{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0001","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-classes；資料={\"n\":4,\"d\":2}。請依 [ideal,cosets,quotient_size,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 2], [[0, 2], [1, 3]], 2, proper_two_sided_ideal_in_commutative_Zn)","explanation_zh":"Wave4收束核對=([0, 2], [[0, 2], [1, 3]], 2, proper_two_sided_ideal_in_commutative_Zn)；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-classes","seed":202610314001,"parameters":{"n":4,"d":2,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-classes","semantic_template":"qf-mod31-quotring-classes/{\"d\":2,\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0002","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-classes；資料={\"n\":6,\"d\":3}。請依 [ideal,cosets,quotient_size,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 3], [[0, 3], [1, 4], [2, 5]], 3, proper_two_sided_ideal_in_commutative_Zn)","explanation_zh":"Wave4收束核對=([0, 3], [[0, 3], [1, 4], [2, 5]], 3, proper_two_sided_ideal_in_commutative_Zn)；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-classes","seed":202610314002,"parameters":{"n":6,"d":3,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-classes","semantic_template":"qf-mod31-quotring-classes/{\"d\":3,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0003","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-classes；資料={\"n\":8,\"d\":4}。請依 [ideal,cosets,quotient_size,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 4], [[0, 4], [1, 5], [2, 6], [3, 7]], 4, proper_two_sided_ideal_in_commutative_Zn)","explanation_zh":"Wave4收束核對=([0, 4], [[0, 4], [1, 5], [2, 6], [3, 7]], 4, proper_two_sided_ideal_in_commutative_Zn)；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-classes","seed":202610314003,"parameters":{"n":8,"d":4,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-classes","semantic_template":"qf-mod31-quotring-classes/{\"d\":4,\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0004","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-classes；資料={\"n\":10,\"d\":5}。請依 [ideal,cosets,quotient_size,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 5], [[0, 5], [1, 6], [2, 7], [3, 8], [4, 9]], 5, proper_two_sided_ideal_in_commutative_Zn)","explanation_zh":"Wave4收束核對=([0, 5], [[0, 5], [1, 6], [2, 7], [3, 8], [4, 9]], 5, proper_two_sided_ideal_in_commutative_Zn)；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-classes","seed":202610314004,"parameters":{"n":10,"d":5,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-classes","semantic_template":"qf-mod31-quotring-classes/{\"d\":5,\"n\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0005","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-classes；資料={\"n\":12,\"d\":6}。請依 [ideal,cosets,quotient_size,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 6], [[0, 6], [1, 7], [2, 8], [3, 9], [4, 10], [5, 11]], 6, proper_two_sided_ideal_in_commutative_Zn)","explanation_zh":"Wave4收束核對=([0, 6], [[0, 6], [1, 7], [2, 8], [3, 9], [4, 10], [5, 11]], 6, proper_two_sided_ideal_in_commutative_Zn)；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-classes","seed":202610314005,"parameters":{"n":12,"d":6,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-classes","semantic_template":"qf-mod31-quotring-classes/{\"d\":6,\"n\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0006","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-classes；資料={\"n\":6,\"d\":2}。請依 [ideal,cosets,quotient_size,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 4], [[0, 2, 4], [1, 3, 5]], 2, proper_two_sided_ideal_in_commutative_Zn)","explanation_zh":"Wave4收束核對=([0, 2, 4], [[0, 2, 4], [1, 3, 5]], 2, proper_two_sided_ideal_in_commutative_Zn)；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-classes","seed":202610314006,"parameters":{"n":6,"d":2,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-classes","semantic_template":"qf-mod31-quotring-classes/{\"d\":2,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0007","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-classes；資料={\"n\":9,\"d\":3}。請依 [ideal,cosets,quotient_size,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 3, 6], [[0, 3, 6], [1, 4, 7], [2, 5, 8]], 3, proper_two_sided_ideal_in_commutative_Zn)","explanation_zh":"Wave4收束核對=([0, 3, 6], [[0, 3, 6], [1, 4, 7], [2, 5, 8]], 3, proper_two_sided_ideal_in_commutative_Zn)；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-classes","seed":202610314007,"parameters":{"n":9,"d":3,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-classes","semantic_template":"qf-mod31-quotring-classes/{\"d\":3,\"n\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0008","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-classes；資料={\"n\":12,\"d\":4}。請依 [ideal,cosets,quotient_size,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 4, 8], [[0, 4, 8], [1, 5, 9], [2, 6, 10], [3, 7, 11]], 4, proper_two_sided_ideal_in_commutative_Zn)","explanation_zh":"Wave4收束核對=([0, 4, 8], [[0, 4, 8], [1, 5, 9], [2, 6, 10], [3, 7, 11]], 4, proper_two_sided_ideal_in_commutative_Zn)；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-classes","seed":202610314008,"parameters":{"n":12,"d":4,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-classes","semantic_template":"qf-mod31-quotring-classes/{\"d\":4,\"n\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0009","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-classes；資料={\"n\":15,\"d\":5}。請依 [ideal,cosets,quotient_size,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 5, 10], [[0, 5, 10], [1, 6, 11], [2, 7, 12], [3, 8, 13], [4, 9, 14]], 5, proper_two_sided_ideal_in_commutative_Zn)","explanation_zh":"Wave4收束核對=([0, 5, 10], [[0, 5, 10], [1, 6, 11], [2, 7, 12], [3, 8, 13], [4, 9, 14]], 5, proper_two_sided_ideal_in_commutative_Zn)；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-classes","seed":202610314009,"parameters":{"n":15,"d":5,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-classes","semantic_template":"qf-mod31-quotring-classes/{\"d\":5,\"n\":15}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0010","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-classes；資料={\"n\":18,\"d\":6}。請依 [ideal,cosets,quotient_size,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 6, 12], [[0, 6, 12], [1, 7, 13], [2, 8, 14], [3, 9, 15], [4, 10, 16], [5, 11, 17]], 6, proper_two_sided_ideal_in_commutative_Zn)","explanation_zh":"Wave4收束核對=([0, 6, 12], [[0, 6, 12], [1, 7, 13], [2, 8, 14], [3, 9, 15], [4, 10, 16], [5, 11, 17]], 6, proper_two_sided_ideal_in_commutative_Zn)；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-classes","seed":202610314010,"parameters":{"n":18,"d":6,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-classes","semantic_template":"qf-mod31-quotring-classes/{\"d\":6,\"n\":18}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0011","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-classes；資料={\"n\":8,\"d\":2}。請依 [ideal,cosets,quotient_size,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 4, 6], [[0, 2, 4, 6], [1, 3, 5, 7]], 2, proper_two_sided_ideal_in_commutative_Zn)","explanation_zh":"Wave4收束核對=([0, 2, 4, 6], [[0, 2, 4, 6], [1, 3, 5, 7]], 2, proper_two_sided_ideal_in_commutative_Zn)；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-classes","seed":202610314011,"parameters":{"n":8,"d":2,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-classes","semantic_template":"qf-mod31-quotring-classes/{\"d\":2,\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0012","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-classes；資料={\"n\":12,\"d\":3}。請依 [ideal,cosets,quotient_size,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 3, 6, 9], [[0, 3, 6, 9], [1, 4, 7, 10], [2, 5, 8, 11]], 3, proper_two_sided_ideal_in_commutative_Zn)","explanation_zh":"Wave4收束核對=([0, 3, 6, 9], [[0, 3, 6, 9], [1, 4, 7, 10], [2, 5, 8, 11]], 3, proper_two_sided_ideal_in_commutative_Zn)；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-classes","seed":202610314012,"parameters":{"n":12,"d":3,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-classes","semantic_template":"qf-mod31-quotring-classes/{\"d\":3,\"n\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0013","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-classes；資料={\"n\":16,\"d\":4}。