{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0001","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-classes；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[0,3],[0,4],[0,5],[1,0],[1,1],[1,2],[1,3],[1,4],[1,5],[2,0],[2,1],[2,2],[2,3],[2,4],[2,5],[3,0],[3,1],[3,2],[3,3],[3,4],[3,5],[4,0],[4,1],[4,2],[4,3],[4,4],[4,5],[5,0],[5,1],[5,2],[5,3],[5,4],[5,5]]}。請依 [classes,count] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5]], 1)","explanation_zh":"Wave4第四層核對=([[0, 1, 2, 3, 4, 5]], 1)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-classes","seed":202610304001,"parameters":{"n":6,"pairs":[[0,0],[0,1],[0,2],[0,3],[0,4],[0,5],[1,0],[1,1],[1,2],[1,3],[1,4],[1,5],[2,0],[2,1],[2,2],[2,3],[2,4],[2,5],[3,0],[3,1],[3,2],[3,3],[3,4],[3,5],[4,0],[4,1],[4,2],[4,3],[4,4],[4,5],[5,0],[5,1],[5,2],[5,3],[5,4],[5,5]],"domain":"Complete_finite_quotient_set","operation":"quotient-classes","semantic_template":"qf-mod30-quotient-classes/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[0,3],[0,4],[0,5],[1,0],[1,1],[1,2],[1,3],[1,4],[1,5],[2,0],[2,1],[2,2],[2,3],[2,4],[2,5],[3,0],[3,1],[3,2],[3,3],[3,4],[3,5],[4,0],[4,1],[4,2],[4,3],[4,4],[4,5],[5,0],[5,1],[5,2],[5,3],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0002","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-classes；資料={\"n\":6,\"pairs\":[[0,0],[1,1],[1,2],[1,3],[1,4],[1,5],[2,1],[2,2],[2,3],[2,4],[2,5],[3,1],[3,2],[3,3],[3,4],[3,5],[4,1],[4,2],[4,3],[4,4],[4,5],[5,1],[5,2],[5,3],[5,4],[5,5]]}。請依 [classes,count] 欄序回答。","answer":{"kind":"expression","value":"([[0], [1, 2, 3, 4, 5]], 2)","explanation_zh":"Wave4第四層核對=([[0], [1, 2, 3, 4, 5]], 2)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-classes","seed":202610304002,"parameters":{"n":6,"pairs":[[0,0],[1,1],[1,2],[1,3],[1,4],[1,5],[2,1],[2,2],[2,3],[2,4],[2,5],[3,1],[3,2],[3,3],[3,4],[3,5],[4,1],[4,2],[4,3],[4,4],[4,5],[5,1],[5,2],[5,3],[5,4],[5,5]],"domain":"Complete_finite_quotient_set","operation":"quotient-classes","semantic_template":"qf-mod30-quotient-classes/{\"n\":6,\"pairs\":[[0,0],[1,1],[1,2],[1,3],[1,4],[1,5],[2,1],[2,2],[2,3],[2,4],[2,5],[3,1],[3,2],[3,3],[3,4],[3,5],[4,1],[4,2],[4,3],[4,4],[4,5],[5,1],[5,2],[5,3],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0003","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-classes；資料={\"n\":6,\"pairs\":[[0,0],[0,2],[0,3],[0,4],[0,5],[1,1],[2,0],[2,2],[2,3],[2,4],[2,5],[3,0],[3,2],[3,3],[3,4],[3,5],[4,0],[4,2],[4,3],[4,4],[4,5],[5,0],[5,2],[5,3],[5,4],[5,5]]}。請依 [classes,count] 欄序回答。","answer":{"kind":"expression","value":"([[0, 2, 3, 4, 5], [1]], 2)","explanation_zh":"Wave4第四層核對=([[0, 2, 3, 4, 5], [1]], 2)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-classes","seed":202610304003,"parameters":{"n":6,"pairs":[[0,0],[0,2],[0,3],[0,4],[0,5],[1,1],[2,0],[2,2],[2,3],[2,4],[2,5],[3,0],[3,2],[3,3],[3,4],[3,5],[4,0],[4,2],[4,3],[4,4],[4,5],[5,0],[5,2],[5,3],[5,4],[5,5]],"domain":"Complete_finite_quotient_set","operation":"quotient-classes","semantic_template":"qf-mod30-quotient-classes/{\"n\":6,\"pairs\":[[0,0],[0,2],[0,3],[0,4],[0,5],[1,1],[2,0],[2,2],[2,3],[2,4],[2,5],[3,0],[3,2],[3,3],[3,4],[3,5],[4,0],[4,2],[4,3],[4,4],[4,5],[5,0],[5,2],[5,3],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0004","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-classes；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[2,3],[2,4],[2,5],[3,2],[3,3],[3,4],[3,5],[4,2],[4,3],[4,4],[4,5],[5,2],[5,3],[5,4],[5,5]]}。請依 [classes,count] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1], [2, 3, 4, 5]], 2)","explanation_zh":"Wave4第四層核對=([[0, 1], [2, 3, 4, 5]], 2)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-classes","seed":202610304004,"parameters":{"n":6,"pairs":[[0,0],[0,1],[1,0],[1,1],[2,2],[2,3],[2,4],[2,5],[3,2],[3,3],[3,4],[3,5],[4,2],[4,3],[4,4],[4,5],[5,2],[5,3],[5,4],[5,5]],"domain":"Complete_finite_quotient_set","operation":"quotient-classes","semantic_template":"qf-mod30-quotient-classes/{\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[2,3],[2,4],[2,5],[3,2],[3,3],[3,4],[3,5],[4,2],[4,3],[4,4],[4,5],[5,2],[5,3],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0005","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-classes；資料={\"n\":6,\"pairs\":[[0,0],[1,1],[2,2],[2,3],[2,4],[2,5],[3,2],[3,3],[3,4],[3,5],[4,2],[4,3],[4,4],[4,5],[5,2],[5,3],[5,4],[5,5]]}。請依 [classes,count] 欄序回答。","answer":{"kind":"expression","value":"([[0], [1], [2, 3, 4, 5]], 3)","explanation_zh":"Wave4第四層核對=([[0], [1], [2, 3, 4, 5]], 3)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-classes","seed":202610304005,"parameters":{"n":6,"pairs":[[0,0],[1,1],[2,2],[2,3],[2,4],[2,5],[3,2],[3,3],[3,4],[3,5],[4,2],[4,3],[4,4],[4,5],[5,2],[5,3],[5,4],[5,5]],"domain":"Complete_finite_quotient_set","operation":"quotient-classes","semantic_template":"qf-mod30-quotient-classes/{\"n\":6,\"pairs\":[[0,0],[1,1],[2,2],[2,3],[2,4],[2,5],[3,2],[3,3],[3,4],[3,5],[4,2],[4,3],[4,4],[4,5],[5,2],[5,3],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0006","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-classes；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[0,3],[0,4],[0,5],[1,0],[1,1],[1,3],[1,4],[1,5],[2,2],[3,0],[3,1],[3,3],[3,4],[3,5],[4,0],[4,1],[4,3],[4,4],[4,5],[5,0],[5,1],[5,3],[5,4],[5,5]]}。請依 [classes,count] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 3, 4, 5], [2]], 2)","explanation_zh":"Wave4第四層核對=([[0, 1, 3, 4, 5], [2]], 2)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-classes","seed":202610304006,"parameters":{"n":6,"pairs":[[0,0],[0,1],[0,3],[0,4],[0,5],[1,0],[1,1],[1,3],[1,4],[1,5],[2,2],[3,0],[3,1],[3,3],[3,4],[3,5],[4,0],[4,1],[4,3],[4,4],[4,5],[5,0],[5,1],[5,3],[5,4],[5,5]],"domain":"Complete_finite_quotient_set","operation":"quotient-classes","semantic_template":"qf-mod30-quotient-classes/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,3],[0,4],[0,5],[1,0],[1,1],[1,3],[1,4],[1,5],[2,2],[3,0],[3,1],[3,3],[3,4],[3,5],[4,0],[4,1],[4,3],[4,4],[4,5],[5,0],[5,1],[5,3],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0007","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-classes；資料={\"n\":6,\"pairs\":[[0,0],[0,2],[1,1],[1,3],[1,4],[1,5],[2,0],[2,2],[3,1],[3,3],[3,4],[3,5],[4,1],[4,3],[4,4],[4,5],[5,1],[5,3],[5,4],[5,5]]}。請依 [classes,count] 欄序回答。","answer":{"kind":"expression","value":"([[0, 2], [1, 3, 4, 5]], 2)","explanation_zh":"Wave4第四層核對=([[0, 2], [1, 3, 4, 5]], 2)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-classes","seed":202610304007,"parameters":{"n":6,"pairs":[[0,0],[0,2],[1,1],[1,3],[1,4],[1,5],[2,0],[2,2],[3,1],[3,3],[3,4],[3,5],[4,1],[4,3],[4,4],[4,5],[5,1],[5,3],[5,4],[5,5]],"domain":"Complete_finite_quotient_set","operation":"quotient-classes","semantic_template":"qf-mod30-quotient-classes/{\"n\":6,\"pairs\":[[0,0],[0,2],[1,1],[1,3],[1,4],[1,5],[2,0],[2,2],[3,1],[3,3],[3,4],[3,5],[4,1],[4,3],[4,4],[4,5],[5,1],[5,3],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0008","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-classes；資料={\"n\":6,\"pairs\":[[0,0],[1,1],[1,3],[1,4],[1,5],[2,2],[3,1],[3,3],[3,4],[3,5],[4,1],[4,3],[4,4],[4,5],[5,1],[5,3],[5,4],[5,5]]}。