6],[5,4,0,1,7,6,2,3],[6,7,5,4,3,2,0,1],[7,5,4,0,6,3,1,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,5,5,6,0,2],[0,3,5,1,7,2,6,4],[0,4,5,7,1,2,6,3],[0,5,6,2,2,6,0,5],[0,6,0,6,6,0,0,6],[0,7,2,4,3,5,6,1]],\"n\":8,\"subset\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0044","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-additive-subgroup；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,5,6,0,4],[2,3,7,4,6,0,1,5],[3,7,4,5,0,1,2,6],[4,5,6,0,2,3,7,1],[5,6,0,1,3,7,4,2],[6,0,1,2,7,4,5,3],[7,4,5,6,1,2,3,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,5,2,7,5,0],[0,3,5,1,6,2,4,7],[0,4,2,6,1,5,3,7],[0,5,7,2,5,7,2,0],[0,6,5,4,3,2,1,7],[0,7,0,7,7,0,7,0]],\"subset\":[0,2,5,7]}。請依 [subset,zero,missing_additions,missing_inverses,additive_subgroup] 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 5, 7], 0, [], [], 是)","explanation_zh":"Wave4第二層核對=([0, 2, 5, 7], 0, [], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-additive-subgroup","seed":202610282044,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,7,5,6,0,4],[2,3,7,4,6,0,1,5],[3,7,4,5,0,1,2,6],[4,5,6,0,2,3,7,1],[5,6,0,1,3,7,4,2],[6,0,1,2,7,4,5,3],[7,4,5,6,1,2,3,0]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,5,2,7,5,0],[0,3,5,1,6,2,4,7],[0,4,2,6,1,5,3,7],[0,5,7,2,5,7,2,0],[0,6,5,4,3,2,1,7],[0,7,0,7,7,0,7,0]],"subset":[0,2,5,7],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-additive-subgroup","semantic_template":"qf-mod28-ideal-additive-subgroup/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,5,6,0,4],[2,3,7,4,6,0,1,5],[3,7,4,5,0,1,2,6],[4,5,6,0,2,3,7,1],[5,6,0,1,3,7,4,2],[6,0,1,2,7,4,5,3],[7,4,5,6,1,2,3,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,5,2,7,5,0],[0,3,5,1,6,2,4,7],[0,4,2,6,1,5,3,7],[0,5,7,2,5,7,2,0],[0,6,5,4,3,2,1,7],[0,7,0,7,7,0,7,0]],\"n\":8,\"subset\":[0,2,5,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0045","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-additive-subgroup；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,6,0,5,4],[2,3,7,4,5,1,0,6],[3,7,4,6,0,2,1,5],[4,6,5,0,2,7,3,1],[5,0,1,2,7,6,4,3],[6,5,0,1,3,4,7,2],[7,4,6,5,1,3,2,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,6,2,6,7,0],[0,3,6,1,5,4,2,7],[0,4,2,5,1,3,6,7],[0,5,6,4,3,1,2,7],[0,6,7,2,6,2,7,0],[0,7,0,7,7,7,0,0]],\"subset\":[0,7]}。請依 [subset,zero,missing_additions,missing_inverses,additive_subgroup] 欄序回答。","answer":{"kind":"expression","value":"([0, 7], 0, [], [], 是)","explanation_zh":"Wave4第二層核對=([0, 7], 0, [], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-additive-subgroup","seed":202610282045,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,7,6,0,5,4],[2,3,7,4,5,1,0,6],[3,7,4,6,0,2,1,5],[4,6,5,0,2,7,3,1],[5,0,1,2,7,6,4,3],[6,5,0,1,3,4,7,2],[7,4,6,5,1,3,2,0]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,6,2,6,7,0],[0,3,6,1,5,4,2,7],[0,4,2,5,1,3,6,7],[0,5,6,4,3,1,2,7],[0,6,7,2,6,2,7,0],[0,7,0,7,7,7,0,0]],"subset":[0,7],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-additive-subgroup","semantic_template":"qf-mod28-ideal-additive-subgroup/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,6,0,5,4],[2,3,7,4,5,1,0,6],[3,7,4,6,0,2,1,5],[4,6,5,0,2,7,3,1],[5,0,1,2,7,6,4,3],[6,5,0,1,3,4,7,2],[7,4,6,5,1,3,2,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,6,2,6,7,0],[0,3,6,1,5,4,2,7],[0,4,2,5,1,3,6,7],[0,5,6,4,3,1,2,7],[0,6,7,2,6,2,7,0],[0,7,0,7,7,7,0,0]],\"n\":8,\"subset\":[0,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0046","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-additive-subgroup；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,6,4,0,5],[2,3,7,5,0,6,1,4],[3,7,5,4,1,0,2,6],[4,6,0,1,7,3,5,2],[5,4,6,0,3,2,7,1],[6,0,1,2,5,7,4,3],[7,5,4,6,2,1,3,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,4,7,2,4,0],[0,3,4,1,2,6,5,7],[0,4,7,2,7,4,2,0],[0,5,2,6,4,1,3,7],[0,6,4,5,2,3,1,7],[0,7,0,7,0,7,7,0]],\"subset\":[0,2,4,7]}。請依 [subset,zero,missing_additions,missing_inverses,additive_subgroup] 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 4, 7], 0, [], [], 是)","explanation_zh":"Wave4第二層核對=([0, 2, 4, 7], 0, [], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-additive-subgroup","seed":202610282046,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,7,6,4,0,5],[2,3,7,5,0,6,1,4],[3,7,5,4,1,0,2,6],[4,6,0,1,7,3,5,2],[5,4,6,0,3,2,7,1],[6,0,1,2,5,7,4,3],[7,5,4,6,2,1,3,0]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,4,7,2,4,0],[0,3,4,1,2,6,5,7],[0,4,7,2,7,4,2,0],[0,5,2,6,4,1,3,7],[0,6,4,5,2,3,1,7],[0,7,0,7,0,7,7,0]],"subset":[0,2,4,7],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-additive-subgroup","semantic_template":"qf-mod28-ideal-additive-subgroup/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,6,4,0,5],[2,3,7,5,0,6,1,4],[3,7,5,4,1,0,2,6],[4,6,0,1,7,3,5,2],[5,4,6,0,3,2,7,1],[6,0,1,2,5,7,4,3],[7,5,4,6,2,1,3,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,4,7,2,4,0],[0,3,4,1,2,6,5,7],[0,4,7,2,7,4,2,0],[0,5,2,6,4,1,3,7],[0,6,4,5,2,3,1,7],[0,7,0,7,0,7,7,0]],\"n\":8,\"subset\":[0,2,4,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0047","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-additive-subgroup；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,0,6,4,5],[2,3,7,5,1,4,0,6],[3,7,5,6,2,0,1,4],[4,0,1,2,6,7,5,3],[5,6,4,0,7,2,3,1],[6,4,0,1,5,3,7,2],[7,5,6,4,3,1,2,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,6,6,2,7,0],[0,3,6,1,5,4,2,7],[0,4,6,5,1,3,2,7],[0,5,2,4,3,1,6,7],[0,6,7,2,2,6,7,0],[0,7,0,7,7,7,0,0]],\"subset\":[0,7]}。請依 [subset,zero,missing_additions,missing_inverses,additive_subgroup] 欄序回答。","answer":{"kind":"expression","value":"([0, 7], 0, [], [], 是)","explanation_zh":"Wave4第二層核對=([0, 7], 0, [], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-additive-subgroup","seed":202610282047,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,7,0,6,4,5],[2,3,7,5,1,4,0,6],[3,7,5,6,2,0,1,4],[4,0,1,2,6,7,5,3],[5,6,4,0,7,2,3,1],[6,4,0,1,5,3,7,2],[7,5,6,4,3,1,2,0]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,6,6,2,7,0],[0,3,6,1,5,4,2,7],[0,4,6,5,1,3,2,7],[0,5,2,4,3,1,6,7],[0,6,7,2,2,6,7,0],[0,7,0,7,7,7,0,0]],"subset":[0,7],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-additive-subgroup","semantic_template":"qf-mod28-ideal-additive-subgroup/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,0,6,4,5],[2,3,7,5,1,4,0,6],[3,7,5,6,2,0,1,4],[4,0,1,2,6,7,5,3],[5,6,4,0,7,2,3,1],[6,4,0,1,5,3,7,2],[7,5,6,4,3,1,2,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,6,6,2,7,0],[0,3,6,1,5,4,2,7],[0,4,6,5,1,3,2,7],[0,5,2,4,3,1,6,7],[0,6,7,2,2,6,7,0],[0,7,0,7,7,7,0,0]],\"n\":8,\"subset\":[0,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0048","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-additive-subgroup；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,5,0,4,6],[2,3,7,6,0,1,5,4],[3,7,6,4,1,2,0,5],[4,5,0,1,7,6,3,2],[5,0,1,2,6,4,7,3],[6,4,5,0,3,7,2,1],[7,6,4,5,2,3,1,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,4,7,4,2,0],[0,3,4,1,2,6,5,7],[0,4,7,2,7,2,4,0],[0,5,4,6,2,1,3,7],[0,6,2,5,4,3,1,7],[0,7,0,7,0,7,7,0]],\"subset\":[0]}。請依 [subset,zero,missing_additions,missing_inverses,additive_subgroup] 欄序回答。","answer":{"kind":"expression","value":"([0], 0, [], [], 是)","explanation_zh":"Wave4第二層核對=([0], 0, [], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-additive-subgroup","seed":202610282048,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,7,5,0,4,6],[2,3,7,6,0,1,5,4],[3,7,6,4,1,2,0,5],[4,5,0,1,7,6,3,2],[5,0,1,2,6,4,7,3],[6,4,5,0,3,7,2,1],[7,6,4,5,2,3,1,0]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,4,7,4,2,0],[0,3,4,1,2,6,5,7],[0,4,7,2,7,2,4,0],[0,5,4,6,2,1,3,7],[0,6,2,5,4,3,1,7],[0,7,0,7,0,7,7,0]],"subset":[0],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-additive-subgroup","semantic_template":"qf-mod28-ideal-additive-subgroup/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,5,0,4,6],[2,3,7,6,0,1,5,4],[3,7,6,4,1,2,0,5],[4,5,0,1,7,6,3,2],[5,0,1,2,6,4,7,3],[6,4,5,0,3,7,2,1],[7,6,4,5,2,3,1,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,4,7,4,2,0],[0,3,4,1,2,6,5,7],[0,4,7,2,7,2,4,0],[0,5,4,6,2,1,3,7],[0,6,2,5,4,3,1,7],[0,7,0,7,0,7,7,0]],\"n\":8,\"subset\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0049","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-additive-subgroup；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,0,4,5,6],[2,3,7,6,1,0,4,5],[3,7,6,5,2,1,0,4],[4,0,1,2,5,6,7,3],[5,4,0,1,6,7,3,2],[6,5,4,0,7,3,2,1],[7,6,5,4,3,2,1,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,5,5,7,2,0],[0,3,5,1,6,2,4,7],[0,4,5,6,1,2,3,7],[0,5,7,2,2,7,5,0],[0,6,2,4,3,5,1,7],[0,7,0,7,7,0,7,0]],\"subset\":[0,2,5,7]}。請依 [subset,zero,missing_additions,missing_inverses,additive_subgroup] 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 5, 7], 0, [], [], 是)","explanation_zh":"Wave4第二層核對=([0, 2, 5, 7], 0, [], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-additive-subgroup","seed":202610282049,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,7,0,4,5,6],[2,3,7,6,1,0,4,5],[3,7,6,5,2,1,0,4],[4,0,1,2,5,6,7,3],[5,4,0,1,6,7,3,2],[6,5,4,0,7,3,2,1],[7,6,5,4,3,2,1,0]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,5,5,7,2,0],[0,3,5,1,6,2,4,7],[0,4,5,6,1,2,3,7],[0,5,7,2,2,7,5,0],[0,6,2,4,3,5,1,7],[0,7,0,7,7,0,7,0]],"subset":[0,2,5,7],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-additive-subgroup","semantic_template":"qf-mod28-ideal-additive-subgroup/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,0,4,5,6],[2,3,7,6,1,0,4,5],[3,7,6,5,2,1,0,4],[4,0,1,2,5,6,7,3],[5,4,0,1,6,7,3,2],[6,5,4,0,7,3,2,1],[7,6,5,4,3,2,1,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,5,5,7,2,0],[0,3,5,1,6,2,4,7],[0,4,5,6,1,2,3,7],[0,5,7,2,2,7,5,0],[0,6,2,4,3,5,1,7],[0,7,0,7,7,0,7,0]],\"n\":8,\"subset\":[0,2,5,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0050","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-additive-subgroup；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,4,5,3,6,7,0],[2,4,3,6,5,7,0,1],[3,5,6,0,7,1,2,4],[4,3,5,7,6,0,1,2],[5,6,7,1,0,2,4,3],[6,7,0,2,1,4,3,5],[7,0,1,4,2,3,5,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,3,0,6,2,3,6],[0,3,0,0,3,3,0,3],[0,4,6,3,1,7,2,5],[0,5,2,3,7,1,6,4],[0,6,3,0,2,6,3,2],[0,7,6,3,5,4,2,1]],\"subset\":[0,3]}。請依 [subset,zero,missing_additions,missing_inverses,additive_subgroup] 欄序回答。","answer":{"kind":"expression","value":"([0, 3], 0, [], [], 是)","explanation_zh":"Wave4第二層核對=([0, 3], 0, [], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-additive-subgroup","seed":202610282050,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,4,5,3,6,7,0],[2,4,3,6,5,7,0,1],[3,5,6,0,7,1,2,4],[4,3,5,7,6,0,1,2],[5,6,7,1,0,2,4,3],[6,7,0,2,1,4,3,5],[7,0,1,4,2,3,5,6]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,3,0,6,2,3,6],[0,3,0,0,3,3,0,3],[0,4,6,3,1,7,2,5],[0,5,2,3,7,1,6,4],[0,6,3,0,2,6,3,2],[0,7,6,3,5,4,2,1]],"subset":[0,3],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-additive-subgroup","semantic_template":"qf-mod28-ideal-additive-subgroup/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,4,5,3,6,7,0],[2,4,3,6,5,7,0,1],[3,5,6,0,7,1,2,4],[4,3,5,7,6,0,1,2],[5,6,7,1,0,2,4,3],[6,7,0,2,1,4,3,5],[7,0,1,4,2,3,5,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,3,0,6,2,3,6],[0,3,0,0,3,3,0,3],[0,4,6,3,1,7,2,5],[0,5,2,3,7,1,6,4],[0,6,3,0,2,6,3,2],[0,7,6,3,5,4,2,1]],\"n\":8,\"subset\":[0,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0051","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-two-sided-absorption；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,6,0,2,4,6],[0,3,6,1,4,7,2,5],[0,4,0,4,0,4,0,4],[0,5,2,7,4,1,6,3],[0,6,4,2,0,6,4,2],[0,7,6,5,4,3,2,1]],\"subset\":[0,2,4,6]}。