請依 [ideal,cosets,quotient_size,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 4, 8, 12], [[0, 4, 8, 12], [1, 5, 9, 13], [2, 6, 10, 14], [3, 7, 11, 15]], 4, proper_two_sided_ideal_in_commutative_Zn)","explanation_zh":"Wave4收束核對=([0, 4, 8, 12], [[0, 4, 8, 12], [1, 5, 9, 13], [2, 6, 10, 14], [3, 7, 11, 15]], 4, proper_two_sided_ideal_in_commutative_Zn)；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-classes","seed":202610314013,"parameters":{"n":16,"d":4,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-classes","semantic_template":"qf-mod31-quotring-classes/{\"d\":4,\"n\":16}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0014","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-classes；資料={\"n\":20,\"d\":5}。請依 [ideal,cosets,quotient_size,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 5, 10, 15], [[0, 5, 10, 15], [1, 6, 11, 16], [2, 7, 12, 17], [3, 8, 13, 18], [4, 9, 14, 19]], 5, proper_two_sided_ideal_in_commutative_Zn)","explanation_zh":"Wave4收束核對=([0, 5, 10, 15], [[0, 5, 10, 15], [1, 6, 11, 16], [2, 7, 12, 17], [3, 8, 13, 18], [4, 9, 14, 19]], 5, proper_two_sided_ideal_in_commutative_Zn)；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-classes","seed":202610314014,"parameters":{"n":20,"d":5,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-classes","semantic_template":"qf-mod31-quotring-classes/{\"d\":5,\"n\":20}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0015","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-classes；資料={\"n\":24,\"d\":6}。請依 [ideal,cosets,quotient_size,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 6, 12, 18], [[0, 6, 12, 18], [1, 7, 13, 19], [2, 8, 14, 20], [3, 9, 15, 21], [4, 10, 16, 22], [5, 11, 17, 23]], 6, proper_two_sided_ideal_in_commutative_Zn)","explanation_zh":"Wave4收束核對=([0, 6, 12, 18], [[0, 6, 12, 18], [1, 7, 13, 19], [2, 8, 14, 20], [3, 9, 15, 21], [4, 10, 16, 22], [5, 11, 17, 23]], 6, proper_two_sided_ideal_in_commutative_Zn)；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-classes","seed":202610314015,"parameters":{"n":24,"d":6,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-classes","semantic_template":"qf-mod31-quotring-classes/{\"d\":6,\"n\":24}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0016","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-classes；資料={\"n\":10,\"d\":2}。請依 [ideal,cosets,quotient_size,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8], [[0, 2, 4, 6, 8], [1, 3, 5, 7, 9]], 2, proper_two_sided_ideal_in_commutative_Zn)","explanation_zh":"Wave4收束核對=([0, 2, 4, 6, 8], [[0, 2, 4, 6, 8], [1, 3, 5, 7, 9]], 2, proper_two_sided_ideal_in_commutative_Zn)；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-classes","seed":202610314016,"parameters":{"n":10,"d":2,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-classes","semantic_template":"qf-mod31-quotring-classes/{\"d\":2,\"n\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0017","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-classes；資料={\"n\":15,\"d\":3}。請依 [ideal,cosets,quotient_size,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 3, 6, 9, 12], [[0, 3, 6, 9, 12], [1, 4, 7, 10, 13], [2, 5, 8, 11, 14]], 3, proper_two_sided_ideal_in_commutative_Zn)","explanation_zh":"Wave4收束核對=([0, 3, 6, 9, 12], [[0, 3, 6, 9, 12], [1, 4, 7, 10, 13], [2, 5, 8, 11, 14]], 3, proper_two_sided_ideal_in_commutative_Zn)；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-classes","seed":202610314017,"parameters":{"n":15,"d":3,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-classes","semantic_template":"qf-mod31-quotring-classes/{\"d\":3,\"n\":15}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0018","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-classes；資料={\"n\":20,\"d\":4}。請依 [ideal,cosets,quotient_size,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 4, 8, 12, 16], [[0, 4, 8, 12, 16], [1, 5, 9, 13, 17], [2, 6, 10, 14, 18], [3, 7, 11, 15, 19]], 4, proper_two_sided_ideal_in_commutative_Zn)","explanation_zh":"Wave4收束核對=([0, 4, 8, 12, 16], [[0, 4, 8, 12, 16], [1, 5, 9, 13, 17], [2, 6, 10, 14, 18], [3, 7, 11, 15, 19]], 4, proper_two_sided_ideal_in_commutative_Zn)；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-classes","seed":202610314018,"parameters":{"n":20,"d":4,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-classes","semantic_template":"qf-mod31-quotring-classes/{\"d\":4,\"n\":20}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0019","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-classes；資料={\"n\":25,\"d\":5}。請依 [ideal,cosets,quotient_size,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 5, 10, 15, 20], [[0, 5, 10, 15, 20], [1, 6, 11, 16, 21], [2, 7, 12, 17, 22], [3, 8, 13, 18, 23], [4, 9, 14, 19, 24]], 5, proper_two_sided_ideal_in_commutative_Zn)","explanation_zh":"Wave4收束核對=([0, 5, 10, 15, 20], [[0, 5, 10, 15, 20], [1, 6, 11, 16, 21], [2, 7, 12, 17, 22], [3, 8, 13, 18, 23], [4, 9, 14, 19, 24]], 5, proper_two_sided_ideal_in_commutative_Zn)；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-classes","seed":202610314019,"parameters":{"n":25,"d":5,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-classes","semantic_template":"qf-mod31-quotring-classes/{\"d\":5,\"n\":25}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0020","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-classes；資料={\"n\":30,\"d\":6}。請依 [ideal,cosets,quotient_size,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 6, 12, 18, 24], [[0, 6, 12, 18, 24], [1, 7, 13, 19, 25], [2, 8, 14, 20, 26], [3, 9, 15, 21, 27], [4, 10, 16, 22, 28], [5, 11, 17, 23, 29]], 6, proper_two_sided_ideal_in_commutative_Zn)","explanation_zh":"Wave4收束核對=([0, 6, 12, 18, 24], [[0, 6, 12, 18, 24], [1, 7, 13, 19, 25], [2, 8, 14, 20, 26], [3, 9, 15, 21, 27], [4, 10, 16, 22, 28], [5, 11, 17, 23, 29]], 6, proper_two_sided_ideal_in_commutative_Zn)；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-classes","seed":202610314020,"parameters":{"n":30,"d":6,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-classes","semantic_template":"qf-mod31-quotring-classes/{\"d\":6,\"n\":30}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0021","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-classes；資料={\"n\":12,\"d\":2}。請依 [ideal,cosets,quotient_size,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10], [[0, 2, 4, 6, 8, 10], [1, 3, 5, 7, 9, 11]], 2, proper_two_sided_ideal_in_commutative_Zn)","explanation_zh":"Wave4收束核對=([0, 2, 4, 6, 8, 10], [[0, 2, 4, 6, 8, 10], [1, 3, 5, 7, 9, 11]], 2, proper_two_sided_ideal_in_commutative_Zn)；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-classes","seed":202610314021,"parameters":{"n":12,"d":2,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-classes","semantic_template":"qf-mod31-quotring-classes/{\"d\":2,\"n\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0022","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-classes；資料={\"n\":18,\"d\":3}。請依 [ideal,cosets,quotient_size,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 3, 6, 9, 12, 15], [[0, 3, 6, 9, 12, 15], [1, 4, 7, 10, 13, 16], [2, 5, 8, 11, 14, 17]], 3, proper_two_sided_ideal_in_commutative_Zn)","explanation_zh":"Wave4收束核對=([0, 3, 6, 9, 12, 15], [[0, 3, 6, 9, 12, 15], [1, 4, 7, 10, 13, 16], [2, 5, 8, 11, 14, 17]], 3, proper_two_sided_ideal_in_commutative_Zn)；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-classes","seed":202610314022,"parameters":{"n":18,"d":3,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-classes","semantic_template":"qf-mod31-quotring-classes/{\"d\":3,\"n\":18}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0023","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-classes；資料={\"n\":24,\"d\":4}。請依 [ideal,cosets,quotient_size,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 4, 8, 12, 16, 20], [[0, 4, 8, 12, 16, 20], [1, 5, 9, 13, 17, 21], [2, 6, 10, 14, 18, 22], [3, 7, 11, 15, 19, 23]], 4, proper_two_sided_ideal_in_commutative_Zn)","explanation_zh":"Wave4收束核對=([0, 4, 8, 12, 16, 20], [[0, 4, 8, 12, 16, 20], [1, 5, 9, 13, 17, 21], [2, 6, 10, 14, 18, 22], [3, 7, 11, 15, 19, 23]], 4, proper_two_sided_ideal_in_commutative_Zn)；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-classes","seed":202610314023,"parameters":{"n":24,"d":4,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-classes","semantic_template":"qf-mod31-quotring-classes/{\"d\":4,\"n\":24}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0024","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-classes；資料={\"n\":30,\"d\":5}。