請依 [classes,count] 欄序回答。","answer":{"kind":"expression","value":"([[0], [1, 3, 4, 5], [2]], 3)","explanation_zh":"Wave4第四層核對=([[0], [1, 3, 4, 5], [2]], 3)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-classes","seed":202610304008,"parameters":{"n":6,"pairs":[[0,0],[1,1],[1,3],[1,4],[1,5],[2,2],[3,1],[3,3],[3,4],[3,5],[4,1],[4,3],[4,4],[4,5],[5,1],[5,3],[5,4],[5,5]],"domain":"Complete_finite_quotient_set","operation":"quotient-classes","semantic_template":"qf-mod30-quotient-classes/{\"n\":6,\"pairs\":[[0,0],[1,1],[1,3],[1,4],[1,5],[2,2],[3,1],[3,3],[3,4],[3,5],[4,1],[4,3],[4,4],[4,5],[5,1],[5,3],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0009","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-classes；資料={\"n\":6,\"pairs\":[[0,0],[0,3],[0,4],[0,5],[1,1],[1,2],[2,1],[2,2],[3,0],[3,3],[3,4],[3,5],[4,0],[4,3],[4,4],[4,5],[5,0],[5,3],[5,4],[5,5]]}。請依 [classes,count] 欄序回答。","answer":{"kind":"expression","value":"([[0, 3, 4, 5], [1, 2]], 2)","explanation_zh":"Wave4第四層核對=([[0, 3, 4, 5], [1, 2]], 2)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-classes","seed":202610304009,"parameters":{"n":6,"pairs":[[0,0],[0,3],[0,4],[0,5],[1,1],[1,2],[2,1],[2,2],[3,0],[3,3],[3,4],[3,5],[4,0],[4,3],[4,4],[4,5],[5,0],[5,3],[5,4],[5,5]],"domain":"Complete_finite_quotient_set","operation":"quotient-classes","semantic_template":"qf-mod30-quotient-classes/{\"n\":6,\"pairs\":[[0,0],[0,3],[0,4],[0,5],[1,1],[1,2],[2,1],[2,2],[3,0],[3,3],[3,4],[3,5],[4,0],[4,3],[4,4],[4,5],[5,0],[5,3],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0010","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-classes；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[1,0],[1,1],[1,2],[2,0],[2,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]]}。請依 [classes,count] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 2], [3, 4, 5]], 2)","explanation_zh":"Wave4第四層核對=([[0, 1, 2], [3, 4, 5]], 2)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-classes","seed":202610304010,"parameters":{"n":6,"pairs":[[0,0],[0,1],[0,2],[1,0],[1,1],[1,2],[2,0],[2,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]],"domain":"Complete_finite_quotient_set","operation":"quotient-classes","semantic_template":"qf-mod30-quotient-classes/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[1,0],[1,1],[1,2],[2,0],[2,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0011","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-classes；資料={\"n\":6,\"pairs\":[[0,0],[1,1],[1,2],[2,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]]}。請依 [classes,count] 欄序回答。","answer":{"kind":"expression","value":"([[0], [1, 2], [3, 4, 5]], 3)","explanation_zh":"Wave4第四層核對=([[0], [1, 2], [3, 4, 5]], 3)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-classes","seed":202610304011,"parameters":{"n":6,"pairs":[[0,0],[1,1],[1,2],[2,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]],"domain":"Complete_finite_quotient_set","operation":"quotient-classes","semantic_template":"qf-mod30-quotient-classes/{\"n\":6,\"pairs\":[[0,0],[1,1],[1,2],[2,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0012","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-classes；資料={\"n\":6,\"pairs\":[[0,0],[0,3],[0,4],[0,5],[1,1],[2,2],[3,0],[3,3],[3,4],[3,5],[4,0],[4,3],[4,4],[4,5],[5,0],[5,3],[5,4],[5,5]]}。請依 [classes,count] 欄序回答。","answer":{"kind":"expression","value":"([[0, 3, 4, 5], [1], [2]], 3)","explanation_zh":"Wave4第四層核對=([[0, 3, 4, 5], [1], [2]], 3)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-classes","seed":202610304012,"parameters":{"n":6,"pairs":[[0,0],[0,3],[0,4],[0,5],[1,1],[2,2],[3,0],[3,3],[3,4],[3,5],[4,0],[4,3],[4,4],[4,5],[5,0],[5,3],[5,4],[5,5]],"domain":"Complete_finite_quotient_set","operation":"quotient-classes","semantic_template":"qf-mod30-quotient-classes/{\"n\":6,\"pairs\":[[0,0],[0,3],[0,4],[0,5],[1,1],[2,2],[3,0],[3,3],[3,4],[3,5],[4,0],[4,3],[4,4],[4,5],[5,0],[5,3],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0013","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-classes；資料={\"n\":6,\"pairs\":[[0,0],[0,2],[1,1],[2,0],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]]}。請依 [classes,count] 欄序回答。","answer":{"kind":"expression","value":"([[0, 2], [1], [3, 4, 5]], 3)","explanation_zh":"Wave4第四層核對=([[0, 2], [1], [3, 4, 5]], 3)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-classes","seed":202610304013,"parameters":{"n":6,"pairs":[[0,0],[0,2],[1,1],[2,0],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]],"domain":"Complete_finite_quotient_set","operation":"quotient-classes","semantic_template":"qf-mod30-quotient-classes/{\"n\":6,\"pairs\":[[0,0],[0,2],[1,1],[2,0],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0014","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-classes；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]]}。請依 [classes,count] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1], [2], [3, 4, 5]], 3)","explanation_zh":"Wave4第四層核對=([[0, 1], [2], [3, 4, 5]], 3)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-classes","seed":202610304014,"parameters":{"n":6,"pairs":[[0,0],[0,1],[1,0],[1,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]],"domain":"Complete_finite_quotient_set","operation":"quotient-classes","semantic_template":"qf-mod30-quotient-classes/{\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0015","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-classes；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[0,4],[0,5],[1,0],[1,1],[1,2],[1,4],[1,5],[2,0],[2,1],[2,2],[2,4],[2,5],[3,3],[4,0],[4,1],[4,2],[4,4],[4,5],[5,0],[5,1],[5,2],[5,4],[5,5]]}。請依 [classes,count] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 2, 4, 5], [3]], 2)","explanation_zh":"Wave4第四層核對=([[0, 1, 2, 4, 5], [3]], 2)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-classes","seed":202610304015,"parameters":{"n":6,"pairs":[[0,0],[0,1],[0,2],[0,4],[0,5],[1,0],[1,1],[1,2],[1,4],[1,5],[2,0],[2,1],[2,2],[2,4],[2,5],[3,3],[4,0],[4,1],[4,2],[4,4],[4,5],[5,0],[5,1],[5,2],[5,4],[5,5]],"domain":"Complete_finite_quotient_set","operation":"quotient-classes","semantic_template":"qf-mod30-quotient-classes/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[0,4],[0,5],[1,0],[1,1],[1,2],[1,4],[1,5],[2,0],[2,1],[2,2],[2,4],[2,5],[3,3],[4,0],[4,1],[4,2],[4,4],[4,5],[5,0],[5,1],[5,2],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0016","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-classes；資料={\"n\":6,\"pairs\":[[0,0],[0,3],[1,1],[1,2],[1,4],[1,5],[2,1],[2,2],[2,4],[2,5],[3,0],[3,3],[4,1],[4,2],[4,4],[4,5],[5,1],[5,2],[5,4],[5,5]]}。請依 [classes,count] 欄序回答。","answer":{"kind":"expression","value":"([[0, 3], [1, 2, 4, 5]], 2)","explanation_zh":"Wave4第四層核對=([[0, 3], [1, 2, 4, 5]], 2)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-classes","seed":202610304016,"parameters":{"n":6,"pairs":[[0,0],[0,3],[1,1],[1,2],[1,4],[1,5],[2,1],[2,2],[2,4],[2,5],[3,0],[3,3],[4,1],[4,2],[4,4],[4,5],[5,1],[5,2],[5,4],[5,5]],"domain":"Complete_finite_quotient_set","operation":"quotient-classes","semantic_template":"qf-mod30-quotient-classes/{\"n\":6,\"pairs\":[[0,0],[0,3],[1,1],[1,2],[1,4],[1,5],[2,1],[2,2],[2,4],[2,5],[3,0],[3,3],[4,1],[4,2],[4,4],[4,5],[5,1],[5,2],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0017","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-classes；資料={\"n\":6,\"pairs\":[[0,0],[1,1],[1,2],[1,4],[1,5],[2,1],[2,2],[2,4],[2,5],[3,3],[4,1],[4,2],[4,4],[4,5],[5,1],[5,2],[5,4],[5,5]]}。