請依 [subset,missing_left,missing_right,left_absorbing,right_absorbing] 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 4, 6], [], [], 是, 是)","explanation_zh":"Wave4第二層核對=([0, 2, 4, 6], [], [], 是, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-two-sided-absorption","seed":202610282051,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,6,0,2,4,6],[0,3,6,1,4,7,2,5],[0,4,0,4,0,4,0,4],[0,5,2,7,4,1,6,3],[0,6,4,2,0,6,4,2],[0,7,6,5,4,3,2,1]],"subset":[0,2,4,6],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-two-sided-absorption","semantic_template":"qf-mod28-ideal-two-sided-absorption/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,6,0,2,4,6],[0,3,6,1,4,7,2,5],[0,4,0,4,0,4,0,4],[0,5,2,7,4,1,6,3],[0,6,4,2,0,6,4,2],[0,7,6,5,4,3,2,1]],\"n\":8,\"subset\":[0,2,4,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0052","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-two-sided-absorption；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,7,0,6],[2,3,4,5,7,6,1,0],[3,4,5,7,6,0,2,1],[4,5,7,6,0,1,3,2],[5,7,6,0,1,2,4,3],[6,0,1,2,3,4,7,5],[7,6,0,1,2,3,5,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,7,0,2,7,4],[0,3,7,1,4,6,5,2],[0,4,0,4,0,4,4,0],[0,5,2,6,4,1,3,7],[0,6,7,5,4,3,1,2],[0,7,4,2,0,7,2,4]],\"subset\":[0,4]}。請依 [subset,missing_left,missing_right,left_absorbing,right_absorbing] 欄序回答。","answer":{"kind":"expression","value":"([0, 4], [], [], 是, 是)","explanation_zh":"Wave4第二層核對=([0, 4], [], [], 是, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-two-sided-absorption","seed":202610282052,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,7,0,6],[2,3,4,5,7,6,1,0],[3,4,5,7,6,0,2,1],[4,5,7,6,0,1,3,2],[5,7,6,0,1,2,4,3],[6,0,1,2,3,4,7,5],[7,6,0,1,2,3,5,4]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,7,0,2,7,4],[0,3,7,1,4,6,5,2],[0,4,0,4,0,4,4,0],[0,5,2,6,4,1,3,7],[0,6,7,5,4,3,1,2],[0,7,4,2,0,7,2,4]],"subset":[0,4],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-two-sided-absorption","semantic_template":"qf-mod28-ideal-two-sided-absorption/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,7,0,6],[2,3,4,5,7,6,1,0],[3,4,5,7,6,0,2,1],[4,5,7,6,0,1,3,2],[5,7,6,0,1,2,4,3],[6,0,1,2,3,4,7,5],[7,6,0,1,2,3,5,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,7,0,2,7,4],[0,3,7,1,4,6,5,2],[0,4,0,4,0,4,4,0],[0,5,2,6,4,1,3,7],[0,6,7,5,4,3,1,2],[0,7,4,2,0,7,2,4]],\"n\":8,\"subset\":[0,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0053","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-two-sided-absorption；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,6,7,5,0],[2,3,4,6,5,0,7,1],[3,4,6,5,7,1,0,2],[4,6,5,7,0,2,1,3],[5,7,0,1,2,4,3,6],[6,5,7,0,1,3,2,4],[7,0,1,2,3,6,4,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,5,0,4,2,5],[0,3,5,1,4,2,7,6],[0,4,0,4,0,0,4,4],[0,5,4,2,0,4,5,2],[0,6,2,7,4,5,1,3],[0,7,5,6,4,2,3,1]],\"subset\":[0]}。請依 [subset,missing_left,missing_right,left_absorbing,right_absorbing] 欄序回答。","answer":{"kind":"expression","value":"([0], [], [], 是, 是)","explanation_zh":"Wave4第二層核對=([0], [], [], 是, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-two-sided-absorption","seed":202610282053,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,4,6,7,5,0],[2,3,4,6,5,0,7,1],[3,4,6,5,7,1,0,2],[4,6,5,7,0,2,1,3],[5,7,0,1,2,4,3,6],[6,5,7,0,1,3,2,4],[7,0,1,2,3,6,4,5]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,5,0,4,2,5],[0,3,5,1,4,2,7,6],[0,4,0,4,0,0,4,4],[0,5,4,2,0,4,5,2],[0,6,2,7,4,5,1,3],[0,7,5,6,4,2,3,1]],"subset":[0],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-two-sided-absorption","semantic_template":"qf-mod28-ideal-two-sided-absorption/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,6,7,5,0],[2,3,4,6,5,0,7,1],[3,4,6,5,7,1,0,2],[4,6,5,7,0,2,1,3],[5,7,0,1,2,4,3,6],[6,5,7,0,1,3,2,4],[7,0,1,2,3,6,4,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,5,0,4,2,5],[0,3,5,1,4,2,7,6],[0,4,0,4,0,0,4,4],[0,5,4,2,0,4,5,2],[0,6,2,7,4,5,1,3],[0,7,5,6,4,2,3,1]],\"n\":8,\"subset\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0054","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-two-sided-absorption；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,6,0,7,5],[2,3,4,6,7,1,5,0],[3,4,6,7,5,2,0,1],[4,6,7,5,0,3,1,2],[5,0,1,2,3,7,4,6],[6,7,5,0,1,4,2,3],[7,5,0,1,2,6,3,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,7,0,7,2,4],[0,3,7,1,4,6,5,2],[0,4,0,4,0,4,4,0],[0,5,7,6,4,1,3,2],[0,6,2,5,4,3,1,7],[0,7,4,2,0,2,7,4]],\"subset\":[0,2,4,7]}。請依 [subset,missing_left,missing_right,left_absorbing,right_absorbing] 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 4, 7], [], [], 是, 是)","explanation_zh":"Wave4第二層核對=([0, 2, 4, 7], [], [], 是, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-two-sided-absorption","seed":202610282054,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,4,6,0,7,5],[2,3,4,6,7,1,5,0],[3,4,6,7,5,2,0,1],[4,6,7,5,0,3,1,2],[5,0,1,2,3,7,4,6],[6,7,5,0,1,4,2,3],[7,5,0,1,2,6,3,4]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,7,0,7,2,4],[0,3,7,1,4,6,5,2],[0,4,0,4,0,4,4,0],[0,5,7,6,4,1,3,2],[0,6,2,5,4,3,1,7],[0,7,4,2,0,2,7,4]],"subset":[0,2,4,7],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-two-sided-absorption","semantic_template":"qf-mod28-ideal-two-sided-absorption/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,6,0,7,5],[2,3,4,6,7,1,5,0],[3,4,6,7,5,2,0,1],[4,6,7,5,0,3,1,2],[5,0,1,2,3,7,4,6],[6,7,5,0,1,4,2,3],[7,5,0,1,2,6,3,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,7,0,7,2,4],[0,3,7,1,4,6,5,2],[0,4,0,4,0,4,4,0],[0,5,7,6,4,1,3,2],[0,6,2,5,4,3,1,7],[0,7,4,2,0,2,7,4]],\"n\":8,\"subset\":[0,2,4,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0055","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-two-sided-absorption；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,7,6,0,5],[2,3,4,7,5,0,1,6],[3,4,7,5,6,1,2,0],[4,7,5,6,0,2,3,1],[5,6,0,1,2,4,7,3],[6,0,1,2,3,7,5,4],[7,5,6,0,1,3,4,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,5,0,4,5,2],[0,3,5,1,4,2,7,6],[0,4,0,4,0,0,4,4],[0,5,4,2,0,4,2,5],[0,6,5,7,4,2,1,3],[0,7,2,6,4,5,3,1]],\"subset\":[0,4]}。請依 [subset,missing_left,missing_right,left_absorbing,right_absorbing] 欄序回答。","answer":{"kind":"expression","value":"([0, 4], [], [], 是, 是)","explanation_zh":"Wave4第二層核對=([0, 4], [], [], 是, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-two-sided-absorption","seed":202610282055,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,4,7,6,0,5],[2,3,4,7,5,0,1,6],[3,4,7,5,6,1,2,0],[4,7,5,6,0,2,3,1],[5,6,0,1,2,4,7,3],[6,0,1,2,3,7,5,4],[7,5,6,0,1,3,4,2]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,5,0,4,5,2],[0,3,5,1,4,2,7,6],[0,4,0,4,0,0,4,4],[0,5,4,2,0,4,2,5],[0,6,5,7,4,2,1,3],[0,7,2,6,4,5,3,1]],"subset":[0,4],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-two-sided-absorption","semantic_template":"qf-mod28-ideal-two-sided-absorption/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,7,6,0,5],[2,3,4,7,5,0,1,6],[3,4,7,5,6,1,2,0],[4,7,5,6,0,2,3,1],[5,6,0,1,2,4,7,3],[6,0,1,2,3,7,5,4],[7,5,6,0,1,3,4,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,5,0,4,5,2],[0,3,5,1,4,2,7,6],[0,4,0,4,0,0,4,4],[0,5,4,2,0,4,2,5],[0,6,5,7,4,2,1,3],[0,7,2,6,4,5,3,1]],\"n\":8,\"subset\":[0,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0056","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-two-sided-absorption；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,7,0,5,6],[2,3,4,7,6,1,0,5],[3,4,7,6,5,2,1,0],[4,7,6,5,0,3,2,1],[5,0,1,2,3,6,7,4],[6,5,0,1,2,7,4,3],[7,6,5,0,1,4,3,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,6,0,6,4,2],[0,3,6,1,4,7,2,5],[0,4,0,4,0,4,0,4],[0,5,6,7,4,1,2,3],[0,6,4,2,0,2,4,6],[0,7,2,5,4,3,6,1]],\"subset\":[0,2,4,6]}。請依 [subset,missing_left,missing_right,left_absorbing,right_absorbing] 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 4, 6], [], [], 是, 是)","explanation_zh":"Wave4第二層核對=([0, 2, 4, 6], [], [], 是, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-two-sided-absorption","seed":202610282056,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,4,7,0,5,6],[2,3,4,7,6,1,0,5],[3,4,7,6,5,2,1,0],[4,7,6,5,0,3,2,1],[5,0,1,2,3,6,7,4],[6,5,0,1,2,7,4,3],[7,6,5,0,1,4,3,2]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,6,0,6,4,2],[0,3,6,1,4,7,2,5],[0,4,0,4,0,4,0,4],[0,5,6,7,4,1,2,3],[0,6,4,2,0,2,4,6],[0,7,2,5,4,3,6,1]],"subset":[0,2,4,6],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-two-sided-absorption","semantic_template":"qf-mod28-ideal-two-sided-absorption/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,7,0,5,6],[2,3,4,7,6,1,0,5],[3,4,7,6,5,2,1,0],[4,7,6,5,0,3,2,1],[5,0,1,2,3,6,7,4],[6,5,0,1,2,7,4,3],[7,6,5,0,1,4,3,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,6,0,6,4,2],[0,3,6,1,4,7,2,5],[0,4,0,4,0,4,0,4],[0,5,6,7,4,1,2,3],[0,6,4,2,0,2,4,6],[0,7,2,5,4,3,6,1]],\"n\":8,\"subset\":[0,2,4,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0057","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-two-sided-absorption；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,6,4,7,0],[2,3,5,4,7,6,0,1],[3,5,4,6,0,7,1,2],[4,6,7,0,2,1,3,5],[5,4,6,7,1,0,2,3],[6,7,0,1,3,2,5,4],[7,0,1,2,5,3,4,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,6,2,0,5,6],[0,3,6,1,7,5,2,4],[0,4,2,7,1,5,6,3],[0,5,0,5,5,0,0,5],[0,6,5,2,6,0,5,2],[0,7,6,4,3,5,2,1]],\"subset\":[0,5]}。請依 [subset,missing_left,missing_right,left_absorbing,right_absorbing] 欄序回答。","answer":{"kind":"expression","value":"([0, 5], [], [], 是, 是)","explanation_zh":"Wave4第二層核對=([0, 5], [], [], 是, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-two-sided-absorption","seed":202610282057,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,5,6,4,7,0],[2,3,5,4,7,6,0,1],[3,5,4,6,0,7,1,2],[4,6,7,0,2,1,3,5],[5,4,6,7,1,0,2,3],[6,7,0,1,3,2,5,4],[7,0,1,2,5,3,4,6]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,6,2,0,5,6],[0,3,6,1,7,5,2,4],[0,4,2,7,1,5,6,3],[0,5,0,5,5,0,0,5],[0,6,5,2,6,0,5,2],[0,7,6,4,3,5,2,1]],"subset":[0,5],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-two-sided-absorption","semantic_template":"qf-mod28-ideal-two-sided-absorption/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,6,4,7,0],[2,3,5,4,7,6,0,1],[3,5,4,6,0,7,1,2],[4,6,7,0,2,1,3,5],[5,4,6,7,1,0,2,3],[6,7,0,1,3,2,5,4],[7,0,1,2,5,3,4,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,6,2,0,5,6],[0,3,6,1,7,5,2,4],[0,4,2,7,1,5,6,3],[0,5,0,5,5,0,0,5],[0,6,5,2,6,0,5,2],[0,7,6,4,3,5,2,1]],\"n\":8,\"subset\":[0,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0058","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-two-sided-absorption；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,7,4,0,6],[2,3,5,4,6,7,1,0],[3,5,4,7,0,6,2,1],[4,7,6,0,2,1,5,3],[5,4,7,6,1,0,3,2],[6,0,1,2,5,3,7,4],[7,6,0,1,3,2,4,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,7,2,0,7,5],[0,3,7,1,6,5,4,2],[0,4,2,6,1,5,3,7],[0,5,0,5,5,0,5,0],[0,6,7,4,3,5,1,2],[0,7,5,2,7,0,2,5]],\"subset\":[0]}。