請依 [ideal,cosets,quotient_size,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 5, 10, 15, 20, 25], [[0, 5, 10, 15, 20, 25], [1, 6, 11, 16, 21, 26], [2, 7, 12, 17, 22, 27], [3, 8, 13, 18, 23, 28], [4, 9, 14, 19, 24, 29]], 5, proper_two_sided_ideal_in_commutative_Zn)","explanation_zh":"Wave4收束核對=([0, 5, 10, 15, 20, 25], [[0, 5, 10, 15, 20, 25], [1, 6, 11, 16, 21, 26], [2, 7, 12, 17, 22, 27], [3, 8, 13, 18, 23, 28], [4, 9, 14, 19, 24, 29]], 5, proper_two_sided_ideal_in_commutative_Zn)；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-classes","seed":202610314024,"parameters":{"n":30,"d":5,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-classes","semantic_template":"qf-mod31-quotring-classes/{\"d\":5,\"n\":30}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0025","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-classes；資料={\"n\":36,\"d\":6}。請依 [ideal,cosets,quotient_size,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 6, 12, 18, 24, 30], [[0, 6, 12, 18, 24, 30], [1, 7, 13, 19, 25, 31], [2, 8, 14, 20, 26, 32], [3, 9, 15, 21, 27, 33], [4, 10, 16, 22, 28, 34], [5, 11, 17, 23, 29, 35]], 6, proper_two_sided_ideal_in_commutative_Zn)","explanation_zh":"Wave4收束核對=([0, 6, 12, 18, 24, 30], [[0, 6, 12, 18, 24, 30], [1, 7, 13, 19, 25, 31], [2, 8, 14, 20, 26, 32], [3, 9, 15, 21, 27, 33], [4, 10, 16, 22, 28, 34], [5, 11, 17, 23, 29, 35]], 6, proper_two_sided_ideal_in_commutative_Zn)；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-classes","seed":202610314025,"parameters":{"n":36,"d":6,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-classes","semantic_template":"qf-mod31-quotring-classes/{\"d\":6,\"n\":36}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0026","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-add；資料={\"n\":4,\"d\":2,\"a\":0,\"b\":1}。請依 [class_a,class_b,parent_result,result_class,law] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 1, 1, [a]+[b]=[a+b])","explanation_zh":"Wave4收束核對=(0, 1, 1, 1, [a]+[b]=[a+b])；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-add","seed":202610314026,"parameters":{"n":4,"d":2,"a":0,"b":1,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-add","semantic_template":"qf-mod31-quotring-add/{\"a\":0,\"b\":1,\"d\":2,\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0027","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-add；資料={\"n\":6,\"d\":3,\"a\":1,\"b\":3}。請依 [class_a,class_b,parent_result,result_class,law] 欄序回答。","answer":{"kind":"expression","value":"(1, 0, 4, 1, [a]+[b]=[a+b])","explanation_zh":"Wave4收束核對=(1, 0, 4, 1, [a]+[b]=[a+b])；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-add","seed":202610314027,"parameters":{"n":6,"d":3,"a":1,"b":3,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-add","semantic_template":"qf-mod31-quotring-add/{\"a\":1,\"b\":3,\"d\":3,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0028","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-add；資料={\"n\":8,\"d\":4,\"a\":2,\"b\":5}。請依 [class_a,class_b,parent_result,result_class,law] 欄序回答。","answer":{"kind":"expression","value":"(2, 1, 7, 3, [a]+[b]=[a+b])","explanation_zh":"Wave4收束核對=(2, 1, 7, 3, [a]+[b]=[a+b])；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-add","seed":202610314028,"parameters":{"n":8,"d":4,"a":2,"b":5,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-add","semantic_template":"qf-mod31-quotring-add/{\"a\":2,\"b\":5,\"d\":4,\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0029","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-add；資料={\"n\":10,\"d\":5,\"a\":3,\"b\":7}。請依 [class_a,class_b,parent_result,result_class,law] 欄序回答。","answer":{"kind":"expression","value":"(3, 2, 0, 0, [a]+[b]=[a+b])","explanation_zh":"Wave4收束核對=(3, 2, 0, 0, [a]+[b]=[a+b])；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-add","seed":202610314029,"parameters":{"n":10,"d":5,"a":3,"b":7,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-add","semantic_template":"qf-mod31-quotring-add/{\"a\":3,\"b\":7,\"d\":5,\"n\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0030","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-add；資料={\"n\":12,\"d\":6,\"a\":4,\"b\":9}。請依 [class_a,class_b,parent_result,result_class,law] 欄序回答。","answer":{"kind":"expression","value":"(4, 3, 1, 1, [a]+[b]=[a+b])","explanation_zh":"Wave4收束核對=(4, 3, 1, 1, [a]+[b]=[a+b])；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-add","seed":202610314030,"parameters":{"n":12,"d":6,"a":4,"b":9,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-add","semantic_template":"qf-mod31-quotring-add/{\"a\":4,\"b\":9,\"d\":6,\"n\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0031","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-add；資料={\"n\":6,\"d\":2,\"a\":5,\"b\":5}。請依 [class_a,class_b,parent_result,result_class,law] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, 4, 0, [a]+[b]=[a+b])","explanation_zh":"Wave4收束核對=(1, 1, 4, 0, [a]+[b]=[a+b])；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-add","seed":202610314031,"parameters":{"n":6,"d":2,"a":5,"b":5,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-add","semantic_template":"qf-mod31-quotring-add/{\"a\":5,\"b\":5,\"d\":2,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0032","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-add；資料={\"n\":9,\"d\":3,\"a\":6,\"b\":4}。請依 [class_a,class_b,parent_result,result_class,law] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 1, 1, [a]+[b]=[a+b])","explanation_zh":"Wave4收束核對=(0, 1, 1, 1, [a]+[b]=[a+b])；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-add","seed":202610314032,"parameters":{"n":9,"d":3,"a":6,"b":4,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-add","semantic_template":"qf-mod31-quotring-add/{\"a\":6,\"b\":4,\"d\":3,\"n\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0033","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-add；資料={\"n\":12,\"d\":4,\"a\":7,\"b\":3}。請依 [class_a,class_b,parent_result,result_class,law] 欄序回答。","answer":{"kind":"expression","value":"(3, 3, 10, 2, [a]+[b]=[a+b])","explanation_zh":"Wave4收束核對=(3, 3, 10, 2, [a]+[b]=[a+b])；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-add","seed":202610314033,"parameters":{"n":12,"d":4,"a":7,"b":3,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-add","semantic_template":"qf-mod31-quotring-add/{\"a\":7,\"b\":3,\"d\":4,\"n\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0034","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-add；資料={\"n\":15,\"d\":5,\"a\":8,\"b\":2}。請依 [class_a,class_b,parent_result,result_class,law] 欄序回答。","answer":{"kind":"expression","value":"(3, 2, 10, 0, [a]+[b]=[a+b])","explanation_zh":"Wave4收束核對=(3, 2, 10, 0, [a]+[b]=[a+b])；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-add","seed":202610314034,"parameters":{"n":15,"d":5,"a":8,"b":2,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-add","semantic_template":"qf-mod31-quotring-add/{\"a\":8,\"b\":2,\"d\":5,\"n\":15}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0035","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-add；資料={\"n\":18,\"d\":6,\"a\":9,\"b\":1}。請依 [class_a,class_b,parent_result,result_class,law] 欄序回答。","answer":{"kind":"expression","value":"(3, 1, 10, 4, [a]+[b]=[a+b])","explanation_zh":"Wave4收束核對=(3, 1, 10, 4, [a]+[b]=[a+b])；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-add","seed":202610314035,"parameters":{"n":18,"d":6,"a":9,"b":1,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-add","semantic_template":"qf-mod31-quotring-add/{\"a\":9,\"b\":1,\"d\":6,\"n\":18}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0036","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-add；資料={\"n\":8,\"d\":2,\"a\":2,\"b\":5}。請依 [class_a,class_b,parent_result,result_class,law] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 7, 1, [a]+[b]=[a+b])","explanation_zh":"Wave4收束核對=(0, 1, 7, 1, [a]+[b]=[a+b])；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-add","seed":202610314036,"parameters":{"n":8,"d":2,"a":2,"b":5,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-add","semantic_template":"qf-mod31-quotring-add/{\"a\":2,\"b\":5,\"d\":2,\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0037","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-add；資料={\"n\":12,\"d\":3,\"a\":11,\"b\":11}。請依 [class_a,class_b,parent_result,result_class,law] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 10, 1, [a]+[b]=[a+b])","explanation_zh":"Wave4收束核對=(2, 2, 10, 1, [a]+[b]=[a+b])；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-add","seed":202610314037,"parameters":{"n":12,"d":3,"a":11,"b":11,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-add","semantic_template":"qf-mod31-quotring-add/{\"a\":11,\"b\":11,\"d\":3,\"n\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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"}}