請依 [classes,count] 欄序回答。","answer":{"kind":"expression","value":"([[0], [1, 2, 4, 5], [3]], 3)","explanation_zh":"Wave4第四層核對=([[0], [1, 2, 4, 5], [3]], 3)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-classes","seed":202610304017,"parameters":{"n":6,"pairs":[[0,0],[1,1],[1,2],[1,4],[1,5],[2,1],[2,2],[2,4],[2,5],[3,3],[4,1],[4,2],[4,4],[4,5],[5,1],[5,2],[5,4],[5,5]],"domain":"Complete_finite_quotient_set","operation":"quotient-classes","semantic_template":"qf-mod30-quotient-classes/{\"n\":6,\"pairs\":[[0,0],[1,1],[1,2],[1,4],[1,5],[2,1],[2,2],[2,4],[2,5],[3,3],[4,1],[4,2],[4,4],[4,5],[5,1],[5,2],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0018","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-classes；資料={\"n\":6,\"pairs\":[[0,0],[0,2],[0,4],[0,5],[1,1],[1,3],[2,0],[2,2],[2,4],[2,5],[3,1],[3,3],[4,0],[4,2],[4,4],[4,5],[5,0],[5,2],[5,4],[5,5]]}。請依 [classes,count] 欄序回答。","answer":{"kind":"expression","value":"([[0, 2, 4, 5], [1, 3]], 2)","explanation_zh":"Wave4第四層核對=([[0, 2, 4, 5], [1, 3]], 2)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-classes","seed":202610304018,"parameters":{"n":6,"pairs":[[0,0],[0,2],[0,4],[0,5],[1,1],[1,3],[2,0],[2,2],[2,4],[2,5],[3,1],[3,3],[4,0],[4,2],[4,4],[4,5],[5,0],[5,2],[5,4],[5,5]],"domain":"Complete_finite_quotient_set","operation":"quotient-classes","semantic_template":"qf-mod30-quotient-classes/{\"n\":6,\"pairs\":[[0,0],[0,2],[0,4],[0,5],[1,1],[1,3],[2,0],[2,2],[2,4],[2,5],[3,1],[3,3],[4,0],[4,2],[4,4],[4,5],[5,0],[5,2],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0019","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-classes；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[0,3],[1,0],[1,1],[1,3],[2,2],[2,4],[2,5],[3,0],[3,1],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]]}。請依 [classes,count] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 3], [2, 4, 5]], 2)","explanation_zh":"Wave4第四層核對=([[0, 1, 3], [2, 4, 5]], 2)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-classes","seed":202610304019,"parameters":{"n":6,"pairs":[[0,0],[0,1],[0,3],[1,0],[1,1],[1,3],[2,2],[2,4],[2,5],[3,0],[3,1],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]],"domain":"Complete_finite_quotient_set","operation":"quotient-classes","semantic_template":"qf-mod30-quotient-classes/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,3],[1,0],[1,1],[1,3],[2,2],[2,4],[2,5],[3,0],[3,1],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0020","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-classes；資料={\"n\":6,\"pairs\":[[0,0],[1,1],[1,3],[2,2],[2,4],[2,5],[3,1],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]]}。請依 [classes,count] 欄序回答。","answer":{"kind":"expression","value":"([[0], [1, 3], [2, 4, 5]], 3)","explanation_zh":"Wave4第四層核對=([[0], [1, 3], [2, 4, 5]], 3)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-classes","seed":202610304020,"parameters":{"n":6,"pairs":[[0,0],[1,1],[1,3],[2,2],[2,4],[2,5],[3,1],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]],"domain":"Complete_finite_quotient_set","operation":"quotient-classes","semantic_template":"qf-mod30-quotient-classes/{\"n\":6,\"pairs\":[[0,0],[1,1],[1,3],[2,2],[2,4],[2,5],[3,1],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0021","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-classes；資料={\"n\":6,\"pairs\":[[0,0],[0,2],[0,4],[0,5],[1,1],[2,0],[2,2],[2,4],[2,5],[3,3],[4,0],[4,2],[4,4],[4,5],[5,0],[5,2],[5,4],[5,5]]}。請依 [classes,count] 欄序回答。","answer":{"kind":"expression","value":"([[0, 2, 4, 5], [1], [3]], 3)","explanation_zh":"Wave4第四層核對=([[0, 2, 4, 5], [1], [3]], 3)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-classes","seed":202610304021,"parameters":{"n":6,"pairs":[[0,0],[0,2],[0,4],[0,5],[1,1],[2,0],[2,2],[2,4],[2,5],[3,3],[4,0],[4,2],[4,4],[4,5],[5,0],[5,2],[5,4],[5,5]],"domain":"Complete_finite_quotient_set","operation":"quotient-classes","semantic_template":"qf-mod30-quotient-classes/{\"n\":6,\"pairs\":[[0,0],[0,2],[0,4],[0,5],[1,1],[2,0],[2,2],[2,4],[2,5],[3,3],[4,0],[4,2],[4,4],[4,5],[5,0],[5,2],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0022","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-classes；資料={\"n\":6,\"pairs\":[[0,0],[0,3],[1,1],[2,2],[2,4],[2,5],[3,0],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]]}。請依 [classes,count] 欄序回答。","answer":{"kind":"expression","value":"([[0, 3], [1], [2, 4, 5]], 3)","explanation_zh":"Wave4第四層核對=([[0, 3], [1], [2, 4, 5]], 3)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-classes","seed":202610304022,"parameters":{"n":6,"pairs":[[0,0],[0,3],[1,1],[2,2],[2,4],[2,5],[3,0],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]],"domain":"Complete_finite_quotient_set","operation":"quotient-classes","semantic_template":"qf-mod30-quotient-classes/{\"n\":6,\"pairs\":[[0,0],[0,3],[1,1],[2,2],[2,4],[2,5],[3,0],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0023","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-classes；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[2,4],[2,5],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]]}。請依 [classes,count] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1], [2, 4, 5], [3]], 3)","explanation_zh":"Wave4第四層核對=([[0, 1], [2, 4, 5], [3]], 3)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-classes","seed":202610304023,"parameters":{"n":6,"pairs":[[0,0],[0,1],[1,0],[1,1],[2,2],[2,4],[2,5],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]],"domain":"Complete_finite_quotient_set","operation":"quotient-classes","semantic_template":"qf-mod30-quotient-classes/{\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[2,4],[2,5],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0024","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-classes；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[0,4],[0,5],[1,0],[1,1],[1,4],[1,5],[2,2],[2,3],[3,2],[3,3],[4,0],[4,1],[4,4],[4,5],[5,0],[5,1],[5,4],[5,5]]}。請依 [classes,count] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 4, 5], [2, 3]], 2)","explanation_zh":"Wave4第四層核對=([[0, 1, 4, 5], [2, 3]], 2)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-classes","seed":202610304024,"parameters":{"n":6,"pairs":[[0,0],[0,1],[0,4],[0,5],[1,0],[1,1],[1,4],[1,5],[2,2],[2,3],[3,2],[3,3],[4,0],[4,1],[4,4],[4,5],[5,0],[5,1],[5,4],[5,5]],"domain":"Complete_finite_quotient_set","operation":"quotient-classes","semantic_template":"qf-mod30-quotient-classes/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,4],[0,5],[1,0],[1,1],[1,4],[1,5],[2,2],[2,3],[3,2],[3,3],[4,0],[4,1],[4,4],[4,5],[5,0],[5,1],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0025","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-classes；資料={\"n\":6,\"pairs\":[[0,0],[0,2],[0,3],[1,1],[1,4],[1,5],[2,0],[2,2],[2,3],[3,0],[3,2],[3,3],[4,1],[4,4],[4,5],[5,1],[5,4],[5,5]]}。