請依 [subset,missing_left,missing_right,left_absorbing,right_absorbing] 欄序回答。","answer":{"kind":"expression","value":"([0], [], [], 是, 是)","explanation_zh":"Wave4第二層核對=([0], [], [], 是, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-two-sided-absorption","seed":202610282058,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,5,7,4,0,6],[2,3,5,4,6,7,1,0],[3,5,4,7,0,6,2,1],[4,7,6,0,2,1,5,3],[5,4,7,6,1,0,3,2],[6,0,1,2,5,3,7,4],[7,6,0,1,3,2,4,5]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,7,2,0,7,5],[0,3,7,1,6,5,4,2],[0,4,2,6,1,5,3,7],[0,5,0,5,5,0,5,0],[0,6,7,4,3,5,1,2],[0,7,5,2,7,0,2,5]],"subset":[0],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-two-sided-absorption","semantic_template":"qf-mod28-ideal-two-sided-absorption/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,7,4,0,6],[2,3,5,4,6,7,1,0],[3,5,4,7,0,6,2,1],[4,7,6,0,2,1,5,3],[5,4,7,6,1,0,3,2],[6,0,1,2,5,3,7,4],[7,6,0,1,3,2,4,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,7,2,0,7,5],[0,3,7,1,6,5,4,2],[0,4,2,6,1,5,3,7],[0,5,0,5,5,0,5,0],[0,6,7,4,3,5,1,2],[0,7,5,2,7,0,2,5]],\"n\":8,\"subset\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0059","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-two-sided-absorption；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,7,6,4,0],[2,3,5,6,0,4,7,1],[3,5,6,4,1,7,0,2],[4,7,0,1,5,2,3,6],[5,6,4,7,2,0,1,3],[6,4,7,0,3,1,2,5],[7,0,1,2,6,3,5,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,4,5,0,2,4],[0,3,4,1,2,5,7,6],[0,4,5,2,5,0,4,2],[0,5,0,5,0,0,5,5],[0,6,2,7,4,5,1,3],[0,7,4,6,2,5,3,1]],\"subset\":[0,2,4,5]}。請依 [subset,missing_left,missing_right,left_absorbing,right_absorbing] 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 4, 5], [], [], 是, 是)","explanation_zh":"Wave4第二層核對=([0, 2, 4, 5], [], [], 是, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-two-sided-absorption","seed":202610282059,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,5,7,6,4,0],[2,3,5,6,0,4,7,1],[3,5,6,4,1,7,0,2],[4,7,0,1,5,2,3,6],[5,6,4,7,2,0,1,3],[6,4,7,0,3,1,2,5],[7,0,1,2,6,3,5,4]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,4,5,0,2,4],[0,3,4,1,2,5,7,6],[0,4,5,2,5,0,4,2],[0,5,0,5,0,0,5,5],[0,6,2,7,4,5,1,3],[0,7,4,6,2,5,3,1]],"subset":[0,2,4,5],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-two-sided-absorption","semantic_template":"qf-mod28-ideal-two-sided-absorption/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,7,6,4,0],[2,3,5,6,0,4,7,1],[3,5,6,4,1,7,0,2],[4,7,0,1,5,2,3,6],[5,6,4,7,2,0,1,3],[6,4,7,0,3,1,2,5],[7,0,1,2,6,3,5,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,4,5,0,2,4],[0,3,4,1,2,5,7,6],[0,4,5,2,5,0,4,2],[0,5,0,5,0,0,5,5],[0,6,2,7,4,5,1,3],[0,7,4,6,2,5,3,1]],\"n\":8,\"subset\":[0,2,4,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0060","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-two-sided-absorption；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,0,6,7,4],[2,3,5,6,1,7,4,0],[3,5,6,7,2,4,0,1],[4,0,1,2,7,3,5,6],[5,6,7,4,3,0,1,2],[6,7,4,0,5,1,2,3],[7,4,0,1,6,2,3,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,7,7,0,2,5],[0,3,7,1,6,5,4,2],[0,4,7,6,1,5,3,2],[0,5,0,5,5,0,5,0],[0,6,2,4,3,5,1,7],[0,7,5,2,2,0,7,5]],\"subset\":[0,5]}。請依 [subset,missing_left,missing_right,left_absorbing,right_absorbing] 欄序回答。","answer":{"kind":"expression","value":"([0, 5], [], [], 是, 是)","explanation_zh":"Wave4第二層核對=([0, 5], [], [], 是, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-two-sided-absorption","seed":202610282060,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,5,0,6,7,4],[2,3,5,6,1,7,4,0],[3,5,6,7,2,4,0,1],[4,0,1,2,7,3,5,6],[5,6,7,4,3,0,1,2],[6,7,4,0,5,1,2,3],[7,4,0,1,6,2,3,5]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,7,7,0,2,5],[0,3,7,1,6,5,4,2],[0,4,7,6,1,5,3,2],[0,5,0,5,5,0,5,0],[0,6,2,4,3,5,1,7],[0,7,5,2,2,0,7,5]],"subset":[0,5],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-two-sided-absorption","semantic_template":"qf-mod28-ideal-two-sided-absorption/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,0,6,7,4],[2,3,5,6,1,7,4,0],[3,5,6,7,2,4,0,1],[4,0,1,2,7,3,5,6],[5,6,7,4,3,0,1,2],[6,7,4,0,5,1,2,3],[7,4,0,1,6,2,3,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,7,7,0,2,5],[0,3,7,1,6,5,4,2],[0,4,7,6,1,5,3,2],[0,5,0,5,5,0,5,0],[0,6,2,4,3,5,1,7],[0,7,5,2,2,0,7,5]],\"n\":8,\"subset\":[0,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0061","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-two-sided-absorption；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,6,7,0,4],[2,3,5,7,0,4,1,6],[3,5,7,4,1,6,2,0],[4,6,0,1,5,2,7,3],[5,7,4,6,2,0,3,1],[6,0,1,2,7,3,4,5],[7,4,6,0,3,1,5,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,4,5,0,4,2],[0,3,4,1,2,5,7,6],[0,4,5,2,5,0,2,4],[0,5,0,5,0,0,5,5],[0,6,4,7,2,5,1,3],[0,7,2,6,4,5,3,1]],\"subset\":[0,2,4,5]}。請依 [subset,missing_left,missing_right,left_absorbing,right_absorbing] 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 4, 5], [], [], 是, 是)","explanation_zh":"Wave4第二層核對=([0, 2, 4, 5], [], [], 是, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-two-sided-absorption","seed":202610282061,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,5,6,7,0,4],[2,3,5,7,0,4,1,6],[3,5,7,4,1,6,2,0],[4,6,0,1,5,2,7,3],[5,7,4,6,2,0,3,1],[6,0,1,2,7,3,4,5],[7,4,6,0,3,1,5,2]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,4,5,0,4,2],[0,3,4,1,2,5,7,6],[0,4,5,2,5,0,2,4],[0,5,0,5,0,0,5,5],[0,6,4,7,2,5,1,3],[0,7,2,6,4,5,3,1]],"subset":[0,2,4,5],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-two-sided-absorption","semantic_template":"qf-mod28-ideal-two-sided-absorption/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,6,7,0,4],[2,3,5,7,0,4,1,6],[3,5,7,4,1,6,2,0],[4,6,0,1,5,2,7,3],[5,7,4,6,2,0,3,1],[6,0,1,2,7,3,4,5],[7,4,6,0,3,1,5,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,4,5,0,4,2],[0,3,4,1,2,5,7,6],[0,4,5,2,5,0,2,4],[0,5,0,5,0,0,5,5],[0,6,4,7,2,5,1,3],[0,7,2,6,4,5,3,1]],\"n\":8,\"subset\":[0,2,4,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0062","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-two-sided-absorption；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,0,7,4,6],[2,3,5,7,1,6,0,4],[3,5,7,6,2,4,1,0],[4,0,1,2,6,3,7,5],[5,7,6,4,3,0,2,1],[6,4,0,1,7,2,5,3],[7,6,4,0,5,1,3,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,6,6,0,5,2],[0,3,6,1,7,5,2,4],[0,4,6,7,1,5,2,3],[0,5,0,5,5,0,0,5],[0,6,5,2,2,0,5,6],[0,7,2,4,3,5,6,1]],\"subset\":[0,5]}。請依 [subset,missing_left,missing_right,left_absorbing,right_absorbing] 欄序回答。","answer":{"kind":"expression","value":"([0, 5], [], [], 是, 是)","explanation_zh":"Wave4第二層核對=([0, 5], [], [], 是, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-two-sided-absorption","seed":202610282062,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,5,0,7,4,6],[2,3,5,7,1,6,0,4],[3,5,7,6,2,4,1,0],[4,0,1,2,6,3,7,5],[5,7,6,4,3,0,2,1],[6,4,0,1,7,2,5,3],[7,6,4,0,5,1,3,2]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,6,6,0,5,2],[0,3,6,1,7,5,2,4],[0,4,6,7,1,5,2,3],[0,5,0,5,5,0,0,5],[0,6,5,2,2,0,5,6],[0,7,2,4,3,5,6,1]],"subset":[0,5],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-two-sided-absorption","semantic_template":"qf-mod28-ideal-two-sided-absorption/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,0,7,4,6],[2,3,5,7,1,6,0,4],[3,5,7,6,2,4,1,0],[4,0,1,2,6,3,7,5],[5,7,6,4,3,0,2,1],[6,4,0,1,7,2,5,3],[7,6,4,0,5,1,3,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,6,6,0,5,2],[0,3,6,1,7,5,2,4],[0,4,6,7,1,5,2,3],[0,5,0,5,5,0,0,5],[0,6,5,2,2,0,5,6],[0,7,2,4,3,5,6,1]],\"n\":8,\"subset\":[0,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0063","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-two-sided-absorption；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,5,7,4,0],[2,3,6,4,7,0,5,1],[3,6,4,5,0,1,7,2],[4,5,7,0,2,3,1,6],[5,7,0,1,3,6,2,4],[6,4,5,7,1,2,0,3],[7,0,1,2,6,4,3,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,5,2,6,0,5],[0,3,5,1,7,2,6,4],[0,4,2,7,1,5,6,3],[0,5,6,2,5,6,0,2],[0,6,0,6,6,0,0,6],[0,7,5,4,3,2,6,1]],\"subset\":[0]}。請依 [subset,missing_left,missing_right,left_absorbing,right_absorbing] 欄序回答。","answer":{"kind":"expression","value":"([0], [], [], 是, 是)","explanation_zh":"Wave4第二層核對=([0], [], [], 是, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-two-sided-absorption","seed":202610282063,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,6,5,7,4,0],[2,3,6,4,7,0,5,1],[3,6,4,5,0,1,7,2],[4,5,7,0,2,3,1,6],[5,7,0,1,3,6,2,4],[6,4,5,7,1,2,0,3],[7,0,1,2,6,4,3,5]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,5,2,6,0,5],[0,3,5,1,7,2,6,4],[0,4,2,7,1,5,6,3],[0,5,6,2,5,6,0,2],[0,6,0,6,6,0,0,6],[0,7,5,4,3,2,6,1]],"subset":[0],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-two-sided-absorption","semantic_template":"qf-mod28-ideal-two-sided-absorption/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,5,7,4,0],[2,3,6,4,7,0,5,1],[3,6,4,5,0,1,7,2],[4,5,7,0,2,3,1,6],[5,7,0,1,3,6,2,4],[6,4,5,7,1,2,0,3],[7,0,1,2,6,4,3,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,5,2,6,0,5],[0,3,5,1,7,2,6,4],[0,4,2,7,1,5,6,3],[0,5,6,2,5,6,0,2],[0,6,0,6,6,0,0,6],[0,7,5,4,3,2,6,1]],\"n\":8,\"subset\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0064","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-two-sided-absorption；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,7,0,4,5],[2,3,6,4,5,1,7,0],[3,6,4,7,0,2,5,1],[4,7,5,0,2,6,1,3],[5,0,1,2,6,7,3,4],[6,4,7,5,1,3,0,2],[7,5,0,1,3,4,2,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,7,2,7,0,6],[0,3,7,1,5,4,6,2],[0,4,2,5,1,3,6,7],[0,5,7,4,3,1,6,2],[0,6,0,6,6,6,0,0],[0,7,6,2,7,2,0,6]],\"subset\":[0,2,6,7]}。請依 [subset,missing_left,missing_right,left_absorbing,right_absorbing] 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 6, 7], [], [], 是, 是)","explanation_zh":"Wave4第二層核對=([0, 2, 6, 7], [], [], 是, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-two-sided-absorption","seed":202610282064,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,6,7,0,4,5],[2,3,6,4,5,1,7,0],[3,6,4,7,0,2,5,1],[4,7,5,0,2,6,1,3],[5,0,1,2,6,7,3,4],[6,4,7,5,1,3,0,2],[7,5,0,1,3,4,2,6]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,7,2,7,0,6],[0,3,7,1,5,4,6,2],[0,4,2,5,1,3,6,7],[0,5,7,4,3,1,6,2],[0,6,0,6,6,6,0,0],[0,7,6,2,7,2,0,6]],"subset":[0,2,6,7],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-two-sided-absorption","semantic_template":"qf-mod28-ideal-two-sided-absorption/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,7,0,4,5],[2,3,6,4,5,1,7,0],[3,6,4,7,0,2,5,1],[4,7,5,0,2,6,1,3],[5,0,1,2,6,7,3,4],[6,4,7,5,1,3,0,2],[7,5,0,1,3,4,2,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,7,2,7,0,6],[0,3,7,1,5,4,6,2],[0,4,2,5,1,3,6,7],[0,5,7,4,3,1,6,2],[0,6,0,6,6,6,0,0],[0,7,6,2,7,2,0,6]],\"n\":8,\"subset\":[0,2,6,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0065","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-two-sided-absorption；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,7,4,5,0],[2,3,6,5,0,7,4,1],[3,6,5,4,1,0,7,2],[4,7,0,1,6,3,2,5],[5,4,7,0,3,2,1,6],[6,5,4,7,2,1,0,3],[7,0,1,2,5,6,3,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,4,6,2,0,4],[0,3,4,1,2,7,6,5],[0,4,6,2,6,4,0,2],[0,5,2,7,4,1,6,3],[0,6,0,6,0,6,0,6],[0,7,4,5,2,3,6,1]],\"subset\":[0,6]}。請依 [subset,missing_left,missing_right,left_absorbing,right_absorbing] 欄序回答。","answer":{"kind":"expression","value":"([0, 6], [], [], 是, 是)","explanation_zh":"Wave4第二層核對=([0, 6], [], [], 是, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-two-sided-absorption","seed":202610282065,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,6,7,4,5,0],[2,3,6,5,0,7,4,1],[3,6,5,4,1,0,7,2],[4,7,0,1,6,3,2,5],[5,4,7,0,3,2,1,6],[6,5,4,7,2,1,0,3],[7,0,1,2,5,6,3,4]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,4,6,2,0,4],[0,3,4,1,2,7,6,5],[0,4,6,2,6,4,0,2],[0,5,2,7,4,1,6,3],[0,6,0,6,0,6,0,6],[0,7,4,5,2,3,6,1]],"subset":[0,6],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-two-sided-absorption","semantic_template":"qf-mod28-ideal-two-sided-absorption/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,7,4,5,0],[2,3,6,5,0,7,4,1],[3,6,5,4,1,0,7,2],[4,7,0,1,6,3,2,5],[5,4,7,0,3,2,1,6],[6,5,4,7,2,1,0,3],[7,0,1,2,5,6,3,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,4,6,2,0,4],[0,3,4,1,2,7,6,5],[0,4,6,2,6,4,0,2],[0,5,2,7,4,1,6,3],[0,6,0,6,0,6,0,6],[0,7,4,5,2,3,6,1]],\"n\":8,\"subset\":[0,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0066","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-two-sided-absorption；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,0,7,5,4],[2,3,6,5,1,4,7,0],[3,6,5,7,2,0,4,1],[4,0,1,2,7,6,3,5],[5,7,4,0,6,2,1,3],[6,5,7,4,3,1,0,2],[7,4,0,1,5,3,2,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,7,7,2,0,6],[0,3,7,1,5,4,6,2],[0,4,7,5,1,3,6,2],[0,5,2,4,3,1,6,7],[0,6,0,6,6,6,0,0],[0,7,6,2,2,7,0,6]],\"subset\":[0,2,6,7]}。