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{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0039","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-add；資料={\"n\":20,\"d\":5,\"a\":13,\"b\":7}。請依 [class_a,class_b,parent_result,result_class,law] 欄序回答。","answer":{"kind":"expression","value":"(3, 2, 0, 0, [a]+[b]=[a+b])","explanation_zh":"Wave4收束核對=(3, 2, 0, 0, [a]+[b]=[a+b])；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-add","seed":202610314039,"parameters":{"n":20,"d":5,"a":13,"b":7,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-add","semantic_template":"qf-mod31-quotring-add/{\"a\":13,\"b\":7,\"d\":5,\"n\":20}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0040","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-add；資料={\"n\":24,\"d\":6,\"a\":14,\"b\":5}。請依 [class_a,class_b,parent_result,result_class,law] 欄序回答。","answer":{"kind":"expression","value":"(2, 5, 19, 1, [a]+[b]=[a+b])","explanation_zh":"Wave4收束核對=(2, 5, 19, 1, [a]+[b]=[a+b])；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-add","seed":202610314040,"parameters":{"n":24,"d":6,"a":14,"b":5,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-add","semantic_template":"qf-mod31-quotring-add/{\"a\":14,\"b\":5,\"d\":6,\"n\":24}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0041","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-add；資料={\"n\":10,\"d\":2,\"a\":5,\"b\":1}。請依 [class_a,class_b,parent_result,result_class,law] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, 6, 0, [a]+[b]=[a+b])","explanation_zh":"Wave4收束核對=(1, 1, 6, 0, [a]+[b]=[a+b])；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-add","seed":202610314041,"parameters":{"n":10,"d":2,"a":5,"b":1,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-add","semantic_template":"qf-mod31-quotring-add/{\"a\":5,\"b\":1,\"d\":2,\"n\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0042","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-add；資料={\"n\":15,\"d\":3,\"a\":1,\"b\":3}。請依 [class_a,class_b,parent_result,result_class,law] 欄序回答。","answer":{"kind":"expression","value":"(1, 0, 4, 1, [a]+[b]=[a+b])","explanation_zh":"Wave4收束核對=(1, 0, 4, 1, [a]+[b]=[a+b])；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-add","seed":202610314042,"parameters":{"n":15,"d":3,"a":1,"b":3,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-add","semantic_template":"qf-mod31-quotring-add/{\"a\":1,\"b\":3,\"d\":3,\"n\":15}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0043","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-add；資料={\"n\":20,\"d\":4,\"a\":17,\"b\":15}。請依 [class_a,class_b,parent_result,result_class,law] 欄序回答。","answer":{"kind":"expression","value":"(1, 3, 12, 0, [a]+[b]=[a+b])","explanation_zh":"Wave4收束核對=(1, 3, 12, 0, [a]+[b]=[a+b])；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-add","seed":202610314043,"parameters":{"n":20,"d":4,"a":17,"b":15,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-add","semantic_template":"qf-mod31-quotring-add/{\"a\":17,\"b\":15,\"d\":4,\"n\":20}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0044","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-add；資料={\"n\":25,\"d\":5,\"a\":18,\"b\":12}。請依 [class_a,class_b,parent_result,result_class,law] 欄序回答。","answer":{"kind":"expression","value":"(3, 2, 5, 0, [a]+[b]=[a+b])","explanation_zh":"Wave4收束核對=(3, 2, 5, 0, [a]+[b]=[a+b])；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-add","seed":202610314044,"parameters":{"n":25,"d":5,"a":18,"b":12,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-add","semantic_template":"qf-mod31-quotring-add/{\"a\":18,\"b\":12,\"d\":5,\"n\":25}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0045","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-add；資料={\"n\":30,\"d\":6,\"a\":19,\"b\":9}。請依 [class_a,class_b,parent_result,result_class,law] 欄序回答。","answer":{"kind":"expression","value":"(1, 3, 28, 4, [a]+[b]=[a+b])","explanation_zh":"Wave4收束核對=(1, 3, 28, 4, [a]+[b]=[a+b])；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-add","seed":202610314045,"parameters":{"n":30,"d":6,"a":19,"b":9,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-add","semantic_template":"qf-mod31-quotring-add/{\"a\":19,\"b\":9,\"d\":6,\"n\":30}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0046","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-add；資料={\"n\":12,\"d\":2,\"a\":8,\"b\":5}。請依 [class_a,class_b,parent_result,result_class,law] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 1, 1, [a]+[b]=[a+b])","explanation_zh":"Wave4收束核對=(0, 1, 1, 1, [a]+[b]=[a+b])；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-add","seed":202610314046,"parameters":{"n":12,"d":2,"a":8,"b":5,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-add","semantic_template":"qf-mod31-quotring-add/{\"a\":8,\"b\":5,\"d\":2,\"n\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0047","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-add；資料={\"n\":18,\"d\":3,\"a\":3,\"b\":7}。請依 [class_a,class_b,parent_result,result_class,law] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 10, 1, [a]+[b]=[a+b])","explanation_zh":"Wave4收束核對=(0, 1, 10, 1, [a]+[b]=[a+b])；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-add","seed":202610314047,"parameters":{"n":18,"d":3,"a":3,"b":7,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-add","semantic_template":"qf-mod31-quotring-add/{\"a\":3,\"b\":7,\"d\":3,\"n\":18}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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"}}
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{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r04-finite-quotient-ring-0049","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-add；資料={\"n\":30,\"d\":5,\"a\":23,\"b\":17}。請依 [class_a,class_b,parent_result,result_class,law] 欄序回答。","answer":{"kind":"expression","value":"(3, 2, 10, 0, [a]+[b]=[a+b])","explanation_zh":"Wave4收束核對=(3, 2, 10, 0, [a]+[b]=[a+b])；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-add","seed":202610314049,"parameters":{"n":30,"d":5,"a":23,"b":17,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-add","semantic_template":"qf-mod31-quotring-add/{\"a\":23,\"b\":17,\"d\":5,\"n\":30}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0050","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-add；資料={\"n\":36,\"d\":6,\"a\":24,\"b\":13}。請依 [class_a,class_b,parent_result,result_class,law] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 1, 1, [a]+[b]=[a+b])","explanation_zh":"Wave4收束核對=(0, 1, 1, 1, [a]+[b]=[a+b])；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-add","seed":202610314050,"parameters":{"n":36,"d":6,"a":24,"b":13,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-add","semantic_template":"qf-mod31-quotring-add/{\"a\":24,\"b\":13,\"d\":6,\"n\":36}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0051","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-multiply；資料={\"n\":4,\"d\":2,\"a\":0,\"b\":1}。請依 [class_a,class_b,parent_result,result_class,law] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 0, 0, [a][b]=[ab])","explanation_zh":"Wave4收束核對=(0, 1, 0, 0, [a][b]=[ab])；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-multiply","seed":202610314051,"parameters":{"n":4,"d":2,"a":0,"b":1,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-multiply","semantic_template":"qf-mod31-quotring-multiply/{\"a\":0,\"b\":1,\"d\":2,\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0052","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-multiply；資料={\"n\":6,\"d\":3,\"a\":1,\"b\":3}。請依 [class_a,class_b,parent_result,result_class,law] 欄序回答。","answer":{"kind":"expression","value":"(1, 0, 3, 0, [a][b]=[ab])","explanation_zh":"Wave4收束核對=(1, 0, 3, 0, [a][b]=[ab])；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-multiply","seed":202610314052,"parameters":{"n":6,"d":3,"a":1,"b":3,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-multiply","semantic_template":"qf-mod31-quotring-multiply/{\"a\":1,\"b\":3,\"d\":3,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0053","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-multiply；資料={\"n\":8,\"d\":4,\"a\":2,\"b\":5}。請依 [class_a,class_b,parent_result,result_class,law] 欄序回答。","answer":{"kind":"expression","value":"(2, 1, 2, 2, [a][b]=[ab])","explanation_zh":"Wave4收束核對=(2, 1, 2, 2, [a][b]=[ab])；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-multiply","seed":202610314053,"parameters":{"n":8,"d":4,"a":2,"b":5,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-multiply","semantic_template":"qf-mod31-quotring-multiply/{\"a\":2,\"b\":5,\"d\":4,\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0054","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-multiply；資料={\"n\":10,\"d\":5,\"a\":3,\"b\":7}。