請依 [classes,count] 欄序回答。","answer":{"kind":"expression","value":"([[0, 2, 3], [1, 4, 5]], 2)","explanation_zh":"Wave4第四層核對=([[0, 2, 3], [1, 4, 5]], 2)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-classes","seed":202610304025,"parameters":{"n":6,"pairs":[[0,0],[0,2],[0,3],[1,1],[1,4],[1,5],[2,0],[2,2],[2,3],[3,0],[3,2],[3,3],[4,1],[4,4],[4,5],[5,1],[5,4],[5,5]],"domain":"Complete_finite_quotient_set","operation":"quotient-classes","semantic_template":"qf-mod30-quotient-classes/{\"n\":6,\"pairs\":[[0,0],[0,2],[0,3],[1,1],[1,4],[1,5],[2,0],[2,2],[2,3],[3,0],[3,2],[3,3],[4,1],[4,4],[4,5],[5,1],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0026","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-projection；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[0,3],[0,4],[0,5],[1,0],[1,1],[1,2],[1,3],[1,4],[1,5],[2,0],[2,1],[2,2],[2,3],[2,4],[2,5],[3,0],[3,1],[3,2],[3,3],[3,4],[3,5],[4,0],[4,1],[4,2],[4,3],[4,4],[4,5],[5,0],[5,1],[5,2],[5,3],[5,4],[5,5]],\"element\":0}。請依 [element,class] 欄序回答。","answer":{"kind":"expression","value":"(0, [0, 1, 2, 3, 4, 5])","explanation_zh":"Wave4第四層核對=(0, [0, 1, 2, 3, 4, 5])；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-projection","seed":202610304026,"parameters":{"n":6,"pairs":[[0,0],[0,1],[0,2],[0,3],[0,4],[0,5],[1,0],[1,1],[1,2],[1,3],[1,4],[1,5],[2,0],[2,1],[2,2],[2,3],[2,4],[2,5],[3,0],[3,1],[3,2],[3,3],[3,4],[3,5],[4,0],[4,1],[4,2],[4,3],[4,4],[4,5],[5,0],[5,1],[5,2],[5,3],[5,4],[5,5]],"element":0,"domain":"Complete_finite_quotient_set","operation":"quotient-projection","semantic_template":"qf-mod30-quotient-projection/{\"element\":0,\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[0,3],[0,4],[0,5],[1,0],[1,1],[1,2],[1,3],[1,4],[1,5],[2,0],[2,1],[2,2],[2,3],[2,4],[2,5],[3,0],[3,1],[3,2],[3,3],[3,4],[3,5],[4,0],[4,1],[4,2],[4,3],[4,4],[4,5],[5,0],[5,1],[5,2],[5,3],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0027","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-projection；資料={\"n\":6,\"pairs\":[[0,0],[1,1],[1,2],[1,3],[1,4],[1,5],[2,1],[2,2],[2,3],[2,4],[2,5],[3,1],[3,2],[3,3],[3,4],[3,5],[4,1],[4,2],[4,3],[4,4],[4,5],[5,1],[5,2],[5,3],[5,4],[5,5]],\"element\":1}。請依 [element,class] 欄序回答。","answer":{"kind":"expression","value":"(1, [1, 2, 3, 4, 5])","explanation_zh":"Wave4第四層核對=(1, [1, 2, 3, 4, 5])；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-projection","seed":202610304027,"parameters":{"n":6,"pairs":[[0,0],[1,1],[1,2],[1,3],[1,4],[1,5],[2,1],[2,2],[2,3],[2,4],[2,5],[3,1],[3,2],[3,3],[3,4],[3,5],[4,1],[4,2],[4,3],[4,4],[4,5],[5,1],[5,2],[5,3],[5,4],[5,5]],"element":1,"domain":"Complete_finite_quotient_set","operation":"quotient-projection","semantic_template":"qf-mod30-quotient-projection/{\"element\":1,\"n\":6,\"pairs\":[[0,0],[1,1],[1,2],[1,3],[1,4],[1,5],[2,1],[2,2],[2,3],[2,4],[2,5],[3,1],[3,2],[3,3],[3,4],[3,5],[4,1],[4,2],[4,3],[4,4],[4,5],[5,1],[5,2],[5,3],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0028","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-projection；資料={\"n\":6,\"pairs\":[[0,0],[0,2],[0,3],[0,4],[0,5],[1,1],[2,0],[2,2],[2,3],[2,4],[2,5],[3,0],[3,2],[3,3],[3,4],[3,5],[4,0],[4,2],[4,3],[4,4],[4,5],[5,0],[5,2],[5,3],[5,4],[5,5]],\"element\":2}。請依 [element,class] 欄序回答。","answer":{"kind":"expression","value":"(2, [0, 2, 3, 4, 5])","explanation_zh":"Wave4第四層核對=(2, [0, 2, 3, 4, 5])；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-projection","seed":202610304028,"parameters":{"n":6,"pairs":[[0,0],[0,2],[0,3],[0,4],[0,5],[1,1],[2,0],[2,2],[2,3],[2,4],[2,5],[3,0],[3,2],[3,3],[3,4],[3,5],[4,0],[4,2],[4,3],[4,4],[4,5],[5,0],[5,2],[5,3],[5,4],[5,5]],"element":2,"domain":"Complete_finite_quotient_set","operation":"quotient-projection","semantic_template":"qf-mod30-quotient-projection/{\"element\":2,\"n\":6,\"pairs\":[[0,0],[0,2],[0,3],[0,4],[0,5],[1,1],[2,0],[2,2],[2,3],[2,4],[2,5],[3,0],[3,2],[3,3],[3,4],[3,5],[4,0],[4,2],[4,3],[4,4],[4,5],[5,0],[5,2],[5,3],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0029","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-projection；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[2,3],[2,4],[2,5],[3,2],[3,3],[3,4],[3,5],[4,2],[4,3],[4,4],[4,5],[5,2],[5,3],[5,4],[5,5]],\"element\":3}。請依 [element,class] 欄序回答。","answer":{"kind":"expression","value":"(3, [2, 3, 4, 5])","explanation_zh":"Wave4第四層核對=(3, [2, 3, 4, 5])；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-projection","seed":202610304029,"parameters":{"n":6,"pairs":[[0,0],[0,1],[1,0],[1,1],[2,2],[2,3],[2,4],[2,5],[3,2],[3,3],[3,4],[3,5],[4,2],[4,3],[4,4],[4,5],[5,2],[5,3],[5,4],[5,5]],"element":3,"domain":"Complete_finite_quotient_set","operation":"quotient-projection","semantic_template":"qf-mod30-quotient-projection/{\"element\":3,\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[2,3],[2,4],[2,5],[3,2],[3,3],[3,4],[3,5],[4,2],[4,3],[4,4],[4,5],[5,2],[5,3],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0030","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-projection；資料={\"n\":6,\"pairs\":[[0,0],[1,1],[2,2],[2,3],[2,4],[2,5],[3,2],[3,3],[3,4],[3,5],[4,2],[4,3],[4,4],[4,5],[5,2],[5,3],[5,4],[5,5]],\"element\":4}。請依 [element,class] 欄序回答。","answer":{"kind":"expression","value":"(4, [2, 3, 4, 5])","explanation_zh":"Wave4第四層核對=(4, [2, 3, 4, 5])；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-projection","seed":202610304030,"parameters":{"n":6,"pairs":[[0,0],[1,1],[2,2],[2,3],[2,4],[2,5],[3,2],[3,3],[3,4],[3,5],[4,2],[4,3],[4,4],[4,5],[5,2],[5,3],[5,4],[5,5]],"element":4,"domain":"Complete_finite_quotient_set","operation":"quotient-projection","semantic_template":"qf-mod30-quotient-projection/{\"element\":4,\"n\":6,\"pairs\":[[0,0],[1,1],[2,2],[2,3],[2,4],[2,5],[3,2],[3,3],[3,4],[3,5],[4,2],[4,3],[4,4],[4,5],[5,2],[5,3],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0031","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-projection；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[0,3],[0,4],[0,5],[1,0],[1,1],[1,3],[1,4],[1,5],[2,2],[3,0],[3,1],[3,3],[3,4],[3,5],[4,0],[4,1],[4,3],[4,4],[4,5],[5,0],[5,1],[5,3],[5,4],[5,5]],\"element\":5}。請依 [element,class] 欄序回答。","answer":{"kind":"expression","value":"(5, [0, 1, 3, 4, 5])","explanation_zh":"Wave4第四層核對=(5, [0, 1, 3, 4, 5])；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-projection","seed":202610304031,"parameters":{"n":6,"pairs":[[0,0],[0,1],[0,3],[0,4],[0,5],[1,0],[1,1],[1,3],[1,4],[1,5],[2,2],[3,0],[3,1],[3,3],[3,4],[3,5],[4,0],[4,1],[4,3],[4,4],[4,5],[5,0],[5,1],[5,3],[5,4],[5,5]],"element":5,"domain":"Complete_finite_quotient_set","operation":"quotient-projection","semantic_template":"qf-mod30-quotient-projection/{\"element\":5,\"n\":6,\"pairs\":[[0,0],[0,1],[0,3],[0,4],[0,5],[1,0],[1,1],[1,3],[1,4],[1,5],[2,2],[3,0],[3,1],[3,3],[3,4],[3,5],[4,0],[4,1],[4,3],[4,4],[4,5],[5,0],[5,1],[5,3],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0032","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-projection；資料={\"n\":6,\"pairs\":[[0,0],[0,2],[1,1],[1,3],[1,4],[1,5],[2,0],[2,2],[3,1],[3,3],[3,4],[3,5],[4,1],[4,3],[4,4],[4,5],[5,1],[5,3],[5,4],[5,5]],\"element\":0}。請依 [element,class] 欄序回答。","answer":{"kind":"expression","value":"(0, [0, 2])","explanation_zh":"Wave4第四層核對=(0, [0, 2])；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-projection","seed":202610304032,"parameters":{"n":6,"pairs":[[0,0],[0,2],[1,1],[1,3],[1,4],[1,5],[2,0],[2,2],[3,1],[3,3],[3,4],[3,5],[4,1],[4,3],[4,4],[4,5],[5,1],[5,3],[5,4],[5,5]],"element":0,"domain":"Complete_finite_quotient_set","operation":"quotient-projection","semantic_template":"qf-mod30-quotient-projection/{\"element\":0,\"n\":6,\"pairs\":[[0,0],[0,2],[1,1],[1,3],[1,4],[1,5],[2,0],[2,2],[3,1],[3,3],[3,4],[3,5],[4,1],[4,3],[4,4],[4,5],[5,1],[5,3],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0033","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-projection；資料={\"n\":6,\"pairs\":[[0,0],[1,1],[1,3],[1,4],[1,5],[2,2],[3,1],[3,3],[3,4],[3,5],[4,1],[4,3],[4,4],[4,5],[5,1],[5,3],[5,4],[5,5]],\"element\":1}。