請依 [subset,missing_left,missing_right,left_absorbing,right_absorbing] 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 6, 7], [], [], 是, 是)","explanation_zh":"Wave4第二層核對=([0, 2, 6, 7], [], [], 是, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-two-sided-absorption","seed":202610282066,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,6,0,7,5,4],[2,3,6,5,1,4,7,0],[3,6,5,7,2,0,4,1],[4,0,1,2,7,6,3,5],[5,7,4,0,6,2,1,3],[6,5,7,4,3,1,0,2],[7,4,0,1,5,3,2,6]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,7,7,2,0,6],[0,3,7,1,5,4,6,2],[0,4,7,5,1,3,6,2],[0,5,2,4,3,1,6,7],[0,6,0,6,6,6,0,0],[0,7,6,2,2,7,0,6]],"subset":[0,2,6,7],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-two-sided-absorption","semantic_template":"qf-mod28-ideal-two-sided-absorption/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,0,7,5,4],[2,3,6,5,1,4,7,0],[3,6,5,7,2,0,4,1],[4,0,1,2,7,6,3,5],[5,7,4,0,6,2,1,3],[6,5,7,4,3,1,0,2],[7,4,0,1,5,3,2,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,7,7,2,0,6],[0,3,7,1,5,4,6,2],[0,4,7,5,1,3,6,2],[0,5,2,4,3,1,6,7],[0,6,0,6,6,6,0,0],[0,7,6,2,2,7,0,6]],\"n\":8,\"subset\":[0,2,6,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0067","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-two-sided-absorption；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,5,0,7,4],[2,3,6,7,0,1,4,5],[3,6,7,4,1,2,5,0],[4,5,0,1,6,7,2,3],[5,0,1,2,7,4,3,6],[6,7,4,5,2,3,0,1],[7,4,5,0,3,6,1,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,4,6,4,0,2],[0,3,4,1,2,7,6,5],[0,4,6,2,6,2,0,4],[0,5,4,7,2,1,6,3],[0,6,0,6,0,6,0,6],[0,7,2,5,4,3,6,1]],\"subset\":[0,6]}。請依 [subset,missing_left,missing_right,left_absorbing,right_absorbing] 欄序回答。","answer":{"kind":"expression","value":"([0, 6], [], [], 是, 是)","explanation_zh":"Wave4第二層核對=([0, 6], [], [], 是, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-two-sided-absorption","seed":202610282067,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,6,5,0,7,4],[2,3,6,7,0,1,4,5],[3,6,7,4,1,2,5,0],[4,5,0,1,6,7,2,3],[5,0,1,2,7,4,3,6],[6,7,4,5,2,3,0,1],[7,4,5,0,3,6,1,2]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,4,6,4,0,2],[0,3,4,1,2,7,6,5],[0,4,6,2,6,2,0,4],[0,5,4,7,2,1,6,3],[0,6,0,6,0,6,0,6],[0,7,2,5,4,3,6,1]],"subset":[0,6],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-two-sided-absorption","semantic_template":"qf-mod28-ideal-two-sided-absorption/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,5,0,7,4],[2,3,6,7,0,1,4,5],[3,6,7,4,1,2,5,0],[4,5,0,1,6,7,2,3],[5,0,1,2,7,4,3,6],[6,7,4,5,2,3,0,1],[7,4,5,0,3,6,1,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,4,6,4,0,2],[0,3,4,1,2,7,6,5],[0,4,6,2,6,2,0,4],[0,5,4,7,2,1,6,3],[0,6,0,6,0,6,0,6],[0,7,2,5,4,3,6,1]],\"n\":8,\"subset\":[0,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0068","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-two-sided-absorption；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,0,4,7,5],[2,3,6,7,1,0,5,4],[3,6,7,5,2,1,4,0],[4,0,1,2,5,7,3,6],[5,4,0,1,7,6,2,3],[6,7,5,4,3,2,0,1],[7,5,4,0,6,3,1,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,5,5,6,0,2],[0,3,5,1,7,2,6,4],[0,4,5,7,1,2,6,3],[0,5,6,2,2,6,0,5],[0,6,0,6,6,0,0,6],[0,7,2,4,3,5,6,1]],\"subset\":[0]}。請依 [subset,missing_left,missing_right,left_absorbing,right_absorbing] 欄序回答。","answer":{"kind":"expression","value":"([0], [], [], 是, 是)","explanation_zh":"Wave4第二層核對=([0], [], [], 是, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-two-sided-absorption","seed":202610282068,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,6,0,4,7,5],[2,3,6,7,1,0,5,4],[3,6,7,5,2,1,4,0],[4,0,1,2,5,7,3,6],[5,4,0,1,7,6,2,3],[6,7,5,4,3,2,0,1],[7,5,4,0,6,3,1,2]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,5,5,6,0,2],[0,3,5,1,7,2,6,4],[0,4,5,7,1,2,6,3],[0,5,6,2,2,6,0,5],[0,6,0,6,6,0,0,6],[0,7,2,4,3,5,6,1]],"subset":[0],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-two-sided-absorption","semantic_template":"qf-mod28-ideal-two-sided-absorption/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,0,4,7,5],[2,3,6,7,1,0,5,4],[3,6,7,5,2,1,4,0],[4,0,1,2,5,7,3,6],[5,4,0,1,7,6,2,3],[6,7,5,4,3,2,0,1],[7,5,4,0,6,3,1,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,5,5,6,0,2],[0,3,5,1,7,2,6,4],[0,4,5,7,1,2,6,3],[0,5,6,2,2,6,0,5],[0,6,0,6,6,0,0,6],[0,7,2,4,3,5,6,1]],\"n\":8,\"subset\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0069","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-two-sided-absorption；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,5,6,0,4],[2,3,7,4,6,0,1,5],[3,7,4,5,0,1,2,6],[4,5,6,0,2,3,7,1],[5,6,0,1,3,7,4,2],[6,0,1,2,7,4,5,3],[7,4,5,6,1,2,3,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,5,2,7,5,0],[0,3,5,1,6,2,4,7],[0,4,2,6,1,5,3,7],[0,5,7,2,5,7,2,0],[0,6,5,4,3,2,1,7],[0,7,0,7,7,0,7,0]],\"subset\":[0,2,5,7]}。請依 [subset,missing_left,missing_right,left_absorbing,right_absorbing] 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 5, 7], [], [], 是, 是)","explanation_zh":"Wave4第二層核對=([0, 2, 5, 7], [], [], 是, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-two-sided-absorption","seed":202610282069,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,7,5,6,0,4],[2,3,7,4,6,0,1,5],[3,7,4,5,0,1,2,6],[4,5,6,0,2,3,7,1],[5,6,0,1,3,7,4,2],[6,0,1,2,7,4,5,3],[7,4,5,6,1,2,3,0]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,5,2,7,5,0],[0,3,5,1,6,2,4,7],[0,4,2,6,1,5,3,7],[0,5,7,2,5,7,2,0],[0,6,5,4,3,2,1,7],[0,7,0,7,7,0,7,0]],"subset":[0,2,5,7],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-two-sided-absorption","semantic_template":"qf-mod28-ideal-two-sided-absorption/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,5,6,0,4],[2,3,7,4,6,0,1,5],[3,7,4,5,0,1,2,6],[4,5,6,0,2,3,7,1],[5,6,0,1,3,7,4,2],[6,0,1,2,7,4,5,3],[7,4,5,6,1,2,3,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,5,2,7,5,0],[0,3,5,1,6,2,4,7],[0,4,2,6,1,5,3,7],[0,5,7,2,5,7,2,0],[0,6,5,4,3,2,1,7],[0,7,0,7,7,0,7,0]],\"n\":8,\"subset\":[0,2,5,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0070","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-two-sided-absorption；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,6,0,5,4],[2,3,7,4,5,1,0,6],[3,7,4,6,0,2,1,5],[4,6,5,0,2,7,3,1],[5,0,1,2,7,6,4,3],[6,5,0,1,3,4,7,2],[7,4,6,5,1,3,2,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,6,2,6,7,0],[0,3,6,1,5,4,2,7],[0,4,2,5,1,3,6,7],[0,5,6,4,3,1,2,7],[0,6,7,2,6,2,7,0],[0,7,0,7,7,7,0,0]],\"subset\":[0,7]}。請依 [subset,missing_left,missing_right,left_absorbing,right_absorbing] 欄序回答。","answer":{"kind":"expression","value":"([0, 7], [], [], 是, 是)","explanation_zh":"Wave4第二層核對=([0, 7], [], [], 是, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-two-sided-absorption","seed":202610282070,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,7,6,0,5,4],[2,3,7,4,5,1,0,6],[3,7,4,6,0,2,1,5],[4,6,5,0,2,7,3,1],[5,0,1,2,7,6,4,3],[6,5,0,1,3,4,7,2],[7,4,6,5,1,3,2,0]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,6,2,6,7,0],[0,3,6,1,5,4,2,7],[0,4,2,5,1,3,6,7],[0,5,6,4,3,1,2,7],[0,6,7,2,6,2,7,0],[0,7,0,7,7,7,0,0]],"subset":[0,7],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-two-sided-absorption","semantic_template":"qf-mod28-ideal-two-sided-absorption/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,6,0,5,4],[2,3,7,4,5,1,0,6],[3,7,4,6,0,2,1,5],[4,6,5,0,2,7,3,1],[5,0,1,2,7,6,4,3],[6,5,0,1,3,4,7,2],[7,4,6,5,1,3,2,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,6,2,6,7,0],[0,3,6,1,5,4,2,7],[0,4,2,5,1,3,6,7],[0,5,6,4,3,1,2,7],[0,6,7,2,6,2,7,0],[0,7,0,7,7,7,0,0]],\"n\":8,\"subset\":[0,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0071","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-two-sided-absorption；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,6,4,0,5],[2,3,7,5,0,6,1,4],[3,7,5,4,1,0,2,6],[4,6,0,1,7,3,5,2],[5,4,6,0,3,2,7,1],[6,0,1,2,5,7,4,3],[7,5,4,6,2,1,3,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,4,7,2,4,0],[0,3,4,1,2,6,5,7],[0,4,7,2,7,4,2,0],[0,5,2,6,4,1,3,7],[0,6,4,5,2,3,1,7],[0,7,0,7,0,7,7,0]],\"subset\":[0,2,4,7]}。請依 [subset,missing_left,missing_right,left_absorbing,right_absorbing] 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 4, 7], [], [], 是, 是)","explanation_zh":"Wave4第二層核對=([0, 2, 4, 7], [], [], 是, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-two-sided-absorption","seed":202610282071,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,7,6,4,0,5],[2,3,7,5,0,6,1,4],[3,7,5,4,1,0,2,6],[4,6,0,1,7,3,5,2],[5,4,6,0,3,2,7,1],[6,0,1,2,5,7,4,3],[7,5,4,6,2,1,3,0]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,4,7,2,4,0],[0,3,4,1,2,6,5,7],[0,4,7,2,7,4,2,0],[0,5,2,6,4,1,3,7],[0,6,4,5,2,3,1,7],[0,7,0,7,0,7,7,0]],"subset":[0,2,4,7],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-two-sided-absorption","semantic_template":"qf-mod28-ideal-two-sided-absorption/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,6,4,0,5],[2,3,7,5,0,6,1,4],[3,7,5,4,1,0,2,6],[4,6,0,1,7,3,5,2],[5,4,6,0,3,2,7,1],[6,0,1,2,5,7,4,3],[7,5,4,6,2,1,3,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,4,7,2,4,0],[0,3,4,1,2,6,5,7],[0,4,7,2,7,4,2,0],[0,5,2,6,4,1,3,7],[0,6,4,5,2,3,1,7],[0,7,0,7,0,7,7,0]],\"n\":8,\"subset\":[0,2,4,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0072","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-two-sided-absorption；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,0,6,4,5],[2,3,7,5,1,4,0,6],[3,7,5,6,2,0,1,4],[4,0,1,2,6,7,5,3],[5,6,4,0,7,2,3,1],[6,4,0,1,5,3,7,2],[7,5,6,4,3,1,2,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,6,6,2,7,0],[0,3,6,1,5,4,2,7],[0,4,6,5,1,3,2,7],[0,5,2,4,3,1,6,7],[0,6,7,2,2,6,7,0],[0,7,0,7,7,7,0,0]],\"subset\":[0,7]}。請依 [subset,missing_left,missing_right,left_absorbing,right_absorbing] 欄序回答。","answer":{"kind":"expression","value":"([0, 7], [], [], 是, 是)","explanation_zh":"Wave4第二層核對=([0, 7], [], [], 是, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-two-sided-absorption","seed":202610282072,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,7,0,6,4,5],[2,3,7,5,1,4,0,6],[3,7,5,6,2,0,1,4],[4,0,1,2,6,7,5,3],[5,6,4,0,7,2,3,1],[6,4,0,1,5,3,7,2],[7,5,6,4,3,1,2,0]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,6,6,2,7,0],[0,3,6,1,5,4,2,7],[0,4,6,5,1,3,2,7],[0,5,2,4,3,1,6,7],[0,6,7,2,2,6,7,0],[0,7,0,7,7,7,0,0]],"subset":[0,7],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-two-sided-absorption","semantic_template":"qf-mod28-ideal-two-sided-absorption/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,0,6,4,5],[2,3,7,5,1,4,0,6],[3,7,5,6,2,0,1,4],[4,0,1,2,6,7,5,3],[5,6,4,0,7,2,3,1],[6,4,0,1,5,3,7,2],[7,5,6,4,3,1,2,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,6,6,2,7,0],[0,3,6,1,5,4,2,7],[0,4,6,5,1,3,2,7],[0,5,2,4,3,1,6,7],[0,6,7,2,2,6,7,0],[0,7,0,7,7,7,0,0]],\"n\":8,\"subset\":[0,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0073","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-two-sided-absorption；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,5,0,4,6],[2,3,7,6,0,1,5,4],[3,7,6,4,1,2,0,5],[4,5,0,1,7,6,3,2],[5,0,1,2,6,4,7,3],[6,4,5,0,3,7,2,1],[7,6,4,5,2,3,1,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,4,7,4,2,0],[0,3,4,1,2,6,5,7],[0,4,7,2,7,2,4,0],[0,5,4,6,2,1,3,7],[0,6,2,5,4,3,1,7],[0,7,0,7,0,7,7,0]],\"subset\":[0]}。