請依 [class_a,class_b,parent_result,result_class,law] 欄序回答。","answer":{"kind":"expression","value":"(3, 2, 1, 1, [a][b]=[ab])","explanation_zh":"Wave4收束核對=(3, 2, 1, 1, [a][b]=[ab])；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-multiply","seed":202610314054,"parameters":{"n":10,"d":5,"a":3,"b":7,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-multiply","semantic_template":"qf-mod31-quotring-multiply/{\"a\":3,\"b\":7,\"d\":5,\"n\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0055","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-multiply；資料={\"n\":12,\"d\":6,\"a\":4,\"b\":9}。請依 [class_a,class_b,parent_result,result_class,law] 欄序回答。","answer":{"kind":"expression","value":"(4, 3, 0, 0, [a][b]=[ab])","explanation_zh":"Wave4收束核對=(4, 3, 0, 0, [a][b]=[ab])；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-multiply","seed":202610314055,"parameters":{"n":12,"d":6,"a":4,"b":9,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-multiply","semantic_template":"qf-mod31-quotring-multiply/{\"a\":4,\"b\":9,\"d\":6,\"n\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0056","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-multiply；資料={\"n\":6,\"d\":2,\"a\":5,\"b\":5}。請依 [class_a,class_b,parent_result,result_class,law] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, 1, 1, [a][b]=[ab])","explanation_zh":"Wave4收束核對=(1, 1, 1, 1, [a][b]=[ab])；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-multiply","seed":202610314056,"parameters":{"n":6,"d":2,"a":5,"b":5,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-multiply","semantic_template":"qf-mod31-quotring-multiply/{\"a\":5,\"b\":5,\"d\":2,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0057","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-multiply；資料={\"n\":9,\"d\":3,\"a\":6,\"b\":4}。請依 [class_a,class_b,parent_result,result_class,law] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 6, 0, [a][b]=[ab])","explanation_zh":"Wave4收束核對=(0, 1, 6, 0, [a][b]=[ab])；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-multiply","seed":202610314057,"parameters":{"n":9,"d":3,"a":6,"b":4,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-multiply","semantic_template":"qf-mod31-quotring-multiply/{\"a\":6,\"b\":4,\"d\":3,\"n\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0058","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-multiply；資料={\"n\":12,\"d\":4,\"a\":7,\"b\":3}。請依 [class_a,class_b,parent_result,result_class,law] 欄序回答。","answer":{"kind":"expression","value":"(3, 3, 9, 1, [a][b]=[ab])","explanation_zh":"Wave4收束核對=(3, 3, 9, 1, [a][b]=[ab])；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-multiply","seed":202610314058,"parameters":{"n":12,"d":4,"a":7,"b":3,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-multiply","semantic_template":"qf-mod31-quotring-multiply/{\"a\":7,\"b\":3,\"d\":4,\"n\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0059","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-multiply；資料={\"n\":15,\"d\":5,\"a\":8,\"b\":2}。請依 [class_a,class_b,parent_result,result_class,law] 欄序回答。","answer":{"kind":"expression","value":"(3, 2, 1, 1, [a][b]=[ab])","explanation_zh":"Wave4收束核對=(3, 2, 1, 1, [a][b]=[ab])；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-multiply","seed":202610314059,"parameters":{"n":15,"d":5,"a":8,"b":2,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-multiply","semantic_template":"qf-mod31-quotring-multiply/{\"a\":8,\"b\":2,\"d\":5,\"n\":15}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0060","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-multiply；資料={\"n\":18,\"d\":6,\"a\":9,\"b\":1}。請依 [class_a,class_b,parent_result,result_class,law] 欄序回答。","answer":{"kind":"expression","value":"(3, 1, 9, 3, [a][b]=[ab])","explanation_zh":"Wave4收束核對=(3, 1, 9, 3, [a][b]=[ab])；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-multiply","seed":202610314060,"parameters":{"n":18,"d":6,"a":9,"b":1,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-multiply","semantic_template":"qf-mod31-quotring-multiply/{\"a\":9,\"b\":1,\"d\":6,\"n\":18}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0061","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-multiply；資料={\"n\":8,\"d\":2,\"a\":2,\"b\":5}。請依 [class_a,class_b,parent_result,result_class,law] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 2, 0, [a][b]=[ab])","explanation_zh":"Wave4收束核對=(0, 1, 2, 0, [a][b]=[ab])；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-multiply","seed":202610314061,"parameters":{"n":8,"d":2,"a":2,"b":5,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-multiply","semantic_template":"qf-mod31-quotring-multiply/{\"a\":2,\"b\":5,\"d\":2,\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0062","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-multiply；資料={\"n\":12,\"d\":3,\"a\":11,\"b\":11}。請依 [class_a,class_b,parent_result,result_class,law] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 1, 1, [a][b]=[ab])","explanation_zh":"Wave4收束核對=(2, 2, 1, 1, [a][b]=[ab])；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-multiply","seed":202610314062,"parameters":{"n":12,"d":3,"a":11,"b":11,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-multiply","semantic_template":"qf-mod31-quotring-multiply/{\"a\":11,\"b\":11,\"d\":3,\"n\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0063","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-multiply；資料={\"n\":16,\"d\":4,\"a\":12,\"b\":9}。請依 [class_a,class_b,parent_result,result_class,law] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 12, 0, [a][b]=[ab])","explanation_zh":"Wave4收束核對=(0, 1, 12, 0, [a][b]=[ab])；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-multiply","seed":202610314063,"parameters":{"n":16,"d":4,"a":12,"b":9,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-multiply","semantic_template":"qf-mod31-quotring-multiply/{\"a\":12,\"b\":9,\"d\":4,\"n\":16}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0064","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-multiply；資料={\"n\":20,\"d\":5,\"a\":13,\"b\":7}。請依 [class_a,class_b,parent_result,result_class,law] 欄序回答。","answer":{"kind":"expression","value":"(3, 2, 11, 1, [a][b]=[ab])","explanation_zh":"Wave4收束核對=(3, 2, 11, 1, [a][b]=[ab])；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-multiply","seed":202610314064,"parameters":{"n":20,"d":5,"a":13,"b":7,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-multiply","semantic_template":"qf-mod31-quotring-multiply/{\"a\":13,\"b\":7,\"d\":5,\"n\":20}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0065","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-multiply；資料={\"n\":24,\"d\":6,\"a\":14,\"b\":5}。請依 [class_a,class_b,parent_result,result_class,law] 欄序回答。","answer":{"kind":"expression","value":"(2, 5, 22, 4, [a][b]=[ab])","explanation_zh":"Wave4收束核對=(2, 5, 22, 4, [a][b]=[ab])；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-multiply","seed":202610314065,"parameters":{"n":24,"d":6,"a":14,"b":5,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-multiply","semantic_template":"qf-mod31-quotring-multiply/{\"a\":14,\"b\":5,\"d\":6,\"n\":24}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0066","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-multiply；資料={\"n\":10,\"d\":2,\"a\":5,\"b\":1}。請依 [class_a,class_b,parent_result,result_class,law] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, 5, 1, [a][b]=[ab])","explanation_zh":"Wave4收束核對=(1, 1, 5, 1, [a][b]=[ab])；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-multiply","seed":202610314066,"parameters":{"n":10,"d":2,"a":5,"b":1,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-multiply","semantic_template":"qf-mod31-quotring-multiply/{\"a\":5,\"b\":1,\"d\":2,\"n\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0067","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-multiply；資料={\"n\":15,\"d\":3,\"a\":1,\"b\":3}。請依 [class_a,class_b,parent_result,result_class,law] 欄序回答。","answer":{"kind":"expression","value":"(1, 0, 3, 0, [a][b]=[ab])","explanation_zh":"Wave4收束核對=(1, 0, 3, 0, [a][b]=[ab])；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-multiply","seed":202610314067,"parameters":{"n":15,"d":3,"a":1,"b":3,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-multiply","semantic_template":"qf-mod31-quotring-multiply/{\"a\":1,\"b\":3,\"d\":3,\"n\":15}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0068","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-multiply；資料={\"n\":20,\"d\":4,\"a\":17,\"b\":15}。