請依 [element,class] 欄序回答。","answer":{"kind":"expression","value":"(1, [1, 3, 4, 5])","explanation_zh":"Wave4第四層核對=(1, [1, 3, 4, 5])；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-projection","seed":202610304033,"parameters":{"n":6,"pairs":[[0,0],[1,1],[1,3],[1,4],[1,5],[2,2],[3,1],[3,3],[3,4],[3,5],[4,1],[4,3],[4,4],[4,5],[5,1],[5,3],[5,4],[5,5]],"element":1,"domain":"Complete_finite_quotient_set","operation":"quotient-projection","semantic_template":"qf-mod30-quotient-projection/{\"element\":1,\"n\":6,\"pairs\":[[0,0],[1,1],[1,3],[1,4],[1,5],[2,2],[3,1],[3,3],[3,4],[3,5],[4,1],[4,3],[4,4],[4,5],[5,1],[5,3],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0034","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-projection；資料={\"n\":6,\"pairs\":[[0,0],[0,3],[0,4],[0,5],[1,1],[1,2],[2,1],[2,2],[3,0],[3,3],[3,4],[3,5],[4,0],[4,3],[4,4],[4,5],[5,0],[5,3],[5,4],[5,5]],\"element\":2}。請依 [element,class] 欄序回答。","answer":{"kind":"expression","value":"(2, [1, 2])","explanation_zh":"Wave4第四層核對=(2, [1, 2])；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-projection","seed":202610304034,"parameters":{"n":6,"pairs":[[0,0],[0,3],[0,4],[0,5],[1,1],[1,2],[2,1],[2,2],[3,0],[3,3],[3,4],[3,5],[4,0],[4,3],[4,4],[4,5],[5,0],[5,3],[5,4],[5,5]],"element":2,"domain":"Complete_finite_quotient_set","operation":"quotient-projection","semantic_template":"qf-mod30-quotient-projection/{\"element\":2,\"n\":6,\"pairs\":[[0,0],[0,3],[0,4],[0,5],[1,1],[1,2],[2,1],[2,2],[3,0],[3,3],[3,4],[3,5],[4,0],[4,3],[4,4],[4,5],[5,0],[5,3],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0035","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-projection；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[1,0],[1,1],[1,2],[2,0],[2,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]],\"element\":3}。請依 [element,class] 欄序回答。","answer":{"kind":"expression","value":"(3, [3, 4, 5])","explanation_zh":"Wave4第四層核對=(3, [3, 4, 5])；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-projection","seed":202610304035,"parameters":{"n":6,"pairs":[[0,0],[0,1],[0,2],[1,0],[1,1],[1,2],[2,0],[2,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]],"element":3,"domain":"Complete_finite_quotient_set","operation":"quotient-projection","semantic_template":"qf-mod30-quotient-projection/{\"element\":3,\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[1,0],[1,1],[1,2],[2,0],[2,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0036","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-projection；資料={\"n\":6,\"pairs\":[[0,0],[1,1],[1,2],[2,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]],\"element\":4}。請依 [element,class] 欄序回答。","answer":{"kind":"expression","value":"(4, [3, 4, 5])","explanation_zh":"Wave4第四層核對=(4, [3, 4, 5])；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-projection","seed":202610304036,"parameters":{"n":6,"pairs":[[0,0],[1,1],[1,2],[2,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]],"element":4,"domain":"Complete_finite_quotient_set","operation":"quotient-projection","semantic_template":"qf-mod30-quotient-projection/{\"element\":4,\"n\":6,\"pairs\":[[0,0],[1,1],[1,2],[2,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0037","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-projection；資料={\"n\":6,\"pairs\":[[0,0],[0,3],[0,4],[0,5],[1,1],[2,2],[3,0],[3,3],[3,4],[3,5],[4,0],[4,3],[4,4],[4,5],[5,0],[5,3],[5,4],[5,5]],\"element\":5}。請依 [element,class] 欄序回答。","answer":{"kind":"expression","value":"(5, [0, 3, 4, 5])","explanation_zh":"Wave4第四層核對=(5, [0, 3, 4, 5])；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-projection","seed":202610304037,"parameters":{"n":6,"pairs":[[0,0],[0,3],[0,4],[0,5],[1,1],[2,2],[3,0],[3,3],[3,4],[3,5],[4,0],[4,3],[4,4],[4,5],[5,0],[5,3],[5,4],[5,5]],"element":5,"domain":"Complete_finite_quotient_set","operation":"quotient-projection","semantic_template":"qf-mod30-quotient-projection/{\"element\":5,\"n\":6,\"pairs\":[[0,0],[0,3],[0,4],[0,5],[1,1],[2,2],[3,0],[3,3],[3,4],[3,5],[4,0],[4,3],[4,4],[4,5],[5,0],[5,3],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0038","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-projection；資料={\"n\":6,\"pairs\":[[0,0],[0,2],[1,1],[2,0],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]],\"element\":0}。請依 [element,class] 欄序回答。","answer":{"kind":"expression","value":"(0, [0, 2])","explanation_zh":"Wave4第四層核對=(0, [0, 2])；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-projection","seed":202610304038,"parameters":{"n":6,"pairs":[[0,0],[0,2],[1,1],[2,0],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]],"element":0,"domain":"Complete_finite_quotient_set","operation":"quotient-projection","semantic_template":"qf-mod30-quotient-projection/{\"element\":0,\"n\":6,\"pairs\":[[0,0],[0,2],[1,1],[2,0],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0039","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-projection；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]],\"element\":1}。請依 [element,class] 欄序回答。","answer":{"kind":"expression","value":"(1, [0, 1])","explanation_zh":"Wave4第四層核對=(1, [0, 1])；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-projection","seed":202610304039,"parameters":{"n":6,"pairs":[[0,0],[0,1],[1,0],[1,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]],"element":1,"domain":"Complete_finite_quotient_set","operation":"quotient-projection","semantic_template":"qf-mod30-quotient-projection/{\"element\":1,\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0040","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-projection；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[0,4],[0,5],[1,0],[1,1],[1,2],[1,4],[1,5],[2,0],[2,1],[2,2],[2,4],[2,5],[3,3],[4,0],[4,1],[4,2],[4,4],[4,5],[5,0],[5,1],[5,2],[5,4],[5,5]],\"element\":2}。請依 [element,class] 欄序回答。","answer":{"kind":"expression","value":"(2, [0, 1, 2, 4, 5])","explanation_zh":"Wave4第四層核對=(2, [0, 1, 2, 4, 5])；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-projection","seed":202610304040,"parameters":{"n":6,"pairs":[[0,0],[0,1],[0,2],[0,4],[0,5],[1,0],[1,1],[1,2],[1,4],[1,5],[2,0],[2,1],[2,2],[2,4],[2,5],[3,3],[4,0],[4,1],[4,2],[4,4],[4,5],[5,0],[5,1],[5,2],[5,4],[5,5]],"element":2,"domain":"Complete_finite_quotient_set","operation":"quotient-projection","semantic_template":"qf-mod30-quotient-projection/{\"element\":2,\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[0,4],[0,5],[1,0],[1,1],[1,2],[1,4],[1,5],[2,0],[2,1],[2,2],[2,4],[2,5],[3,3],[4,0],[4,1],[4,2],[4,4],[4,5],[5,0],[5,1],[5,2],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0041","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-projection；資料={\"n\":6,\"pairs\":[[0,0],[0,3],[1,1],[1,2],[1,4],[1,5],[2,1],[2,2],[2,4],[2,5],[3,0],[3,3],[4,1],[4,2],[4,4],[4,5],[5,1],[5,2],[5,4],[5,5]],\"element\":3}。