請依 [subset,missing_left,missing_right,left_absorbing,right_absorbing] 欄序回答。","answer":{"kind":"expression","value":"([0], [], [], 是, 是)","explanation_zh":"Wave4第二層核對=([0], [], [], 是, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-two-sided-absorption","seed":202610282073,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,7,5,0,4,6],[2,3,7,6,0,1,5,4],[3,7,6,4,1,2,0,5],[4,5,0,1,7,6,3,2],[5,0,1,2,6,4,7,3],[6,4,5,0,3,7,2,1],[7,6,4,5,2,3,1,0]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,4,7,4,2,0],[0,3,4,1,2,6,5,7],[0,4,7,2,7,2,4,0],[0,5,4,6,2,1,3,7],[0,6,2,5,4,3,1,7],[0,7,0,7,0,7,7,0]],"subset":[0],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-two-sided-absorption","semantic_template":"qf-mod28-ideal-two-sided-absorption/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,5,0,4,6],[2,3,7,6,0,1,5,4],[3,7,6,4,1,2,0,5],[4,5,0,1,7,6,3,2],[5,0,1,2,6,4,7,3],[6,4,5,0,3,7,2,1],[7,6,4,5,2,3,1,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,4,7,4,2,0],[0,3,4,1,2,6,5,7],[0,4,7,2,7,2,4,0],[0,5,4,6,2,1,3,7],[0,6,2,5,4,3,1,7],[0,7,0,7,0,7,7,0]],\"n\":8,\"subset\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0074","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-two-sided-absorption；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,0,4,5,6],[2,3,7,6,1,0,4,5],[3,7,6,5,2,1,0,4],[4,0,1,2,5,6,7,3],[5,4,0,1,6,7,3,2],[6,5,4,0,7,3,2,1],[7,6,5,4,3,2,1,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,5,5,7,2,0],[0,3,5,1,6,2,4,7],[0,4,5,6,1,2,3,7],[0,5,7,2,2,7,5,0],[0,6,2,4,3,5,1,7],[0,7,0,7,7,0,7,0]],\"subset\":[0,2,5,7]}。請依 [subset,missing_left,missing_right,left_absorbing,right_absorbing] 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 5, 7], [], [], 是, 是)","explanation_zh":"Wave4第二層核對=([0, 2, 5, 7], [], [], 是, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-two-sided-absorption","seed":202610282074,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,7,0,4,5,6],[2,3,7,6,1,0,4,5],[3,7,6,5,2,1,0,4],[4,0,1,2,5,6,7,3],[5,4,0,1,6,7,3,2],[6,5,4,0,7,3,2,1],[7,6,5,4,3,2,1,0]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,5,5,7,2,0],[0,3,5,1,6,2,4,7],[0,4,5,6,1,2,3,7],[0,5,7,2,2,7,5,0],[0,6,2,4,3,5,1,7],[0,7,0,7,7,0,7,0]],"subset":[0,2,5,7],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-two-sided-absorption","semantic_template":"qf-mod28-ideal-two-sided-absorption/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,0,4,5,6],[2,3,7,6,1,0,4,5],[3,7,6,5,2,1,0,4],[4,0,1,2,5,6,7,3],[5,4,0,1,6,7,3,2],[6,5,4,0,7,3,2,1],[7,6,5,4,3,2,1,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,5,5,7,2,0],[0,3,5,1,6,2,4,7],[0,4,5,6,1,2,3,7],[0,5,7,2,2,7,5,0],[0,6,2,4,3,5,1,7],[0,7,0,7,7,0,7,0]],\"n\":8,\"subset\":[0,2,5,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0075","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-two-sided-absorption；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,4,5,3,6,7,0],[2,4,3,6,5,7,0,1],[3,5,6,0,7,1,2,4],[4,3,5,7,6,0,1,2],[5,6,7,1,0,2,4,3],[6,7,0,2,1,4,3,5],[7,0,1,4,2,3,5,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,3,0,6,2,3,6],[0,3,0,0,3,3,0,3],[0,4,6,3,1,7,2,5],[0,5,2,3,7,1,6,4],[0,6,3,0,2,6,3,2],[0,7,6,3,5,4,2,1]],\"subset\":[0,3]}。請依 [subset,missing_left,missing_right,left_absorbing,right_absorbing] 欄序回答。","answer":{"kind":"expression","value":"([0, 3], [], [], 是, 是)","explanation_zh":"Wave4第二層核對=([0, 3], [], [], 是, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-two-sided-absorption","seed":202610282075,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,4,5,3,6,7,0],[2,4,3,6,5,7,0,1],[3,5,6,0,7,1,2,4],[4,3,5,7,6,0,1,2],[5,6,7,1,0,2,4,3],[6,7,0,2,1,4,3,5],[7,0,1,4,2,3,5,6]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,3,0,6,2,3,6],[0,3,0,0,3,3,0,3],[0,4,6,3,1,7,2,5],[0,5,2,3,7,1,6,4],[0,6,3,0,2,6,3,2],[0,7,6,3,5,4,2,1]],"subset":[0,3],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-two-sided-absorption","semantic_template":"qf-mod28-ideal-two-sided-absorption/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,4,5,3,6,7,0],[2,4,3,6,5,7,0,1],[3,5,6,0,7,1,2,4],[4,3,5,7,6,0,1,2],[5,6,7,1,0,2,4,3],[6,7,0,2,1,4,3,5],[7,0,1,4,2,3,5,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,3,0,6,2,3,6],[0,3,0,0,3,3,0,3],[0,4,6,3,1,7,2,5],[0,5,2,3,7,1,6,4],[0,6,3,0,2,6,3,2],[0,7,6,3,5,4,2,1]],\"n\":8,\"subset\":[0,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0076","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-trivial-boundary；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,6,0,2,4,6],[0,3,6,1,4,7,2,5],[0,4,0,4,0,4,0,4],[0,5,2,7,4,1,6,3],[0,6,4,2,0,6,4,2],[0,7,6,5,4,3,2,1]],\"subset\":[0]}。請依 [subset,zero_ideal,whole_ring,qualified,boundary] 欄序回答。","answer":{"kind":"expression","value":"([0], 是, 否, 是, zero_and_whole_are_always_two_sided_ideals)","explanation_zh":"Wave4第二層核對=([0], 是, 否, 是, zero_and_whole_are_always_two_sided_ideals)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-trivial-boundary","seed":202610282076,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,6,0,2,4,6],[0,3,6,1,4,7,2,5],[0,4,0,4,0,4,0,4],[0,5,2,7,4,1,6,3],[0,6,4,2,0,6,4,2],[0,7,6,5,4,3,2,1]],"subset":[0],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-trivial-boundary","semantic_template":"qf-mod28-ideal-trivial-boundary/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,6,0,2,4,6],[0,3,6,1,4,7,2,5],[0,4,0,4,0,4,0,4],[0,5,2,7,4,1,6,3],[0,6,4,2,0,6,4,2],[0,7,6,5,4,3,2,1]],\"n\":8,\"subset\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0077","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-trivial-boundary；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,7,0,6],[2,3,4,5,7,6,1,0],[3,4,5,7,6,0,2,1],[4,5,7,6,0,1,3,2],[5,7,6,0,1,2,4,3],[6,0,1,2,3,4,7,5],[7,6,0,1,2,3,5,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,7,0,2,7,4],[0,3,7,1,4,6,5,2],[0,4,0,4,0,4,4,0],[0,5,2,6,4,1,3,7],[0,6,7,5,4,3,1,2],[0,7,4,2,0,7,2,4]],\"subset\":[0,1,2,3,4,5,6,7]}。請依 [subset,zero_ideal,whole_ring,qualified,boundary] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7], 否, 是, 是, zero_and_whole_are_always_two_sided_ideals)","explanation_zh":"Wave4第二層核對=([0, 1, 2, 3, 4, 5, 6, 7], 否, 是, 是, zero_and_whole_are_always_two_sided_ideals)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-trivial-boundary","seed":202610282077,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,7,0,6],[2,3,4,5,7,6,1,0],[3,4,5,7,6,0,2,1],[4,5,7,6,0,1,3,2],[5,7,6,0,1,2,4,3],[6,0,1,2,3,4,7,5],[7,6,0,1,2,3,5,4]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,7,0,2,7,4],[0,3,7,1,4,6,5,2],[0,4,0,4,0,4,4,0],[0,5,2,6,4,1,3,7],[0,6,7,5,4,3,1,2],[0,7,4,2,0,7,2,4]],"subset":[0,1,2,3,4,5,6,7],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-trivial-boundary","semantic_template":"qf-mod28-ideal-trivial-boundary/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,7,0,6],[2,3,4,5,7,6,1,0],[3,4,5,7,6,0,2,1],[4,5,7,6,0,1,3,2],[5,7,6,0,1,2,4,3],[6,0,1,2,3,4,7,5],[7,6,0,1,2,3,5,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,7,0,2,7,4],[0,3,7,1,4,6,5,2],[0,4,0,4,0,4,4,0],[0,5,2,6,4,1,3,7],[0,6,7,5,4,3,1,2],[0,7,4,2,0,7,2,4]],\"n\":8,\"subset\":[0,1,2,3,4,5,6,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0078","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-trivial-boundary；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,6,7,5,0],[2,3,4,6,5,0,7,1],[3,4,6,5,7,1,0,2],[4,6,5,7,0,2,1,3],[5,7,0,1,2,4,3,6],[6,5,7,0,1,3,2,4],[7,0,1,2,3,6,4,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,5,0,4,2,5],[0,3,5,1,4,2,7,6],[0,4,0,4,0,0,4,4],[0,5,4,2,0,4,5,2],[0,6,2,7,4,5,1,3],[0,7,5,6,4,2,3,1]],\"subset\":[0]}。請依 [subset,zero_ideal,whole_ring,qualified,boundary] 欄序回答。","answer":{"kind":"expression","value":"([0], 是, 否, 是, zero_and_whole_are_always_two_sided_ideals)","explanation_zh":"Wave4第二層核對=([0], 是, 否, 是, zero_and_whole_are_always_two_sided_ideals)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-trivial-boundary","seed":202610282078,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,4,6,7,5,0],[2,3,4,6,5,0,7,1],[3,4,6,5,7,1,0,2],[4,6,5,7,0,2,1,3],[5,7,0,1,2,4,3,6],[6,5,7,0,1,3,2,4],[7,0,1,2,3,6,4,5]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,5,0,4,2,5],[0,3,5,1,4,2,7,6],[0,4,0,4,0,0,4,4],[0,5,4,2,0,4,5,2],[0,6,2,7,4,5,1,3],[0,7,5,6,4,2,3,1]],"subset":[0],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-trivial-boundary","semantic_template":"qf-mod28-ideal-trivial-boundary/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,6,7,5,0],[2,3,4,6,5,0,7,1],[3,4,6,5,7,1,0,2],[4,6,5,7,0,2,1,3],[5,7,0,1,2,4,3,6],[6,5,7,0,1,3,2,4],[7,0,1,2,3,6,4,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,5,0,4,2,5],[0,3,5,1,4,2,7,6],[0,4,0,4,0,0,4,4],[0,5,4,2,0,4,5,2],[0,6,2,7,4,5,1,3],[0,7,5,6,4,2,3,1]],\"n\":8,\"subset\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0079","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-trivial-boundary；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,6,0,7,5],[2,3,4,6,7,1,5,0],[3,4,6,7,5,2,0,1],[4,6,7,5,0,3,1,2],[5,0,1,2,3,7,4,6],[6,7,5,0,1,4,2,3],[7,5,0,1,2,6,3,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,7,0,7,2,4],[0,3,7,1,4,6,5,2],[0,4,0,4,0,4,4,0],[0,5,7,6,4,1,3,2],[0,6,2,5,4,3,1,7],[0,7,4,2,0,2,7,4]],\"subset\":[0,1,2,3,4,5,6,7]}。請依 [subset,zero_ideal,whole_ring,qualified,boundary] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7], 否, 是, 是, zero_and_whole_are_always_two_sided_ideals)","explanation_zh":"Wave4第二層核對=([0, 1, 2, 3, 4, 5, 6, 7], 否, 是, 是, zero_and_whole_are_always_two_sided_ideals)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-trivial-boundary","seed":202610282079,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,4,6,0,7,5],[2,3,4,6,7,1,5,0],[3,4,6,7,5,2,0,1],[4,6,7,5,0,3,1,2],[5,0,1,2,3,7,4,6],[6,7,5,0,1,4,2,3],[7,5,0,1,2,6,3,4]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,7,0,7,2,4],[0,3,7,1,4,6,5,2],[0,4,0,4,0,4,4,0],[0,5,7,6,4,1,3,2],[0,6,2,5,4,3,1,7],[0,7,4,2,0,2,7,4]],"subset":[0,1,2,3,4,5,6,7],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-trivial-boundary","semantic_template":"qf-mod28-ideal-trivial-boundary/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,6,0,7,5],[2,3,4,6,7,1,5,0],[3,4,6,7,5,2,0,1],[4,6,7,5,0,3,1,2],[5,0,1,2,3,7,4,6],[6,7,5,0,1,4,2,3],[7,5,0,1,2,6,3,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,7,0,7,2,4],[0,3,7,1,4,6,5,2],[0,4,0,4,0,4,4,0],[0,5,7,6,4,1,3,2],[0,6,2,5,4,3,1,7],[0,7,4,2,0,2,7,4]],\"n\":8,\"subset\":[0,1,2,3,4,5,6,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0080","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-trivial-boundary；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,7,6,0,5],[2,3,4,7,5,0,1,6],[3,4,7,5,6,1,2,0],[4,7,5,6,0,2,3,1],[5,6,0,1,2,4,7,3],[6,0,1,2,3,7,5,4],[7,5,6,0,1,3,4,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,5,0,4,5,2],[0,3,5,1,4,2,7,6],[0,4,0,4,0,0,4,4],[0,5,4,2,0,4,2,5],[0,6,5,7,4,2,1,3],[0,7,2,6,4,5,3,1]],\"subset\":[0]}。