請依 [class_a,class_b,parent_result,result_class,law] 欄序回答。","answer":{"kind":"expression","value":"(1, 3, 15, 3, [a][b]=[ab])","explanation_zh":"Wave4收束核對=(1, 3, 15, 3, [a][b]=[ab])；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-multiply","seed":202610314068,"parameters":{"n":20,"d":4,"a":17,"b":15,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-multiply","semantic_template":"qf-mod31-quotring-multiply/{\"a\":17,\"b\":15,\"d\":4,\"n\":20}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0069","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-multiply；資料={\"n\":25,\"d\":5,\"a\":18,\"b\":12}。請依 [class_a,class_b,parent_result,result_class,law] 欄序回答。","answer":{"kind":"expression","value":"(3, 2, 16, 1, [a][b]=[ab])","explanation_zh":"Wave4收束核對=(3, 2, 16, 1, [a][b]=[ab])；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-multiply","seed":202610314069,"parameters":{"n":25,"d":5,"a":18,"b":12,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-multiply","semantic_template":"qf-mod31-quotring-multiply/{\"a\":18,\"b\":12,\"d\":5,\"n\":25}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0070","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-multiply；資料={\"n\":30,\"d\":6,\"a\":19,\"b\":9}。請依 [class_a,class_b,parent_result,result_class,law] 欄序回答。","answer":{"kind":"expression","value":"(1, 3, 21, 3, [a][b]=[ab])","explanation_zh":"Wave4收束核對=(1, 3, 21, 3, [a][b]=[ab])；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-multiply","seed":202610314070,"parameters":{"n":30,"d":6,"a":19,"b":9,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-multiply","semantic_template":"qf-mod31-quotring-multiply/{\"a\":19,\"b\":9,\"d\":6,\"n\":30}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0071","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-multiply；資料={\"n\":12,\"d\":2,\"a\":8,\"b\":5}。請依 [class_a,class_b,parent_result,result_class,law] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 4, 0, [a][b]=[ab])","explanation_zh":"Wave4收束核對=(0, 1, 4, 0, [a][b]=[ab])；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-multiply","seed":202610314071,"parameters":{"n":12,"d":2,"a":8,"b":5,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-multiply","semantic_template":"qf-mod31-quotring-multiply/{\"a\":8,\"b\":5,\"d\":2,\"n\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0072","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-multiply；資料={\"n\":18,\"d\":3,\"a\":3,\"b\":7}。請依 [class_a,class_b,parent_result,result_class,law] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 3, 0, [a][b]=[ab])","explanation_zh":"Wave4收束核對=(0, 1, 3, 0, [a][b]=[ab])；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-multiply","seed":202610314072,"parameters":{"n":18,"d":3,"a":3,"b":7,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-multiply","semantic_template":"qf-mod31-quotring-multiply/{\"a\":3,\"b\":7,\"d\":3,\"n\":18}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0073","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-multiply；資料={\"n\":24,\"d\":4,\"a\":22,\"b\":21}。請依 [class_a,class_b,parent_result,result_class,law] 欄序回答。","answer":{"kind":"expression","value":"(2, 1, 6, 2, [a][b]=[ab])","explanation_zh":"Wave4收束核對=(2, 1, 6, 2, [a][b]=[ab])；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-multiply","seed":202610314073,"parameters":{"n":24,"d":4,"a":22,"b":21,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-multiply","semantic_template":"qf-mod31-quotring-multiply/{\"a\":22,\"b\":21,\"d\":4,\"n\":24}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0074","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-multiply；資料={\"n\":30,\"d\":5,\"a\":23,\"b\":17}。請依 [class_a,class_b,parent_result,result_class,law] 欄序回答。","answer":{"kind":"expression","value":"(3, 2, 1, 1, [a][b]=[ab])","explanation_zh":"Wave4收束核對=(3, 2, 1, 1, [a][b]=[ab])；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-multiply","seed":202610314074,"parameters":{"n":30,"d":5,"a":23,"b":17,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-multiply","semantic_template":"qf-mod31-quotring-multiply/{\"a\":23,\"b\":17,\"d\":5,\"n\":30}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0075","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-multiply；資料={\"n\":36,\"d\":6,\"a\":24,\"b\":13}。請依 [class_a,class_b,parent_result,result_class,law] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 24, 0, [a][b]=[ab])","explanation_zh":"Wave4收束核對=(0, 1, 24, 0, [a][b]=[ab])；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-multiply","seed":202610314075,"parameters":{"n":36,"d":6,"a":24,"b":13,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-multiply","semantic_template":"qf-mod31-quotring-multiply/{\"a\":24,\"b\":13,\"d\":6,\"n\":36}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0076","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-well-defined；資料={\"n\":4,\"d\":2,\"a\":0,\"b\":1,\"a_prime\":2,\"b_prime\":3}。請依 [representatives,addition_classes,multiplication_classes,addition_well_defined,multiplication_well_defined,quotient_operations_well_defined] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3], [1, 1], [0, 0], 是, 是, 是)","explanation_zh":"Wave4收束核對=([0, 1, 2, 3], [1, 1], [0, 0], 是, 是, 是)；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-well-defined","seed":202610314076,"parameters":{"n":4,"d":2,"a":0,"b":1,"a_prime":2,"b_prime":3,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-well-defined","semantic_template":"qf-mod31-quotring-well-defined/{\"a\":0,\"a_prime\":2,\"b\":1,\"b_prime\":3,\"d\":2,\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0077","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-well-defined；資料={\"n\":6,\"d\":3,\"a\":1,\"b\":0,\"a_prime\":4,\"b_prime\":3}。請依 [representatives,addition_classes,multiplication_classes,addition_well_defined,multiplication_well_defined,quotient_operations_well_defined] 欄序回答。","answer":{"kind":"expression","value":"([1, 0, 4, 3], [1, 1], [0, 0], 是, 是, 是)","explanation_zh":"Wave4收束核對=([1, 0, 4, 3], [1, 1], [0, 0], 是, 是, 是)；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-well-defined","seed":202610314077,"parameters":{"n":6,"d":3,"a":1,"b":0,"a_prime":4,"b_prime":3,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-well-defined","semantic_template":"qf-mod31-quotring-well-defined/{\"a\":1,\"a_prime\":4,\"b\":0,\"b_prime\":3,\"d\":3,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0078","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-well-defined；資料={\"n\":8,\"d\":4,\"a\":2,\"b\":1,\"a_prime\":6,\"b_prime\":5}。請依 [representatives,addition_classes,multiplication_classes,addition_well_defined,multiplication_well_defined,quotient_operations_well_defined] 欄序回答。","answer":{"kind":"expression","value":"([2, 1, 6, 5], [3, 3], [2, 2], 是, 是, 是)","explanation_zh":"Wave4收束核對=([2, 1, 6, 5], [3, 3], [2, 2], 是, 是, 是)；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-well-defined","seed":202610314078,"parameters":{"n":8,"d":4,"a":2,"b":1,"a_prime":6,"b_prime":5,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-well-defined","semantic_template":"qf-mod31-quotring-well-defined/{\"a\":2,\"a_prime\":6,\"b\":1,\"b_prime\":5,\"d\":4,\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0079","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-well-defined；資料={\"n\":10,\"d\":5,\"a\":3,\"b\":2,\"a_prime\":8,\"b_prime\":7}。請依 [representatives,addition_classes,multiplication_classes,addition_well_defined,multiplication_well_defined,quotient_operations_well_defined] 欄序回答。","answer":{"kind":"expression","value":"([3, 2, 8, 7], [0, 0], [1, 1], 是, 是, 是)","explanation_zh":"Wave4收束核對=([3, 2, 8, 7], [0, 0], [1, 1], 是, 是, 是)；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-well-defined","seed":202610314079,"parameters":{"n":10,"d":5,"a":3,"b":2,"a_prime":8,"b_prime":7,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-well-defined","semantic_template":"qf-mod31-quotring-well-defined/{\"a\":3,\"a_prime\":8,\"b\":2,\"b_prime\":7,\"d\":5,\"n\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0080","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-well-defined；資料={\"n\":12,\"d\":6,\"a\":4,\"b\":3,\"a_prime\":10,\"b_prime\":9}。