請依 [element,class] 欄序回答。","answer":{"kind":"expression","value":"(3, [0, 3])","explanation_zh":"Wave4第四層核對=(3, [0, 3])；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-projection","seed":202610304041,"parameters":{"n":6,"pairs":[[0,0],[0,3],[1,1],[1,2],[1,4],[1,5],[2,1],[2,2],[2,4],[2,5],[3,0],[3,3],[4,1],[4,2],[4,4],[4,5],[5,1],[5,2],[5,4],[5,5]],"element":3,"domain":"Complete_finite_quotient_set","operation":"quotient-projection","semantic_template":"qf-mod30-quotient-projection/{\"element\":3,\"n\":6,\"pairs\":[[0,0],[0,3],[1,1],[1,2],[1,4],[1,5],[2,1],[2,2],[2,4],[2,5],[3,0],[3,3],[4,1],[4,2],[4,4],[4,5],[5,1],[5,2],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0042","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-projection；資料={\"n\":6,\"pairs\":[[0,0],[1,1],[1,2],[1,4],[1,5],[2,1],[2,2],[2,4],[2,5],[3,3],[4,1],[4,2],[4,4],[4,5],[5,1],[5,2],[5,4],[5,5]],\"element\":4}。請依 [element,class] 欄序回答。","answer":{"kind":"expression","value":"(4, [1, 2, 4, 5])","explanation_zh":"Wave4第四層核對=(4, [1, 2, 4, 5])；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-projection","seed":202610304042,"parameters":{"n":6,"pairs":[[0,0],[1,1],[1,2],[1,4],[1,5],[2,1],[2,2],[2,4],[2,5],[3,3],[4,1],[4,2],[4,4],[4,5],[5,1],[5,2],[5,4],[5,5]],"element":4,"domain":"Complete_finite_quotient_set","operation":"quotient-projection","semantic_template":"qf-mod30-quotient-projection/{\"element\":4,\"n\":6,\"pairs\":[[0,0],[1,1],[1,2],[1,4],[1,5],[2,1],[2,2],[2,4],[2,5],[3,3],[4,1],[4,2],[4,4],[4,5],[5,1],[5,2],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0043","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-projection；資料={\"n\":6,\"pairs\":[[0,0],[0,2],[0,4],[0,5],[1,1],[1,3],[2,0],[2,2],[2,4],[2,5],[3,1],[3,3],[4,0],[4,2],[4,4],[4,5],[5,0],[5,2],[5,4],[5,5]],\"element\":5}。請依 [element,class] 欄序回答。","answer":{"kind":"expression","value":"(5, [0, 2, 4, 5])","explanation_zh":"Wave4第四層核對=(5, [0, 2, 4, 5])；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-projection","seed":202610304043,"parameters":{"n":6,"pairs":[[0,0],[0,2],[0,4],[0,5],[1,1],[1,3],[2,0],[2,2],[2,4],[2,5],[3,1],[3,3],[4,0],[4,2],[4,4],[4,5],[5,0],[5,2],[5,4],[5,5]],"element":5,"domain":"Complete_finite_quotient_set","operation":"quotient-projection","semantic_template":"qf-mod30-quotient-projection/{\"element\":5,\"n\":6,\"pairs\":[[0,0],[0,2],[0,4],[0,5],[1,1],[1,3],[2,0],[2,2],[2,4],[2,5],[3,1],[3,3],[4,0],[4,2],[4,4],[4,5],[5,0],[5,2],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0044","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-projection；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[0,3],[1,0],[1,1],[1,3],[2,2],[2,4],[2,5],[3,0],[3,1],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]],\"element\":0}。請依 [element,class] 欄序回答。","answer":{"kind":"expression","value":"(0, [0, 1, 3])","explanation_zh":"Wave4第四層核對=(0, [0, 1, 3])；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-projection","seed":202610304044,"parameters":{"n":6,"pairs":[[0,0],[0,1],[0,3],[1,0],[1,1],[1,3],[2,2],[2,4],[2,5],[3,0],[3,1],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]],"element":0,"domain":"Complete_finite_quotient_set","operation":"quotient-projection","semantic_template":"qf-mod30-quotient-projection/{\"element\":0,\"n\":6,\"pairs\":[[0,0],[0,1],[0,3],[1,0],[1,1],[1,3],[2,2],[2,4],[2,5],[3,0],[3,1],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0045","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-projection；資料={\"n\":6,\"pairs\":[[0,0],[1,1],[1,3],[2,2],[2,4],[2,5],[3,1],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]],\"element\":1}。請依 [element,class] 欄序回答。","answer":{"kind":"expression","value":"(1, [1, 3])","explanation_zh":"Wave4第四層核對=(1, [1, 3])；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-projection","seed":202610304045,"parameters":{"n":6,"pairs":[[0,0],[1,1],[1,3],[2,2],[2,4],[2,5],[3,1],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]],"element":1,"domain":"Complete_finite_quotient_set","operation":"quotient-projection","semantic_template":"qf-mod30-quotient-projection/{\"element\":1,\"n\":6,\"pairs\":[[0,0],[1,1],[1,3],[2,2],[2,4],[2,5],[3,1],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0046","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-projection；資料={\"n\":6,\"pairs\":[[0,0],[0,2],[0,4],[0,5],[1,1],[2,0],[2,2],[2,4],[2,5],[3,3],[4,0],[4,2],[4,4],[4,5],[5,0],[5,2],[5,4],[5,5]],\"element\":2}。請依 [element,class] 欄序回答。","answer":{"kind":"expression","value":"(2, [0, 2, 4, 5])","explanation_zh":"Wave4第四層核對=(2, [0, 2, 4, 5])；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-projection","seed":202610304046,"parameters":{"n":6,"pairs":[[0,0],[0,2],[0,4],[0,5],[1,1],[2,0],[2,2],[2,4],[2,5],[3,3],[4,0],[4,2],[4,4],[4,5],[5,0],[5,2],[5,4],[5,5]],"element":2,"domain":"Complete_finite_quotient_set","operation":"quotient-projection","semantic_template":"qf-mod30-quotient-projection/{\"element\":2,\"n\":6,\"pairs\":[[0,0],[0,2],[0,4],[0,5],[1,1],[2,0],[2,2],[2,4],[2,5],[3,3],[4,0],[4,2],[4,4],[4,5],[5,0],[5,2],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0047","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-projection；資料={\"n\":6,\"pairs\":[[0,0],[0,3],[1,1],[2,2],[2,4],[2,5],[3,0],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]],\"element\":3}。請依 [element,class] 欄序回答。","answer":{"kind":"expression","value":"(3, [0, 3])","explanation_zh":"Wave4第四層核對=(3, [0, 3])；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-projection","seed":202610304047,"parameters":{"n":6,"pairs":[[0,0],[0,3],[1,1],[2,2],[2,4],[2,5],[3,0],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]],"element":3,"domain":"Complete_finite_quotient_set","operation":"quotient-projection","semantic_template":"qf-mod30-quotient-projection/{\"element\":3,\"n\":6,\"pairs\":[[0,0],[0,3],[1,1],[2,2],[2,4],[2,5],[3,0],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0048","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-projection；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[2,4],[2,5],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]],\"element\":4}。請依 [element,class] 欄序回答。","answer":{"kind":"expression","value":"(4, [2, 4, 5])","explanation_zh":"Wave4第四層核對=(4, [2, 4, 5])；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-projection","seed":202610304048,"parameters":{"n":6,"pairs":[[0,0],[0,1],[1,0],[1,1],[2,2],[2,4],[2,5],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]],"element":4,"domain":"Complete_finite_quotient_set","operation":"quotient-projection","semantic_template":"qf-mod30-quotient-projection/{\"element\":4,\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[2,4],[2,5],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0049","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-projection；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[0,4],[0,5],[1,0],[1,1],[1,4],[1,5],[2,2],[2,3],[3,2],[3,3],[4,0],[4,1],[4,4],[4,5],[5,0],[5,1],[5,4],[5,5]],\"element\":5}。