請依 [subset,zero_ideal,whole_ring,qualified,boundary] 欄序回答。","answer":{"kind":"expression","value":"([0], 是, 否, 是, zero_and_whole_are_always_two_sided_ideals)","explanation_zh":"Wave4第二層核對=([0], 是, 否, 是, zero_and_whole_are_always_two_sided_ideals)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-trivial-boundary","seed":202610282080,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,4,7,6,0,5],[2,3,4,7,5,0,1,6],[3,4,7,5,6,1,2,0],[4,7,5,6,0,2,3,1],[5,6,0,1,2,4,7,3],[6,0,1,2,3,7,5,4],[7,5,6,0,1,3,4,2]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,5,0,4,5,2],[0,3,5,1,4,2,7,6],[0,4,0,4,0,0,4,4],[0,5,4,2,0,4,2,5],[0,6,5,7,4,2,1,3],[0,7,2,6,4,5,3,1]],"subset":[0],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-trivial-boundary","semantic_template":"qf-mod28-ideal-trivial-boundary/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,7,6,0,5],[2,3,4,7,5,0,1,6],[3,4,7,5,6,1,2,0],[4,7,5,6,0,2,3,1],[5,6,0,1,2,4,7,3],[6,0,1,2,3,7,5,4],[7,5,6,0,1,3,4,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,5,0,4,5,2],[0,3,5,1,4,2,7,6],[0,4,0,4,0,0,4,4],[0,5,4,2,0,4,2,5],[0,6,5,7,4,2,1,3],[0,7,2,6,4,5,3,1]],\"n\":8,\"subset\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0081","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-trivial-boundary；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,7,0,5,6],[2,3,4,7,6,1,0,5],[3,4,7,6,5,2,1,0],[4,7,6,5,0,3,2,1],[5,0,1,2,3,6,7,4],[6,5,0,1,2,7,4,3],[7,6,5,0,1,4,3,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,6,0,6,4,2],[0,3,6,1,4,7,2,5],[0,4,0,4,0,4,0,4],[0,5,6,7,4,1,2,3],[0,6,4,2,0,2,4,6],[0,7,2,5,4,3,6,1]],\"subset\":[0,1,2,3,4,5,6,7]}。請依 [subset,zero_ideal,whole_ring,qualified,boundary] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7], 否, 是, 是, zero_and_whole_are_always_two_sided_ideals)","explanation_zh":"Wave4第二層核對=([0, 1, 2, 3, 4, 5, 6, 7], 否, 是, 是, zero_and_whole_are_always_two_sided_ideals)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-trivial-boundary","seed":202610282081,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,4,7,0,5,6],[2,3,4,7,6,1,0,5],[3,4,7,6,5,2,1,0],[4,7,6,5,0,3,2,1],[5,0,1,2,3,6,7,4],[6,5,0,1,2,7,4,3],[7,6,5,0,1,4,3,2]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,6,0,6,4,2],[0,3,6,1,4,7,2,5],[0,4,0,4,0,4,0,4],[0,5,6,7,4,1,2,3],[0,6,4,2,0,2,4,6],[0,7,2,5,4,3,6,1]],"subset":[0,1,2,3,4,5,6,7],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-trivial-boundary","semantic_template":"qf-mod28-ideal-trivial-boundary/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,7,0,5,6],[2,3,4,7,6,1,0,5],[3,4,7,6,5,2,1,0],[4,7,6,5,0,3,2,1],[5,0,1,2,3,6,7,4],[6,5,0,1,2,7,4,3],[7,6,5,0,1,4,3,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,6,0,6,4,2],[0,3,6,1,4,7,2,5],[0,4,0,4,0,4,0,4],[0,5,6,7,4,1,2,3],[0,6,4,2,0,2,4,6],[0,7,2,5,4,3,6,1]],\"n\":8,\"subset\":[0,1,2,3,4,5,6,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0082","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-trivial-boundary；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,6,4,7,0],[2,3,5,4,7,6,0,1],[3,5,4,6,0,7,1,2],[4,6,7,0,2,1,3,5],[5,4,6,7,1,0,2,3],[6,7,0,1,3,2,5,4],[7,0,1,2,5,3,4,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,6,2,0,5,6],[0,3,6,1,7,5,2,4],[0,4,2,7,1,5,6,3],[0,5,0,5,5,0,0,5],[0,6,5,2,6,0,5,2],[0,7,6,4,3,5,2,1]],\"subset\":[0]}。請依 [subset,zero_ideal,whole_ring,qualified,boundary] 欄序回答。","answer":{"kind":"expression","value":"([0], 是, 否, 是, zero_and_whole_are_always_two_sided_ideals)","explanation_zh":"Wave4第二層核對=([0], 是, 否, 是, zero_and_whole_are_always_two_sided_ideals)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-trivial-boundary","seed":202610282082,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,5,6,4,7,0],[2,3,5,4,7,6,0,1],[3,5,4,6,0,7,1,2],[4,6,7,0,2,1,3,5],[5,4,6,7,1,0,2,3],[6,7,0,1,3,2,5,4],[7,0,1,2,5,3,4,6]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,6,2,0,5,6],[0,3,6,1,7,5,2,4],[0,4,2,7,1,5,6,3],[0,5,0,5,5,0,0,5],[0,6,5,2,6,0,5,2],[0,7,6,4,3,5,2,1]],"subset":[0],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-trivial-boundary","semantic_template":"qf-mod28-ideal-trivial-boundary/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,6,4,7,0],[2,3,5,4,7,6,0,1],[3,5,4,6,0,7,1,2],[4,6,7,0,2,1,3,5],[5,4,6,7,1,0,2,3],[6,7,0,1,3,2,5,4],[7,0,1,2,5,3,4,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,6,2,0,5,6],[0,3,6,1,7,5,2,4],[0,4,2,7,1,5,6,3],[0,5,0,5,5,0,0,5],[0,6,5,2,6,0,5,2],[0,7,6,4,3,5,2,1]],\"n\":8,\"subset\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0083","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-trivial-boundary；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,7,4,0,6],[2,3,5,4,6,7,1,0],[3,5,4,7,0,6,2,1],[4,7,6,0,2,1,5,3],[5,4,7,6,1,0,3,2],[6,0,1,2,5,3,7,4],[7,6,0,1,3,2,4,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,7,2,0,7,5],[0,3,7,1,6,5,4,2],[0,4,2,6,1,5,3,7],[0,5,0,5,5,0,5,0],[0,6,7,4,3,5,1,2],[0,7,5,2,7,0,2,5]],\"subset\":[0,1,2,3,4,5,6,7]}。請依 [subset,zero_ideal,whole_ring,qualified,boundary] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7], 否, 是, 是, zero_and_whole_are_always_two_sided_ideals)","explanation_zh":"Wave4第二層核對=([0, 1, 2, 3, 4, 5, 6, 7], 否, 是, 是, zero_and_whole_are_always_two_sided_ideals)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-trivial-boundary","seed":202610282083,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,5,7,4,0,6],[2,3,5,4,6,7,1,0],[3,5,4,7,0,6,2,1],[4,7,6,0,2,1,5,3],[5,4,7,6,1,0,3,2],[6,0,1,2,5,3,7,4],[7,6,0,1,3,2,4,5]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,7,2,0,7,5],[0,3,7,1,6,5,4,2],[0,4,2,6,1,5,3,7],[0,5,0,5,5,0,5,0],[0,6,7,4,3,5,1,2],[0,7,5,2,7,0,2,5]],"subset":[0,1,2,3,4,5,6,7],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-trivial-boundary","semantic_template":"qf-mod28-ideal-trivial-boundary/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,7,4,0,6],[2,3,5,4,6,7,1,0],[3,5,4,7,0,6,2,1],[4,7,6,0,2,1,5,3],[5,4,7,6,1,0,3,2],[6,0,1,2,5,3,7,4],[7,6,0,1,3,2,4,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,7,2,0,7,5],[0,3,7,1,6,5,4,2],[0,4,2,6,1,5,3,7],[0,5,0,5,5,0,5,0],[0,6,7,4,3,5,1,2],[0,7,5,2,7,0,2,5]],\"n\":8,\"subset\":[0,1,2,3,4,5,6,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0084","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-trivial-boundary；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,7,6,4,0],[2,3,5,6,0,4,7,1],[3,5,6,4,1,7,0,2],[4,7,0,1,5,2,3,6],[5,6,4,7,2,0,1,3],[6,4,7,0,3,1,2,5],[7,0,1,2,6,3,5,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,4,5,0,2,4],[0,3,4,1,2,5,7,6],[0,4,5,2,5,0,4,2],[0,5,0,5,0,0,5,5],[0,6,2,7,4,5,1,3],[0,7,4,6,2,5,3,1]],\"subset\":[0]}。請依 [subset,zero_ideal,whole_ring,qualified,boundary] 欄序回答。","answer":{"kind":"expression","value":"([0], 是, 否, 是, zero_and_whole_are_always_two_sided_ideals)","explanation_zh":"Wave4第二層核對=([0], 是, 否, 是, zero_and_whole_are_always_two_sided_ideals)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-trivial-boundary","seed":202610282084,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,5,7,6,4,0],[2,3,5,6,0,4,7,1],[3,5,6,4,1,7,0,2],[4,7,0,1,5,2,3,6],[5,6,4,7,2,0,1,3],[6,4,7,0,3,1,2,5],[7,0,1,2,6,3,5,4]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,4,5,0,2,4],[0,3,4,1,2,5,7,6],[0,4,5,2,5,0,4,2],[0,5,0,5,0,0,5,5],[0,6,2,7,4,5,1,3],[0,7,4,6,2,5,3,1]],"subset":[0],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-trivial-boundary","semantic_template":"qf-mod28-ideal-trivial-boundary/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,7,6,4,0],[2,3,5,6,0,4,7,1],[3,5,6,4,1,7,0,2],[4,7,0,1,5,2,3,6],[5,6,4,7,2,0,1,3],[6,4,7,0,3,1,2,5],[7,0,1,2,6,3,5,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,4,5,0,2,4],[0,3,4,1,2,5,7,6],[0,4,5,2,5,0,4,2],[0,5,0,5,0,0,5,5],[0,6,2,7,4,5,1,3],[0,7,4,6,2,5,3,1]],\"n\":8,\"subset\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0085","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-trivial-boundary；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,0,6,7,4],[2,3,5,6,1,7,4,0],[3,5,6,7,2,4,0,1],[4,0,1,2,7,3,5,6],[5,6,7,4,3,0,1,2],[6,7,4,0,5,1,2,3],[7,4,0,1,6,2,3,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,7,7,0,2,5],[0,3,7,1,6,5,4,2],[0,4,7,6,1,5,3,2],[0,5,0,5,5,0,5,0],[0,6,2,4,3,5,1,7],[0,7,5,2,2,0,7,5]],\"subset\":[0,1,2,3,4,5,6,7]}。請依 [subset,zero_ideal,whole_ring,qualified,boundary] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7], 否, 是, 是, zero_and_whole_are_always_two_sided_ideals)","explanation_zh":"Wave4第二層核對=([0, 1, 2, 3, 4, 5, 6, 7], 否, 是, 是, zero_and_whole_are_always_two_sided_ideals)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-trivial-boundary","seed":202610282085,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,5,0,6,7,4],[2,3,5,6,1,7,4,0],[3,5,6,7,2,4,0,1],[4,0,1,2,7,3,5,6],[5,6,7,4,3,0,1,2],[6,7,4,0,5,1,2,3],[7,4,0,1,6,2,3,5]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,7,7,0,2,5],[0,3,7,1,6,5,4,2],[0,4,7,6,1,5,3,2],[0,5,0,5,5,0,5,0],[0,6,2,4,3,5,1,7],[0,7,5,2,2,0,7,5]],"subset":[0,1,2,3,4,5,6,7],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-trivial-boundary","semantic_template":"qf-mod28-ideal-trivial-boundary/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,0,6,7,4],[2,3,5,6,1,7,4,0],[3,5,6,7,2,4,0,1],[4,0,1,2,7,3,5,6],[5,6,7,4,3,0,1,2],[6,7,4,0,5,1,2,3],[7,4,0,1,6,2,3,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,7,7,0,2,5],[0,3,7,1,6,5,4,2],[0,4,7,6,1,5,3,2],[0,5,0,5,5,0,5,0],[0,6,2,4,3,5,1,7],[0,7,5,2,2,0,7,5]],\"n\":8,\"subset\":[0,1,2,3,4,5,6,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0086","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-trivial-boundary；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,6,7,0,4],[2,3,5,7,0,4,1,6],[3,5,7,4,1,6,2,0],[4,6,0,1,5,2,7,3],[5,7,4,6,2,0,3,1],[6,0,1,2,7,3,4,5],[7,4,6,0,3,1,5,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,4,5,0,4,2],[0,3,4,1,2,5,7,6],[0,4,5,2,5,0,2,4],[0,5,0,5,0,0,5,5],[0,6,4,7,2,5,1,3],[0,7,2,6,4,5,3,1]],\"subset\":[0]}。請依 [subset,zero_ideal,whole_ring,qualified,boundary] 欄序回答。","answer":{"kind":"expression","value":"([0], 是, 否, 是, zero_and_whole_are_always_two_sided_ideals)","explanation_zh":"Wave4第二層核對=([0], 是, 否, 是, zero_and_whole_are_always_two_sided_ideals)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-trivial-boundary","seed":202610282086,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,5,6,7,0,4],[2,3,5,7,0,4,1,6],[3,5,7,4,1,6,2,0],[4,6,0,1,5,2,7,3],[5,7,4,6,2,0,3,1],[6,0,1,2,7,3,4,5],[7,4,6,0,3,1,5,2]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,4,5,0,4,2],[0,3,4,1,2,5,7,6],[0,4,5,2,5,0,2,4],[0,5,0,5,0,0,5,5],[0,6,4,7,2,5,1,3],[0,7,2,6,4,5,3,1]],"subset":[0],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-trivial-boundary","semantic_template":"qf-mod28-ideal-trivial-boundary/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,6,7,0,4],[2,3,5,7,0,4,1,6],[3,5,7,4,1,6,2,0],[4,6,0,1,5,2,7,3],[5,7,4,6,2,0,3,1],[6,0,1,2,7,3,4,5],[7,4,6,0,3,1,5,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,4,5,0,4,2],[0,3,4,1,2,5,7,6],[0,4,5,2,5,0,2,4],[0,5,0,5,0,0,5,5],[0,6,4,7,2,5,1,3],[0,7,2,6,4,5,3,1]],\"n\":8,\"subset\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0087","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-trivial-boundary；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,0,7,4,6],[2,3,5,7,1,6,0,4],[3,5,7,6,2,4,1,0],[4,0,1,2,6,3,7,5],[5,7,6,4,3,0,2,1],[6,4,0,1,7,2,5,3],[7,6,4,0,5,1,3,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,6,6,0,5,2],[0,3,6,1,7,5,2,4],[0,4,6,7,1,5,2,3],[0,5,0,5,5,0,0,5],[0,6,5,2,2,0,5,6],[0,7,2,4,3,5,6,1]],\"subset\":[0,1,2,3,4,5,6,7]}。