請依 [representatives,addition_classes,multiplication_classes,addition_well_defined,multiplication_well_defined,quotient_operations_well_defined] 欄序回答。","answer":{"kind":"expression","value":"([4, 3, 10, 9], [1, 1], [0, 0], 是, 是, 是)","explanation_zh":"Wave4收束核對=([4, 3, 10, 9], [1, 1], [0, 0], 是, 是, 是)；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-well-defined","seed":202610314080,"parameters":{"n":12,"d":6,"a":4,"b":3,"a_prime":10,"b_prime":9,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-well-defined","semantic_template":"qf-mod31-quotring-well-defined/{\"a\":4,\"a_prime\":10,\"b\":3,\"b_prime\":9,\"d\":6,\"n\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0081","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-well-defined；資料={\"n\":6,\"d\":2,\"a\":1,\"b\":1,\"a_prime\":3,\"b_prime\":3}。請依 [representatives,addition_classes,multiplication_classes,addition_well_defined,multiplication_well_defined,quotient_operations_well_defined] 欄序回答。","answer":{"kind":"expression","value":"([1, 1, 3, 3], [0, 0], [1, 1], 是, 是, 是)","explanation_zh":"Wave4收束核對=([1, 1, 3, 3], [0, 0], [1, 1], 是, 是, 是)；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-well-defined","seed":202610314081,"parameters":{"n":6,"d":2,"a":1,"b":1,"a_prime":3,"b_prime":3,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-well-defined","semantic_template":"qf-mod31-quotring-well-defined/{\"a\":1,\"a_prime\":3,\"b\":1,\"b_prime\":3,\"d\":2,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0082","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-well-defined；資料={\"n\":9,\"d\":3,\"a\":0,\"b\":1,\"a_prime\":3,\"b_prime\":4}。請依 [representatives,addition_classes,multiplication_classes,addition_well_defined,multiplication_well_defined,quotient_operations_well_defined] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 3, 4], [1, 1], [0, 0], 是, 是, 是)","explanation_zh":"Wave4收束核對=([0, 1, 3, 4], [1, 1], [0, 0], 是, 是, 是)；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-well-defined","seed":202610314082,"parameters":{"n":9,"d":3,"a":0,"b":1,"a_prime":3,"b_prime":4,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-well-defined","semantic_template":"qf-mod31-quotring-well-defined/{\"a\":0,\"a_prime\":3,\"b\":1,\"b_prime\":4,\"d\":3,\"n\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0083","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-well-defined；資料={\"n\":12,\"d\":4,\"a\":3,\"b\":3,\"a_prime\":7,\"b_prime\":7}。請依 [representatives,addition_classes,multiplication_classes,addition_well_defined,multiplication_well_defined,quotient_operations_well_defined] 欄序回答。","answer":{"kind":"expression","value":"([3, 3, 7, 7], [2, 2], [1, 1], 是, 是, 是)","explanation_zh":"Wave4收束核對=([3, 3, 7, 7], [2, 2], [1, 1], 是, 是, 是)；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-well-defined","seed":202610314083,"parameters":{"n":12,"d":4,"a":3,"b":3,"a_prime":7,"b_prime":7,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-well-defined","semantic_template":"qf-mod31-quotring-well-defined/{\"a\":3,\"a_prime\":7,\"b\":3,\"b_prime\":7,\"d\":4,\"n\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0084","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-well-defined；資料={\"n\":15,\"d\":5,\"a\":3,\"b\":2,\"a_prime\":8,\"b_prime\":7}。請依 [representatives,addition_classes,multiplication_classes,addition_well_defined,multiplication_well_defined,quotient_operations_well_defined] 欄序回答。","answer":{"kind":"expression","value":"([3, 2, 8, 7], [0, 0], [1, 1], 是, 是, 是)","explanation_zh":"Wave4收束核對=([3, 2, 8, 7], [0, 0], [1, 1], 是, 是, 是)；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-well-defined","seed":202610314084,"parameters":{"n":15,"d":5,"a":3,"b":2,"a_prime":8,"b_prime":7,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-well-defined","semantic_template":"qf-mod31-quotring-well-defined/{\"a\":3,\"a_prime\":8,\"b\":2,\"b_prime\":7,\"d\":5,\"n\":15}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0085","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-well-defined；資料={\"n\":18,\"d\":6,\"a\":3,\"b\":1,\"a_prime\":9,\"b_prime\":7}。請依 [representatives,addition_classes,multiplication_classes,addition_well_defined,multiplication_well_defined,quotient_operations_well_defined] 欄序回答。","answer":{"kind":"expression","value":"([3, 1, 9, 7], [4, 4], [3, 3], 是, 是, 是)","explanation_zh":"Wave4收束核對=([3, 1, 9, 7], [4, 4], [3, 3], 是, 是, 是)；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-well-defined","seed":202610314085,"parameters":{"n":18,"d":6,"a":3,"b":1,"a_prime":9,"b_prime":7,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-well-defined","semantic_template":"qf-mod31-quotring-well-defined/{\"a\":3,\"a_prime\":9,\"b\":1,\"b_prime\":7,\"d\":6,\"n\":18}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0086","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-well-defined；資料={\"n\":8,\"d\":2,\"a\":0,\"b\":1,\"a_prime\":2,\"b_prime\":3}。請依 [representatives,addition_classes,multiplication_classes,addition_well_defined,multiplication_well_defined,quotient_operations_well_defined] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3], [1, 1], [0, 0], 是, 是, 是)","explanation_zh":"Wave4收束核對=([0, 1, 2, 3], [1, 1], [0, 0], 是, 是, 是)；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-well-defined","seed":202610314086,"parameters":{"n":8,"d":2,"a":0,"b":1,"a_prime":2,"b_prime":3,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-well-defined","semantic_template":"qf-mod31-quotring-well-defined/{\"a\":0,\"a_prime\":2,\"b\":1,\"b_prime\":3,\"d\":2,\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0087","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-well-defined；資料={\"n\":12,\"d\":3,\"a\":2,\"b\":2,\"a_prime\":5,\"b_prime\":5}。請依 [representatives,addition_classes,multiplication_classes,addition_well_defined,multiplication_well_defined,quotient_operations_well_defined] 欄序回答。","answer":{"kind":"expression","value":"([2, 2, 5, 5], [1, 1], [1, 1], 是, 是, 是)","explanation_zh":"Wave4收束核對=([2, 2, 5, 5], [1, 1], [1, 1], 是, 是, 是)；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-well-defined","seed":202610314087,"parameters":{"n":12,"d":3,"a":2,"b":2,"a_prime":5,"b_prime":5,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-well-defined","semantic_template":"qf-mod31-quotring-well-defined/{\"a\":2,\"a_prime\":5,\"b\":2,\"b_prime\":5,\"d\":3,\"n\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0088","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-well-defined；資料={\"n\":16,\"d\":4,\"a\":0,\"b\":1,\"a_prime\":4,\"b_prime\":5}。請依 [representatives,addition_classes,multiplication_classes,addition_well_defined,multiplication_well_defined,quotient_operations_well_defined] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 4, 5], [1, 1], [0, 0], 是, 是, 是)","explanation_zh":"Wave4收束核對=([0, 1, 4, 5], [1, 1], [0, 0], 是, 是, 是)；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-well-defined","seed":202610314088,"parameters":{"n":16,"d":4,"a":0,"b":1,"a_prime":4,"b_prime":5,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-well-defined","semantic_template":"qf-mod31-quotring-well-defined/{\"a\":0,\"a_prime\":4,\"b\":1,\"b_prime\":5,\"d\":4,\"n\":16}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0089","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-well-defined；資料={\"n\":20,\"d\":5,\"a\":3,\"b\":2,\"a_prime\":8,\"b_prime\":7}。請依 [representatives,addition_classes,multiplication_classes,addition_well_defined,multiplication_well_defined,quotient_operations_well_defined] 欄序回答。","answer":{"kind":"expression","value":"([3, 2, 8, 7], [0, 0], [1, 1], 是, 是, 是)","explanation_zh":"Wave4收束核對=([3, 2, 8, 7], [0, 0], [1, 1], 是, 是, 是)；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-well-defined","seed":202610314089,"parameters":{"n":20,"d":5,"a":3,"b":2,"a_prime":8,"b_prime":7,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-well-defined","semantic_template":"qf-mod31-quotring-well-defined/{\"a\":3,\"a_prime\":8,\"b\":2,\"b_prime\":7,\"d\":5,\"n\":20}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0090","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-well-defined；資料={\"n\":24,\"d\":6,\"a\":2,\"b\":5,\"a_prime\":8,\"b_prime\":11}。請依 [representatives,addition_classes,multiplication_classes,addition_well_defined,multiplication_well_defined,quotient_operations_well_defined] 欄序回答。","answer":{"kind":"expression","value":"([2, 5, 8, 11], [1, 1], [4, 4], 是, 是, 是)","explanation_zh":"Wave4收束核對=([2, 5, 8, 11], [1, 1], [4, 4], 是, 是, 是)；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-well-defined","seed":202610314090,"parameters":{"n":24,"d":6,"a":2,"b":5,"a_prime":8,"b_prime":11,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-well-defined","semantic_template":"qf-mod31-quotring-well-defined/{\"a\":2,\"a_prime\":8,\"b\":5,\"b_prime\":11,\"d\":6,\"n\":24}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0091","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-well-defined；資料={\"n\":10,\"d\":2,\"a\":1,\"b\":1,\"a_prime\":3,\"b_prime\":3}。