請依 [element,class] 欄序回答。","answer":{"kind":"expression","value":"(5, [0, 1, 4, 5])","explanation_zh":"Wave4第四層核對=(5, [0, 1, 4, 5])；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-projection","seed":202610304049,"parameters":{"n":6,"pairs":[[0,0],[0,1],[0,4],[0,5],[1,0],[1,1],[1,4],[1,5],[2,2],[2,3],[3,2],[3,3],[4,0],[4,1],[4,4],[4,5],[5,0],[5,1],[5,4],[5,5]],"element":5,"domain":"Complete_finite_quotient_set","operation":"quotient-projection","semantic_template":"qf-mod30-quotient-projection/{\"element\":5,\"n\":6,\"pairs\":[[0,0],[0,1],[0,4],[0,5],[1,0],[1,1],[1,4],[1,5],[2,2],[2,3],[3,2],[3,3],[4,0],[4,1],[4,4],[4,5],[5,0],[5,1],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0050","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-projection；資料={\"n\":6,\"pairs\":[[0,0],[0,2],[0,3],[1,1],[1,4],[1,5],[2,0],[2,2],[2,3],[3,0],[3,2],[3,3],[4,1],[4,4],[4,5],[5,1],[5,4],[5,5]],\"element\":0}。請依 [element,class] 欄序回答。","answer":{"kind":"expression","value":"(0, [0, 2, 3])","explanation_zh":"Wave4第四層核對=(0, [0, 2, 3])；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-projection","seed":202610304050,"parameters":{"n":6,"pairs":[[0,0],[0,2],[0,3],[1,1],[1,4],[1,5],[2,0],[2,2],[2,3],[3,0],[3,2],[3,3],[4,1],[4,4],[4,5],[5,1],[5,4],[5,5]],"element":0,"domain":"Complete_finite_quotient_set","operation":"quotient-projection","semantic_template":"qf-mod30-quotient-projection/{\"element\":0,\"n\":6,\"pairs\":[[0,0],[0,2],[0,3],[1,1],[1,4],[1,5],[2,0],[2,2],[2,3],[3,0],[3,2],[3,3],[4,1],[4,4],[4,5],[5,1],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0051","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-partition；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[0,3],[0,4],[0,5],[1,0],[1,1],[1,2],[1,3],[1,4],[1,5],[2,0],[2,1],[2,2],[2,3],[2,4],[2,5],[3,0],[3,1],[3,2],[3,3],[3,4],[3,5],[4,0],[4,1],[4,2],[4,3],[4,4],[4,5],[5,0],[5,1],[5,2],[5,3],[5,4],[5,5]]}。請依 [classes,covers,disjoint] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5]], 是, 是)","explanation_zh":"Wave4第四層核對=([[0, 1, 2, 3, 4, 5]], 是, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-partition","seed":202610304051,"parameters":{"n":6,"pairs":[[0,0],[0,1],[0,2],[0,3],[0,4],[0,5],[1,0],[1,1],[1,2],[1,3],[1,4],[1,5],[2,0],[2,1],[2,2],[2,3],[2,4],[2,5],[3,0],[3,1],[3,2],[3,3],[3,4],[3,5],[4,0],[4,1],[4,2],[4,3],[4,4],[4,5],[5,0],[5,1],[5,2],[5,3],[5,4],[5,5]],"domain":"Complete_finite_quotient_set","operation":"quotient-partition","semantic_template":"qf-mod30-quotient-partition/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[0,3],[0,4],[0,5],[1,0],[1,1],[1,2],[1,3],[1,4],[1,5],[2,0],[2,1],[2,2],[2,3],[2,4],[2,5],[3,0],[3,1],[3,2],[3,3],[3,4],[3,5],[4,0],[4,1],[4,2],[4,3],[4,4],[4,5],[5,0],[5,1],[5,2],[5,3],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0052","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-partition；資料={\"n\":6,\"pairs\":[[0,0],[1,1],[1,2],[1,3],[1,4],[1,5],[2,1],[2,2],[2,3],[2,4],[2,5],[3,1],[3,2],[3,3],[3,4],[3,5],[4,1],[4,2],[4,3],[4,4],[4,5],[5,1],[5,2],[5,3],[5,4],[5,5]]}。請依 [classes,covers,disjoint] 欄序回答。","answer":{"kind":"expression","value":"([[0], [1, 2, 3, 4, 5]], 是, 是)","explanation_zh":"Wave4第四層核對=([[0], [1, 2, 3, 4, 5]], 是, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-partition","seed":202610304052,"parameters":{"n":6,"pairs":[[0,0],[1,1],[1,2],[1,3],[1,4],[1,5],[2,1],[2,2],[2,3],[2,4],[2,5],[3,1],[3,2],[3,3],[3,4],[3,5],[4,1],[4,2],[4,3],[4,4],[4,5],[5,1],[5,2],[5,3],[5,4],[5,5]],"domain":"Complete_finite_quotient_set","operation":"quotient-partition","semantic_template":"qf-mod30-quotient-partition/{\"n\":6,\"pairs\":[[0,0],[1,1],[1,2],[1,3],[1,4],[1,5],[2,1],[2,2],[2,3],[2,4],[2,5],[3,1],[3,2],[3,3],[3,4],[3,5],[4,1],[4,2],[4,3],[4,4],[4,5],[5,1],[5,2],[5,3],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0053","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-partition；資料={\"n\":6,\"pairs\":[[0,0],[0,2],[0,3],[0,4],[0,5],[1,1],[2,0],[2,2],[2,3],[2,4],[2,5],[3,0],[3,2],[3,3],[3,4],[3,5],[4,0],[4,2],[4,3],[4,4],[4,5],[5,0],[5,2],[5,3],[5,4],[5,5]]}。請依 [classes,covers,disjoint] 欄序回答。","answer":{"kind":"expression","value":"([[0, 2, 3, 4, 5], [1]], 是, 是)","explanation_zh":"Wave4第四層核對=([[0, 2, 3, 4, 5], [1]], 是, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-partition","seed":202610304053,"parameters":{"n":6,"pairs":[[0,0],[0,2],[0,3],[0,4],[0,5],[1,1],[2,0],[2,2],[2,3],[2,4],[2,5],[3,0],[3,2],[3,3],[3,4],[3,5],[4,0],[4,2],[4,3],[4,4],[4,5],[5,0],[5,2],[5,3],[5,4],[5,5]],"domain":"Complete_finite_quotient_set","operation":"quotient-partition","semantic_template":"qf-mod30-quotient-partition/{\"n\":6,\"pairs\":[[0,0],[0,2],[0,3],[0,4],[0,5],[1,1],[2,0],[2,2],[2,3],[2,4],[2,5],[3,0],[3,2],[3,3],[3,4],[3,5],[4,0],[4,2],[4,3],[4,4],[4,5],[5,0],[5,2],[5,3],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0054","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-partition；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[2,3],[2,4],[2,5],[3,2],[3,3],[3,4],[3,5],[4,2],[4,3],[4,4],[4,5],[5,2],[5,3],[5,4],[5,5]]}。請依 [classes,covers,disjoint] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1], [2, 3, 4, 5]], 是, 是)","explanation_zh":"Wave4第四層核對=([[0, 1], [2, 3, 4, 5]], 是, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-partition","seed":202610304054,"parameters":{"n":6,"pairs":[[0,0],[0,1],[1,0],[1,1],[2,2],[2,3],[2,4],[2,5],[3,2],[3,3],[3,4],[3,5],[4,2],[4,3],[4,4],[4,5],[5,2],[5,3],[5,4],[5,5]],"domain":"Complete_finite_quotient_set","operation":"quotient-partition","semantic_template":"qf-mod30-quotient-partition/{\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[2,3],[2,4],[2,5],[3,2],[3,3],[3,4],[3,5],[4,2],[4,3],[4,4],[4,5],[5,2],[5,3],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0055","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-partition；資料={\"n\":6,\"pairs\":[[0,0],[1,1],[2,2],[2,3],[2,4],[2,5],[3,2],[3,3],[3,4],[3,5],[4,2],[4,3],[4,4],[4,5],[5,2],[5,3],[5,4],[5,5]]}。請依 [classes,covers,disjoint] 欄序回答。","answer":{"kind":"expression","value":"([[0], [1], [2, 3, 4, 5]], 是, 是)","explanation_zh":"Wave4第四層核對=([[0], [1], [2, 3, 4, 5]], 是, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-partition","seed":202610304055,"parameters":{"n":6,"pairs":[[0,0],[1,1],[2,2],[2,3],[2,4],[2,5],[3,2],[3,3],[3,4],[3,5],[4,2],[4,3],[4,4],[4,5],[5,2],[5,3],[5,4],[5,5]],"domain":"Complete_finite_quotient_set","operation":"quotient-partition","semantic_template":"qf-mod30-quotient-partition/{\"n\":6,\"pairs\":[[0,0],[1,1],[2,2],[2,3],[2,4],[2,5],[3,2],[3,3],[3,4],[3,5],[4,2],[4,3],[4,4],[4,5],[5,2],[5,3],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0056","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-partition；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[0,3],[0,4],[0,5],[1,0],[1,1],[1,3],[1,4],[1,5],[2,2],[3,0],[3,1],[3,3],[3,4],[3,5],[4,0],[4,1],[4,3],[4,4],[4,5],[5,0],[5,1],[5,3],[5,4],[5,5]]}。請依 [classes,covers,disjoint] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 3, 4, 5], [2]], 是, 是)","explanation_zh":"Wave4第四層核對=([[0, 1, 3, 4, 5], [2]], 是, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-partition","seed":202610304056,"parameters":{"n":6,"pairs":[[0,0],[0,1],[0,3],[0,4],[0,5],[1,0],[1,1],[1,3],[1,4],[1,5],[2,2],[3,0],[3,1],[3,3],[3,4],[3,5],[4,0],[4,1],[4,3],[4,4],[4,5],[5,0],[5,1],[5,3],[5,4],[5,5]],"domain":"Complete_finite_quotient_set","operation":"quotient-partition","semantic_template":"qf-mod30-quotient-partition/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,3],[0,4],[0,5],[1,0],[1,1],[1,3],[1,4],[1,5],[2,2],[3,0],[3,1],[3,3],[3,4],[3,5],[4,0],[4,1],[4,3],[4,4],[4,5],[5,0],[5,1],[5,3],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0057","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-partition；資料={\"n\":6,\"pairs\":[[0,0],[0,2],[1,1],[1,3],[1,4],[1,5],[2,0],[2,2],[3,1],[3,3],[3,4],[3,5],[4,1],[4,3],[4,4],[4,5],[5,1],[5,3],[5,4],[5,5]]}。