請依 [subset,zero_ideal,whole_ring,qualified,boundary] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7], 否, 是, 是, zero_and_whole_are_always_two_sided_ideals)","explanation_zh":"Wave4第二層核對=([0, 1, 2, 3, 4, 5, 6, 7], 否, 是, 是, zero_and_whole_are_always_two_sided_ideals)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-trivial-boundary","seed":202610282087,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,5,0,7,4,6],[2,3,5,7,1,6,0,4],[3,5,7,6,2,4,1,0],[4,0,1,2,6,3,7,5],[5,7,6,4,3,0,2,1],[6,4,0,1,7,2,5,3],[7,6,4,0,5,1,3,2]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,6,6,0,5,2],[0,3,6,1,7,5,2,4],[0,4,6,7,1,5,2,3],[0,5,0,5,5,0,0,5],[0,6,5,2,2,0,5,6],[0,7,2,4,3,5,6,1]],"subset":[0,1,2,3,4,5,6,7],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-trivial-boundary","semantic_template":"qf-mod28-ideal-trivial-boundary/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,0,7,4,6],[2,3,5,7,1,6,0,4],[3,5,7,6,2,4,1,0],[4,0,1,2,6,3,7,5],[5,7,6,4,3,0,2,1],[6,4,0,1,7,2,5,3],[7,6,4,0,5,1,3,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,6,6,0,5,2],[0,3,6,1,7,5,2,4],[0,4,6,7,1,5,2,3],[0,5,0,5,5,0,0,5],[0,6,5,2,2,0,5,6],[0,7,2,4,3,5,6,1]],\"n\":8,\"subset\":[0,1,2,3,4,5,6,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0088","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-trivial-boundary；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,5,7,4,0],[2,3,6,4,7,0,5,1],[3,6,4,5,0,1,7,2],[4,5,7,0,2,3,1,6],[5,7,0,1,3,6,2,4],[6,4,5,7,1,2,0,3],[7,0,1,2,6,4,3,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,5,2,6,0,5],[0,3,5,1,7,2,6,4],[0,4,2,7,1,5,6,3],[0,5,6,2,5,6,0,2],[0,6,0,6,6,0,0,6],[0,7,5,4,3,2,6,1]],\"subset\":[0]}。請依 [subset,zero_ideal,whole_ring,qualified,boundary] 欄序回答。","answer":{"kind":"expression","value":"([0], 是, 否, 是, zero_and_whole_are_always_two_sided_ideals)","explanation_zh":"Wave4第二層核對=([0], 是, 否, 是, zero_and_whole_are_always_two_sided_ideals)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-trivial-boundary","seed":202610282088,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,6,5,7,4,0],[2,3,6,4,7,0,5,1],[3,6,4,5,0,1,7,2],[4,5,7,0,2,3,1,6],[5,7,0,1,3,6,2,4],[6,4,5,7,1,2,0,3],[7,0,1,2,6,4,3,5]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,5,2,6,0,5],[0,3,5,1,7,2,6,4],[0,4,2,7,1,5,6,3],[0,5,6,2,5,6,0,2],[0,6,0,6,6,0,0,6],[0,7,5,4,3,2,6,1]],"subset":[0],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-trivial-boundary","semantic_template":"qf-mod28-ideal-trivial-boundary/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,5,7,4,0],[2,3,6,4,7,0,5,1],[3,6,4,5,0,1,7,2],[4,5,7,0,2,3,1,6],[5,7,0,1,3,6,2,4],[6,4,5,7,1,2,0,3],[7,0,1,2,6,4,3,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,5,2,6,0,5],[0,3,5,1,7,2,6,4],[0,4,2,7,1,5,6,3],[0,5,6,2,5,6,0,2],[0,6,0,6,6,0,0,6],[0,7,5,4,3,2,6,1]],\"n\":8,\"subset\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0089","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-trivial-boundary；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,7,0,4,5],[2,3,6,4,5,1,7,0],[3,6,4,7,0,2,5,1],[4,7,5,0,2,6,1,3],[5,0,1,2,6,7,3,4],[6,4,7,5,1,3,0,2],[7,5,0,1,3,4,2,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,7,2,7,0,6],[0,3,7,1,5,4,6,2],[0,4,2,5,1,3,6,7],[0,5,7,4,3,1,6,2],[0,6,0,6,6,6,0,0],[0,7,6,2,7,2,0,6]],\"subset\":[0,1,2,3,4,5,6,7]}。請依 [subset,zero_ideal,whole_ring,qualified,boundary] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7], 否, 是, 是, zero_and_whole_are_always_two_sided_ideals)","explanation_zh":"Wave4第二層核對=([0, 1, 2, 3, 4, 5, 6, 7], 否, 是, 是, zero_and_whole_are_always_two_sided_ideals)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-trivial-boundary","seed":202610282089,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,6,7,0,4,5],[2,3,6,4,5,1,7,0],[3,6,4,7,0,2,5,1],[4,7,5,0,2,6,1,3],[5,0,1,2,6,7,3,4],[6,4,7,5,1,3,0,2],[7,5,0,1,3,4,2,6]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,7,2,7,0,6],[0,3,7,1,5,4,6,2],[0,4,2,5,1,3,6,7],[0,5,7,4,3,1,6,2],[0,6,0,6,6,6,0,0],[0,7,6,2,7,2,0,6]],"subset":[0,1,2,3,4,5,6,7],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-trivial-boundary","semantic_template":"qf-mod28-ideal-trivial-boundary/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,7,0,4,5],[2,3,6,4,5,1,7,0],[3,6,4,7,0,2,5,1],[4,7,5,0,2,6,1,3],[5,0,1,2,6,7,3,4],[6,4,7,5,1,3,0,2],[7,5,0,1,3,4,2,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,7,2,7,0,6],[0,3,7,1,5,4,6,2],[0,4,2,5,1,3,6,7],[0,5,7,4,3,1,6,2],[0,6,0,6,6,6,0,0],[0,7,6,2,7,2,0,6]],\"n\":8,\"subset\":[0,1,2,3,4,5,6,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0090","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-trivial-boundary；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,7,4,5,0],[2,3,6,5,0,7,4,1],[3,6,5,4,1,0,7,2],[4,7,0,1,6,3,2,5],[5,4,7,0,3,2,1,6],[6,5,4,7,2,1,0,3],[7,0,1,2,5,6,3,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,4,6,2,0,4],[0,3,4,1,2,7,6,5],[0,4,6,2,6,4,0,2],[0,5,2,7,4,1,6,3],[0,6,0,6,0,6,0,6],[0,7,4,5,2,3,6,1]],\"subset\":[0]}。請依 [subset,zero_ideal,whole_ring,qualified,boundary] 欄序回答。","answer":{"kind":"expression","value":"([0], 是, 否, 是, zero_and_whole_are_always_two_sided_ideals)","explanation_zh":"Wave4第二層核對=([0], 是, 否, 是, zero_and_whole_are_always_two_sided_ideals)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-trivial-boundary","seed":202610282090,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,6,7,4,5,0],[2,3,6,5,0,7,4,1],[3,6,5,4,1,0,7,2],[4,7,0,1,6,3,2,5],[5,4,7,0,3,2,1,6],[6,5,4,7,2,1,0,3],[7,0,1,2,5,6,3,4]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,4,6,2,0,4],[0,3,4,1,2,7,6,5],[0,4,6,2,6,4,0,2],[0,5,2,7,4,1,6,3],[0,6,0,6,0,6,0,6],[0,7,4,5,2,3,6,1]],"subset":[0],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-trivial-boundary","semantic_template":"qf-mod28-ideal-trivial-boundary/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,7,4,5,0],[2,3,6,5,0,7,4,1],[3,6,5,4,1,0,7,2],[4,7,0,1,6,3,2,5],[5,4,7,0,3,2,1,6],[6,5,4,7,2,1,0,3],[7,0,1,2,5,6,3,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,4,6,2,0,4],[0,3,4,1,2,7,6,5],[0,4,6,2,6,4,0,2],[0,5,2,7,4,1,6,3],[0,6,0,6,0,6,0,6],[0,7,4,5,2,3,6,1]],\"n\":8,\"subset\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0091","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-trivial-boundary；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,0,7,5,4],[2,3,6,5,1,4,7,0],[3,6,5,7,2,0,4,1],[4,0,1,2,7,6,3,5],[5,7,4,0,6,2,1,3],[6,5,7,4,3,1,0,2],[7,4,0,1,5,3,2,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,7,7,2,0,6],[0,3,7,1,5,4,6,2],[0,4,7,5,1,3,6,2],[0,5,2,4,3,1,6,7],[0,6,0,6,6,6,0,0],[0,7,6,2,2,7,0,6]],\"subset\":[0,1,2,3,4,5,6,7]}。請依 [subset,zero_ideal,whole_ring,qualified,boundary] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7], 否, 是, 是, zero_and_whole_are_always_two_sided_ideals)","explanation_zh":"Wave4第二層核對=([0, 1, 2, 3, 4, 5, 6, 7], 否, 是, 是, zero_and_whole_are_always_two_sided_ideals)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-trivial-boundary","seed":202610282091,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,6,0,7,5,4],[2,3,6,5,1,4,7,0],[3,6,5,7,2,0,4,1],[4,0,1,2,7,6,3,5],[5,7,4,0,6,2,1,3],[6,5,7,4,3,1,0,2],[7,4,0,1,5,3,2,6]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,7,7,2,0,6],[0,3,7,1,5,4,6,2],[0,4,7,5,1,3,6,2],[0,5,2,4,3,1,6,7],[0,6,0,6,6,6,0,0],[0,7,6,2,2,7,0,6]],"subset":[0,1,2,3,4,5,6,7],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-trivial-boundary","semantic_template":"qf-mod28-ideal-trivial-boundary/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,0,7,5,4],[2,3,6,5,1,4,7,0],[3,6,5,7,2,0,4,1],[4,0,1,2,7,6,3,5],[5,7,4,0,6,2,1,3],[6,5,7,4,3,1,0,2],[7,4,0,1,5,3,2,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,7,7,2,0,6],[0,3,7,1,5,4,6,2],[0,4,7,5,1,3,6,2],[0,5,2,4,3,1,6,7],[0,6,0,6,6,6,0,0],[0,7,6,2,2,7,0,6]],\"n\":8,\"subset\":[0,1,2,3,4,5,6,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0092","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-trivial-boundary；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,5,0,7,4],[2,3,6,7,0,1,4,5],[3,6,7,4,1,2,5,0],[4,5,0,1,6,7,2,3],[5,0,1,2,7,4,3,6],[6,7,4,5,2,3,0,1],[7,4,5,0,3,6,1,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,4,6,4,0,2],[0,3,4,1,2,7,6,5],[0,4,6,2,6,2,0,4],[0,5,4,7,2,1,6,3],[0,6,0,6,0,6,0,6],[0,7,2,5,4,3,6,1]],\"subset\":[0]}。請依 [subset,zero_ideal,whole_ring,qualified,boundary] 欄序回答。","answer":{"kind":"expression","value":"([0], 是, 否, 是, zero_and_whole_are_always_two_sided_ideals)","explanation_zh":"Wave4第二層核對=([0], 是, 否, 是, zero_and_whole_are_always_two_sided_ideals)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-trivial-boundary","seed":202610282092,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,6,5,0,7,4],[2,3,6,7,0,1,4,5],[3,6,7,4,1,2,5,0],[4,5,0,1,6,7,2,3],[5,0,1,2,7,4,3,6],[6,7,4,5,2,3,0,1],[7,4,5,0,3,6,1,2]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,4,6,4,0,2],[0,3,4,1,2,7,6,5],[0,4,6,2,6,2,0,4],[0,5,4,7,2,1,6,3],[0,6,0,6,0,6,0,6],[0,7,2,5,4,3,6,1]],"subset":[0],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-trivial-boundary","semantic_template":"qf-mod28-ideal-trivial-boundary/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,5,0,7,4],[2,3,6,7,0,1,4,5],[3,6,7,4,1,2,5,0],[4,5,0,1,6,7,2,3],[5,0,1,2,7,4,3,6],[6,7,4,5,2,3,0,1],[7,4,5,0,3,6,1,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,4,6,4,0,2],[0,3,4,1,2,7,6,5],[0,4,6,2,6,2,0,4],[0,5,4,7,2,1,6,3],[0,6,0,6,0,6,0,6],[0,7,2,5,4,3,6,1]],\"n\":8,\"subset\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0093","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-trivial-boundary；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,0,4,7,5],[2,3,6,7,1,0,5,4],[3,6,7,5,2,1,4,0],[4,0,1,2,5,7,3,6],[5,4,0,1,7,6,2,3],[6,7,5,4,3,2,0,1],[7,5,4,0,6,3,1,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,5,5,6,0,2],[0,3,5,1,7,2,6,4],[0,4,5,7,1,2,6,3],[0,5,6,2,2,6,0,5],[0,6,0,6,6,0,0,6],[0,7,2,4,3,5,6,1]],\"subset\":[0,1,2,3,4,5,6,7]}。請依 [subset,zero_ideal,whole_ring,qualified,boundary] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7], 否, 是, 是, zero_and_whole_are_always_two_sided_ideals)","explanation_zh":"Wave4第二層核對=([0, 1, 2, 3, 4, 5, 6, 7], 否, 是, 是, zero_and_whole_are_always_two_sided_ideals)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-trivial-boundary","seed":202610282093,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,6,0,4,7,5],[2,3,6,7,1,0,5,4],[3,6,7,5,2,1,4,0],[4,0,1,2,5,7,3,6],[5,4,0,1,7,6,2,3],[6,7,5,4,3,2,0,1],[7,5,4,0,6,3,1,2]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,5,5,6,0,2],[0,3,5,1,7,2,6,4],[0,4,5,7,1,2,6,3],[0,5,6,2,2,6,0,5],[0,6,0,6,6,0,0,6],[0,7,2,4,3,5,6,1]],"subset":[0,1,2,3,4,5,6,7],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-trivial-boundary","semantic_template":"qf-mod28-ideal-trivial-boundary/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,0,4,7,5],[2,3,6,7,1,0,5,4],[3,6,7,5,2,1,4,0],[4,0,1,2,5,7,3,6],[5,4,0,1,7,6,2,3],[6,7,5,4,3,2,0,1],[7,5,4,0,6,3,1,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,5,5,6,0,2],[0,3,5,1,7,2,6,4],[0,4,5,7,1,2,6,3],[0,5,6,2,2,6,0,5],[0,6,0,6,6,0,0,6],[0,7,2,4,3,5,6,1]],\"n\":8,\"subset\":[0,1,2,3,4,5,6,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0094","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-trivial-boundary；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,5,6,0,4],[2,3,7,4,6,0,1,5],[3,7,4,5,0,1,2,6],[4,5,6,0,2,3,7,1],[5,6,0,1,3,7,4,2],[6,0,1,2,7,4,5,3],[7,4,5,6,1,2,3,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,5,2,7,5,0],[0,3,5,1,6,2,4,7],[0,4,2,6,1,5,3,7],[0,5,7,2,5,7,2,0],[0,6,5,4,3,2,1,7],[0,7,0,7,7,0,7,0]],\"subset\":[0]}。