請依 [representatives,addition_classes,multiplication_classes,addition_well_defined,multiplication_well_defined,quotient_operations_well_defined] 欄序回答。","answer":{"kind":"expression","value":"([1, 1, 3, 3], [0, 0], [1, 1], 是, 是, 是)","explanation_zh":"Wave4收束核對=([1, 1, 3, 3], [0, 0], [1, 1], 是, 是, 是)；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-well-defined","seed":202610314091,"parameters":{"n":10,"d":2,"a":1,"b":1,"a_prime":3,"b_prime":3,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-well-defined","semantic_template":"qf-mod31-quotring-well-defined/{\"a\":1,\"a_prime\":3,\"b\":1,\"b_prime\":3,\"d\":2,\"n\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0092","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-well-defined；資料={\"n\":15,\"d\":3,\"a\":1,\"b\":0,\"a_prime\":4,\"b_prime\":3}。請依 [representatives,addition_classes,multiplication_classes,addition_well_defined,multiplication_well_defined,quotient_operations_well_defined] 欄序回答。","answer":{"kind":"expression","value":"([1, 0, 4, 3], [1, 1], [0, 0], 是, 是, 是)","explanation_zh":"Wave4收束核對=([1, 0, 4, 3], [1, 1], [0, 0], 是, 是, 是)；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-well-defined","seed":202610314092,"parameters":{"n":15,"d":3,"a":1,"b":0,"a_prime":4,"b_prime":3,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-well-defined","semantic_template":"qf-mod31-quotring-well-defined/{\"a\":1,\"a_prime\":4,\"b\":0,\"b_prime\":3,\"d\":3,\"n\":15}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0093","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-well-defined；資料={\"n\":20,\"d\":4,\"a\":1,\"b\":3,\"a_prime\":5,\"b_prime\":7}。請依 [representatives,addition_classes,multiplication_classes,addition_well_defined,multiplication_well_defined,quotient_operations_well_defined] 欄序回答。","answer":{"kind":"expression","value":"([1, 3, 5, 7], [0, 0], [3, 3], 是, 是, 是)","explanation_zh":"Wave4收束核對=([1, 3, 5, 7], [0, 0], [3, 3], 是, 是, 是)；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-well-defined","seed":202610314093,"parameters":{"n":20,"d":4,"a":1,"b":3,"a_prime":5,"b_prime":7,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-well-defined","semantic_template":"qf-mod31-quotring-well-defined/{\"a\":1,\"a_prime\":5,\"b\":3,\"b_prime\":7,\"d\":4,\"n\":20}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0094","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-well-defined；資料={\"n\":25,\"d\":5,\"a\":3,\"b\":2,\"a_prime\":8,\"b_prime\":7}。請依 [representatives,addition_classes,multiplication_classes,addition_well_defined,multiplication_well_defined,quotient_operations_well_defined] 欄序回答。","answer":{"kind":"expression","value":"([3, 2, 8, 7], [0, 0], [1, 1], 是, 是, 是)","explanation_zh":"Wave4收束核對=([3, 2, 8, 7], [0, 0], [1, 1], 是, 是, 是)；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-well-defined","seed":202610314094,"parameters":{"n":25,"d":5,"a":3,"b":2,"a_prime":8,"b_prime":7,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-well-defined","semantic_template":"qf-mod31-quotring-well-defined/{\"a\":3,\"a_prime\":8,\"b\":2,\"b_prime\":7,\"d\":5,\"n\":25}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0095","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-well-defined；資料={\"n\":30,\"d\":6,\"a\":1,\"b\":3,\"a_prime\":7,\"b_prime\":9}。請依 [representatives,addition_classes,multiplication_classes,addition_well_defined,multiplication_well_defined,quotient_operations_well_defined] 欄序回答。","answer":{"kind":"expression","value":"([1, 3, 7, 9], [4, 4], [3, 3], 是, 是, 是)","explanation_zh":"Wave4收束核對=([1, 3, 7, 9], [4, 4], [3, 3], 是, 是, 是)；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-well-defined","seed":202610314095,"parameters":{"n":30,"d":6,"a":1,"b":3,"a_prime":7,"b_prime":9,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-well-defined","semantic_template":"qf-mod31-quotring-well-defined/{\"a\":1,\"a_prime\":7,\"b\":3,\"b_prime\":9,\"d\":6,\"n\":30}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0096","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-well-defined；資料={\"n\":12,\"d\":2,\"a\":0,\"b\":1,\"a_prime\":2,\"b_prime\":3}。請依 [representatives,addition_classes,multiplication_classes,addition_well_defined,multiplication_well_defined,quotient_operations_well_defined] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3], [1, 1], [0, 0], 是, 是, 是)","explanation_zh":"Wave4收束核對=([0, 1, 2, 3], [1, 1], [0, 0], 是, 是, 是)；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-well-defined","seed":202610314096,"parameters":{"n":12,"d":2,"a":0,"b":1,"a_prime":2,"b_prime":3,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-well-defined","semantic_template":"qf-mod31-quotring-well-defined/{\"a\":0,\"a_prime\":2,\"b\":1,\"b_prime\":3,\"d\":2,\"n\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0097","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-well-defined；資料={\"n\":18,\"d\":3,\"a\":0,\"b\":1,\"a_prime\":3,\"b_prime\":4}。請依 [representatives,addition_classes,multiplication_classes,addition_well_defined,multiplication_well_defined,quotient_operations_well_defined] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 3, 4], [1, 1], [0, 0], 是, 是, 是)","explanation_zh":"Wave4收束核對=([0, 1, 3, 4], [1, 1], [0, 0], 是, 是, 是)；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-well-defined","seed":202610314097,"parameters":{"n":18,"d":3,"a":0,"b":1,"a_prime":3,"b_prime":4,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-well-defined","semantic_template":"qf-mod31-quotring-well-defined/{\"a\":0,\"a_prime\":3,\"b\":1,\"b_prime\":4,\"d\":3,\"n\":18}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0098","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-well-defined；資料={\"n\":24,\"d\":4,\"a\":2,\"b\":1,\"a_prime\":6,\"b_prime\":5}。請依 [representatives,addition_classes,multiplication_classes,addition_well_defined,multiplication_well_defined,quotient_operations_well_defined] 欄序回答。","answer":{"kind":"expression","value":"([2, 1, 6, 5], [3, 3], [2, 2], 是, 是, 是)","explanation_zh":"Wave4收束核對=([2, 1, 6, 5], [3, 3], [2, 2], 是, 是, 是)；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-well-defined","seed":202610314098,"parameters":{"n":24,"d":4,"a":2,"b":1,"a_prime":6,"b_prime":5,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-well-defined","semantic_template":"qf-mod31-quotring-well-defined/{\"a\":2,\"a_prime\":6,\"b\":1,\"b_prime\":5,\"d\":4,\"n\":24}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0099","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-well-defined；資料={\"n\":30,\"d\":5,\"a\":3,\"b\":2,\"a_prime\":8,\"b_prime\":7}。請依 [representatives,addition_classes,multiplication_classes,addition_well_defined,multiplication_well_defined,quotient_operations_well_defined] 欄序回答。","answer":{"kind":"expression","value":"([3, 2, 8, 7], [0, 0], [1, 1], 是, 是, 是)","explanation_zh":"Wave4收束核對=([3, 2, 8, 7], [0, 0], [1, 1], 是, 是, 是)；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-well-defined","seed":202610314099,"parameters":{"n":30,"d":5,"a":3,"b":2,"a_prime":8,"b_prime":7,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-well-defined","semantic_template":"qf-mod31-quotring-well-defined/{\"a\":3,\"a_prime\":8,\"b\":2,\"b_prime\":7,\"d\":5,\"n\":30}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r04-finite-quotient-ring-0100","batch_id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-quotient-ring"],"title_zh":"有限商環","stem_zh":"Z_n與proper雙邊理想的全部加法cosets。操作=quotring-well-defined；資料={\"n\":36,\"d\":6,\"a\":0,\"b\":1,\"a_prime\":6,\"b_prime\":7}。請依 [representatives,addition_classes,multiplication_classes,addition_well_defined,multiplication_well_defined,quotient_operations_well_defined] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 6, 7], [1, 1], [0, 0], 是, 是, 是)","explanation_zh":"Wave4收束核對=([0, 1, 6, 7], [1, 1], [0, 0], 是, 是, 是)；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限商環"]},"variant":{"family_id":"qf-mod31-quotring-well-defined","seed":202610314100,"parameters":{"n":36,"d":6,"a":0,"b":1,"a_prime":6,"b_prime":7,"domain":"Exact_quotient_of_Zn_by_proper_ideal","operation":"quotring-well-defined","semantic_template":"qf-mod31-quotring-well-defined/{\"a\":0,\"a_prime\":6,\"b\":1,\"b_prime\":7,\"d\":6,\"n\":36}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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"}}