請依 [subset,zero_ideal,whole_ring,qualified,boundary] 欄序回答。","answer":{"kind":"expression","value":"([0], 是, 否, 是, zero_and_whole_are_always_two_sided_ideals)","explanation_zh":"Wave4第二層核對=([0], 是, 否, 是, zero_and_whole_are_always_two_sided_ideals)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-trivial-boundary","seed":202610282094,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,7,5,6,0,4],[2,3,7,4,6,0,1,5],[3,7,4,5,0,1,2,6],[4,5,6,0,2,3,7,1],[5,6,0,1,3,7,4,2],[6,0,1,2,7,4,5,3],[7,4,5,6,1,2,3,0]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,5,2,7,5,0],[0,3,5,1,6,2,4,7],[0,4,2,6,1,5,3,7],[0,5,7,2,5,7,2,0],[0,6,5,4,3,2,1,7],[0,7,0,7,7,0,7,0]],"subset":[0],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-trivial-boundary","semantic_template":"qf-mod28-ideal-trivial-boundary/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,5,6,0,4],[2,3,7,4,6,0,1,5],[3,7,4,5,0,1,2,6],[4,5,6,0,2,3,7,1],[5,6,0,1,3,7,4,2],[6,0,1,2,7,4,5,3],[7,4,5,6,1,2,3,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,5,2,7,5,0],[0,3,5,1,6,2,4,7],[0,4,2,6,1,5,3,7],[0,5,7,2,5,7,2,0],[0,6,5,4,3,2,1,7],[0,7,0,7,7,0,7,0]],\"n\":8,\"subset\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0095","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-trivial-boundary；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,6,0,5,4],[2,3,7,4,5,1,0,6],[3,7,4,6,0,2,1,5],[4,6,5,0,2,7,3,1],[5,0,1,2,7,6,4,3],[6,5,0,1,3,4,7,2],[7,4,6,5,1,3,2,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,6,2,6,7,0],[0,3,6,1,5,4,2,7],[0,4,2,5,1,3,6,7],[0,5,6,4,3,1,2,7],[0,6,7,2,6,2,7,0],[0,7,0,7,7,7,0,0]],\"subset\":[0,1,2,3,4,5,6,7]}。請依 [subset,zero_ideal,whole_ring,qualified,boundary] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7], 否, 是, 是, zero_and_whole_are_always_two_sided_ideals)","explanation_zh":"Wave4第二層核對=([0, 1, 2, 3, 4, 5, 6, 7], 否, 是, 是, zero_and_whole_are_always_two_sided_ideals)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-trivial-boundary","seed":202610282095,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,7,6,0,5,4],[2,3,7,4,5,1,0,6],[3,7,4,6,0,2,1,5],[4,6,5,0,2,7,3,1],[5,0,1,2,7,6,4,3],[6,5,0,1,3,4,7,2],[7,4,6,5,1,3,2,0]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,6,2,6,7,0],[0,3,6,1,5,4,2,7],[0,4,2,5,1,3,6,7],[0,5,6,4,3,1,2,7],[0,6,7,2,6,2,7,0],[0,7,0,7,7,7,0,0]],"subset":[0,1,2,3,4,5,6,7],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-trivial-boundary","semantic_template":"qf-mod28-ideal-trivial-boundary/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,6,0,5,4],[2,3,7,4,5,1,0,6],[3,7,4,6,0,2,1,5],[4,6,5,0,2,7,3,1],[5,0,1,2,7,6,4,3],[6,5,0,1,3,4,7,2],[7,4,6,5,1,3,2,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,6,2,6,7,0],[0,3,6,1,5,4,2,7],[0,4,2,5,1,3,6,7],[0,5,6,4,3,1,2,7],[0,6,7,2,6,2,7,0],[0,7,0,7,7,7,0,0]],\"n\":8,\"subset\":[0,1,2,3,4,5,6,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0096","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-trivial-boundary；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,6,4,0,5],[2,3,7,5,0,6,1,4],[3,7,5,4,1,0,2,6],[4,6,0,1,7,3,5,2],[5,4,6,0,3,2,7,1],[6,0,1,2,5,7,4,3],[7,5,4,6,2,1,3,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,4,7,2,4,0],[0,3,4,1,2,6,5,7],[0,4,7,2,7,4,2,0],[0,5,2,6,4,1,3,7],[0,6,4,5,2,3,1,7],[0,7,0,7,0,7,7,0]],\"subset\":[0]}。請依 [subset,zero_ideal,whole_ring,qualified,boundary] 欄序回答。","answer":{"kind":"expression","value":"([0], 是, 否, 是, zero_and_whole_are_always_two_sided_ideals)","explanation_zh":"Wave4第二層核對=([0], 是, 否, 是, zero_and_whole_are_always_two_sided_ideals)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-trivial-boundary","seed":202610282096,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,7,6,4,0,5],[2,3,7,5,0,6,1,4],[3,7,5,4,1,0,2,6],[4,6,0,1,7,3,5,2],[5,4,6,0,3,2,7,1],[6,0,1,2,5,7,4,3],[7,5,4,6,2,1,3,0]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,4,7,2,4,0],[0,3,4,1,2,6,5,7],[0,4,7,2,7,4,2,0],[0,5,2,6,4,1,3,7],[0,6,4,5,2,3,1,7],[0,7,0,7,0,7,7,0]],"subset":[0],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-trivial-boundary","semantic_template":"qf-mod28-ideal-trivial-boundary/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,6,4,0,5],[2,3,7,5,0,6,1,4],[3,7,5,4,1,0,2,6],[4,6,0,1,7,3,5,2],[5,4,6,0,3,2,7,1],[6,0,1,2,5,7,4,3],[7,5,4,6,2,1,3,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,4,7,2,4,0],[0,3,4,1,2,6,5,7],[0,4,7,2,7,4,2,0],[0,5,2,6,4,1,3,7],[0,6,4,5,2,3,1,7],[0,7,0,7,0,7,7,0]],\"n\":8,\"subset\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0097","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-trivial-boundary；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,0,6,4,5],[2,3,7,5,1,4,0,6],[3,7,5,6,2,0,1,4],[4,0,1,2,6,7,5,3],[5,6,4,0,7,2,3,1],[6,4,0,1,5,3,7,2],[7,5,6,4,3,1,2,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,6,6,2,7,0],[0,3,6,1,5,4,2,7],[0,4,6,5,1,3,2,7],[0,5,2,4,3,1,6,7],[0,6,7,2,2,6,7,0],[0,7,0,7,7,7,0,0]],\"subset\":[0,1,2,3,4,5,6,7]}。請依 [subset,zero_ideal,whole_ring,qualified,boundary] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7], 否, 是, 是, zero_and_whole_are_always_two_sided_ideals)","explanation_zh":"Wave4第二層核對=([0, 1, 2, 3, 4, 5, 6, 7], 否, 是, 是, zero_and_whole_are_always_two_sided_ideals)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-trivial-boundary","seed":202610282097,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,7,0,6,4,5],[2,3,7,5,1,4,0,6],[3,7,5,6,2,0,1,4],[4,0,1,2,6,7,5,3],[5,6,4,0,7,2,3,1],[6,4,0,1,5,3,7,2],[7,5,6,4,3,1,2,0]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,6,6,2,7,0],[0,3,6,1,5,4,2,7],[0,4,6,5,1,3,2,7],[0,5,2,4,3,1,6,7],[0,6,7,2,2,6,7,0],[0,7,0,7,7,7,0,0]],"subset":[0,1,2,3,4,5,6,7],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-trivial-boundary","semantic_template":"qf-mod28-ideal-trivial-boundary/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,0,6,4,5],[2,3,7,5,1,4,0,6],[3,7,5,6,2,0,1,4],[4,0,1,2,6,7,5,3],[5,6,4,0,7,2,3,1],[6,4,0,1,5,3,7,2],[7,5,6,4,3,1,2,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,6,6,2,7,0],[0,3,6,1,5,4,2,7],[0,4,6,5,1,3,2,7],[0,5,2,4,3,1,6,7],[0,6,7,2,2,6,7,0],[0,7,0,7,7,7,0,0]],\"n\":8,\"subset\":[0,1,2,3,4,5,6,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0098","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-trivial-boundary；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,5,0,4,6],[2,3,7,6,0,1,5,4],[3,7,6,4,1,2,0,5],[4,5,0,1,7,6,3,2],[5,0,1,2,6,4,7,3],[6,4,5,0,3,7,2,1],[7,6,4,5,2,3,1,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,4,7,4,2,0],[0,3,4,1,2,6,5,7],[0,4,7,2,7,2,4,0],[0,5,4,6,2,1,3,7],[0,6,2,5,4,3,1,7],[0,7,0,7,0,7,7,0]],\"subset\":[0]}。請依 [subset,zero_ideal,whole_ring,qualified,boundary] 欄序回答。","answer":{"kind":"expression","value":"([0], 是, 否, 是, zero_and_whole_are_always_two_sided_ideals)","explanation_zh":"Wave4第二層核對=([0], 是, 否, 是, zero_and_whole_are_always_two_sided_ideals)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-trivial-boundary","seed":202610282098,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,7,5,0,4,6],[2,3,7,6,0,1,5,4],[3,7,6,4,1,2,0,5],[4,5,0,1,7,6,3,2],[5,0,1,2,6,4,7,3],[6,4,5,0,3,7,2,1],[7,6,4,5,2,3,1,0]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,4,7,4,2,0],[0,3,4,1,2,6,5,7],[0,4,7,2,7,2,4,0],[0,5,4,6,2,1,3,7],[0,6,2,5,4,3,1,7],[0,7,0,7,0,7,7,0]],"subset":[0],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-trivial-boundary","semantic_template":"qf-mod28-ideal-trivial-boundary/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,5,0,4,6],[2,3,7,6,0,1,5,4],[3,7,6,4,1,2,0,5],[4,5,0,1,7,6,3,2],[5,0,1,2,6,4,7,3],[6,4,5,0,3,7,2,1],[7,6,4,5,2,3,1,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,4,7,4,2,0],[0,3,4,1,2,6,5,7],[0,4,7,2,7,2,4,0],[0,5,4,6,2,1,3,7],[0,6,2,5,4,3,1,7],[0,7,0,7,0,7,7,0]],\"n\":8,\"subset\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0099","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-trivial-boundary；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,0,4,5,6],[2,3,7,6,1,0,4,5],[3,7,6,5,2,1,0,4],[4,0,1,2,5,6,7,3],[5,4,0,1,6,7,3,2],[6,5,4,0,7,3,2,1],[7,6,5,4,3,2,1,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,5,5,7,2,0],[0,3,5,1,6,2,4,7],[0,4,5,6,1,2,3,7],[0,5,7,2,2,7,5,0],[0,6,2,4,3,5,1,7],[0,7,0,7,7,0,7,0]],\"subset\":[0,1,2,3,4,5,6,7]}。請依 [subset,zero_ideal,whole_ring,qualified,boundary] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7], 否, 是, 是, zero_and_whole_are_always_two_sided_ideals)","explanation_zh":"Wave4第二層核對=([0, 1, 2, 3, 4, 5, 6, 7], 否, 是, 是, zero_and_whole_are_always_two_sided_ideals)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-trivial-boundary","seed":202610282099,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,7,0,4,5,6],[2,3,7,6,1,0,4,5],[3,7,6,5,2,1,0,4],[4,0,1,2,5,6,7,3],[5,4,0,1,6,7,3,2],[6,5,4,0,7,3,2,1],[7,6,5,4,3,2,1,0]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,5,5,7,2,0],[0,3,5,1,6,2,4,7],[0,4,5,6,1,2,3,7],[0,5,7,2,2,7,5,0],[0,6,2,4,3,5,1,7],[0,7,0,7,7,0,7,0]],"subset":[0,1,2,3,4,5,6,7],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-trivial-boundary","semantic_template":"qf-mod28-ideal-trivial-boundary/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,0,4,5,6],[2,3,7,6,1,0,4,5],[3,7,6,5,2,1,0,4],[4,0,1,2,5,6,7,3],[5,4,0,1,6,7,3,2],[6,5,4,0,7,3,2,1],[7,6,5,4,3,2,1,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,5,5,7,2,0],[0,3,5,1,6,2,4,7],[0,4,5,6,1,2,3,7],[0,5,7,2,2,7,5,0],[0,6,2,4,3,5,1,7],[0,7,0,7,7,0,7,0]],\"n\":8,\"subset\":[0,1,2,3,4,5,6,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r02-finite-two-sided-ideal-0100","batch_id":"batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-two-sided-ideal"],"title_zh":"有限雙邊理想","stem_zh":"子集綁定同一完整有限父環。操作=ideal-trivial-boundary；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,4,5,3,6,7,0],[2,4,3,6,5,7,0,1],[3,5,6,0,7,1,2,4],[4,3,5,7,6,0,1,2],[5,6,7,1,0,2,4,3],[6,7,0,2,1,4,3,5],[7,0,1,4,2,3,5,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,3,0,6,2,3,6],[0,3,0,0,3,3,0,3],[0,4,6,3,1,7,2,5],[0,5,2,3,7,1,6,4],[0,6,3,0,2,6,3,2],[0,7,6,3,5,4,2,1]],\"subset\":[0]}。請依 [subset,zero_ideal,whole_ring,qualified,boundary] 欄序回答。","answer":{"kind":"expression","value":"([0], 是, 否, 是, zero_and_whole_are_always_two_sided_ideals)","explanation_zh":"Wave4第二層核對=([0], 是, 否, 是, zero_and_whole_are_always_two_sided_ideals)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限雙邊理想"]},"variant":{"family_id":"qf-mod28-ideal-trivial-boundary","seed":202610282100,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,4,5,3,6,7,0],[2,4,3,6,5,7,0,1],[3,5,6,0,7,1,2,4],[4,3,5,7,6,0,1,2],[5,6,7,1,0,2,4,3],[6,7,0,2,1,4,3,5],[7,0,1,4,2,3,5,6]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,3,0,6,2,3,6],[0,3,0,0,3,3,0,3],[0,4,6,3,1,7,2,5],[0,5,2,3,7,1,6,4],[0,6,3,0,2,6,3,2],[0,7,6,3,5,4,2,1]],"subset":[0],"domain":"Complete_finite_two_sided_ideal","operation":"ideal-trivial-boundary","semantic_template":"qf-mod28-ideal-trivial-boundary/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,4,5,3,6,7,0],[2,4,3,6,5,7,0,1],[3,5,6,0,7,1,2,4],[4,3,5,7,6,0,1,2],[5,6,7,1,0,2,4,3],[6,7,0,2,1,4,3,5],[7,0,1,4,2,3,5,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,3,0,6,2,3,6],[0,3,0,0,3,3,0,3],[0,4,6,3,1,7,2,5],[0,5,2,3,7,1,6,4],[0,6,3,0,2,6,3,2],[0,7,6,3,5,4,2,1]],\"n\":8,\"subset\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}