請依 [classes,covers,disjoint] 欄序回答。","answer":{"kind":"expression","value":"([[0, 2], [1, 3, 4, 5]], 是, 是)","explanation_zh":"Wave4第四層核對=([[0, 2], [1, 3, 4, 5]], 是, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-partition","seed":202610304057,"parameters":{"n":6,"pairs":[[0,0],[0,2],[1,1],[1,3],[1,4],[1,5],[2,0],[2,2],[3,1],[3,3],[3,4],[3,5],[4,1],[4,3],[4,4],[4,5],[5,1],[5,3],[5,4],[5,5]],"domain":"Complete_finite_quotient_set","operation":"quotient-partition","semantic_template":"qf-mod30-quotient-partition/{\"n\":6,\"pairs\":[[0,0],[0,2],[1,1],[1,3],[1,4],[1,5],[2,0],[2,2],[3,1],[3,3],[3,4],[3,5],[4,1],[4,3],[4,4],[4,5],[5,1],[5,3],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0058","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-partition；資料={\"n\":6,\"pairs\":[[0,0],[1,1],[1,3],[1,4],[1,5],[2,2],[3,1],[3,3],[3,4],[3,5],[4,1],[4,3],[4,4],[4,5],[5,1],[5,3],[5,4],[5,5]]}。請依 [classes,covers,disjoint] 欄序回答。","answer":{"kind":"expression","value":"([[0], [1, 3, 4, 5], [2]], 是, 是)","explanation_zh":"Wave4第四層核對=([[0], [1, 3, 4, 5], [2]], 是, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-partition","seed":202610304058,"parameters":{"n":6,"pairs":[[0,0],[1,1],[1,3],[1,4],[1,5],[2,2],[3,1],[3,3],[3,4],[3,5],[4,1],[4,3],[4,4],[4,5],[5,1],[5,3],[5,4],[5,5]],"domain":"Complete_finite_quotient_set","operation":"quotient-partition","semantic_template":"qf-mod30-quotient-partition/{\"n\":6,\"pairs\":[[0,0],[1,1],[1,3],[1,4],[1,5],[2,2],[3,1],[3,3],[3,4],[3,5],[4,1],[4,3],[4,4],[4,5],[5,1],[5,3],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0059","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-partition；資料={\"n\":6,\"pairs\":[[0,0],[0,3],[0,4],[0,5],[1,1],[1,2],[2,1],[2,2],[3,0],[3,3],[3,4],[3,5],[4,0],[4,3],[4,4],[4,5],[5,0],[5,3],[5,4],[5,5]]}。請依 [classes,covers,disjoint] 欄序回答。","answer":{"kind":"expression","value":"([[0, 3, 4, 5], [1, 2]], 是, 是)","explanation_zh":"Wave4第四層核對=([[0, 3, 4, 5], [1, 2]], 是, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-partition","seed":202610304059,"parameters":{"n":6,"pairs":[[0,0],[0,3],[0,4],[0,5],[1,1],[1,2],[2,1],[2,2],[3,0],[3,3],[3,4],[3,5],[4,0],[4,3],[4,4],[4,5],[5,0],[5,3],[5,4],[5,5]],"domain":"Complete_finite_quotient_set","operation":"quotient-partition","semantic_template":"qf-mod30-quotient-partition/{\"n\":6,\"pairs\":[[0,0],[0,3],[0,4],[0,5],[1,1],[1,2],[2,1],[2,2],[3,0],[3,3],[3,4],[3,5],[4,0],[4,3],[4,4],[4,5],[5,0],[5,3],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0060","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-partition；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[1,0],[1,1],[1,2],[2,0],[2,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]]}。請依 [classes,covers,disjoint] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 2], [3, 4, 5]], 是, 是)","explanation_zh":"Wave4第四層核對=([[0, 1, 2], [3, 4, 5]], 是, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-partition","seed":202610304060,"parameters":{"n":6,"pairs":[[0,0],[0,1],[0,2],[1,0],[1,1],[1,2],[2,0],[2,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]],"domain":"Complete_finite_quotient_set","operation":"quotient-partition","semantic_template":"qf-mod30-quotient-partition/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[1,0],[1,1],[1,2],[2,0],[2,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0061","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-partition；資料={\"n\":6,\"pairs\":[[0,0],[1,1],[1,2],[2,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]]}。請依 [classes,covers,disjoint] 欄序回答。","answer":{"kind":"expression","value":"([[0], [1, 2], [3, 4, 5]], 是, 是)","explanation_zh":"Wave4第四層核對=([[0], [1, 2], [3, 4, 5]], 是, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-partition","seed":202610304061,"parameters":{"n":6,"pairs":[[0,0],[1,1],[1,2],[2,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]],"domain":"Complete_finite_quotient_set","operation":"quotient-partition","semantic_template":"qf-mod30-quotient-partition/{\"n\":6,\"pairs\":[[0,0],[1,1],[1,2],[2,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0062","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-partition；資料={\"n\":6,\"pairs\":[[0,0],[0,3],[0,4],[0,5],[1,1],[2,2],[3,0],[3,3],[3,4],[3,5],[4,0],[4,3],[4,4],[4,5],[5,0],[5,3],[5,4],[5,5]]}。請依 [classes,covers,disjoint] 欄序回答。","answer":{"kind":"expression","value":"([[0, 3, 4, 5], [1], [2]], 是, 是)","explanation_zh":"Wave4第四層核對=([[0, 3, 4, 5], [1], [2]], 是, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-partition","seed":202610304062,"parameters":{"n":6,"pairs":[[0,0],[0,3],[0,4],[0,5],[1,1],[2,2],[3,0],[3,3],[3,4],[3,5],[4,0],[4,3],[4,4],[4,5],[5,0],[5,3],[5,4],[5,5]],"domain":"Complete_finite_quotient_set","operation":"quotient-partition","semantic_template":"qf-mod30-quotient-partition/{\"n\":6,\"pairs\":[[0,0],[0,3],[0,4],[0,5],[1,1],[2,2],[3,0],[3,3],[3,4],[3,5],[4,0],[4,3],[4,4],[4,5],[5,0],[5,3],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0063","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-partition；資料={\"n\":6,\"pairs\":[[0,0],[0,2],[1,1],[2,0],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]]}。請依 [classes,covers,disjoint] 欄序回答。","answer":{"kind":"expression","value":"([[0, 2], [1], [3, 4, 5]], 是, 是)","explanation_zh":"Wave4第四層核對=([[0, 2], [1], [3, 4, 5]], 是, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-partition","seed":202610304063,"parameters":{"n":6,"pairs":[[0,0],[0,2],[1,1],[2,0],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]],"domain":"Complete_finite_quotient_set","operation":"quotient-partition","semantic_template":"qf-mod30-quotient-partition/{\"n\":6,\"pairs\":[[0,0],[0,2],[1,1],[2,0],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0064","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-partition；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]]}。請依 [classes,covers,disjoint] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1], [2], [3, 4, 5]], 是, 是)","explanation_zh":"Wave4第四層核對=([[0, 1], [2], [3, 4, 5]], 是, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-partition","seed":202610304064,"parameters":{"n":6,"pairs":[[0,0],[0,1],[1,0],[1,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]],"domain":"Complete_finite_quotient_set","operation":"quotient-partition","semantic_template":"qf-mod30-quotient-partition/{\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0065","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-partition；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[0,4],[0,5],[1,0],[1,1],[1,2],[1,4],[1,5],[2,0],[2,1],[2,2],[2,4],[2,5],[3,3],[4,0],[4,1],[4,2],[4,4],[4,5],[5,0],[5,1],[5,2],[5,4],[5,5]]}。請依 [classes,covers,disjoint] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 2, 4, 5], [3]], 是, 是)","explanation_zh":"Wave4第四層核對=([[0, 1, 2, 4, 5], [3]], 是, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-partition","seed":202610304065,"parameters":{"n":6,"pairs":[[0,0],[0,1],[0,2],[0,4],[0,5],[1,0],[1,1],[1,2],[1,4],[1,5],[2,0],[2,1],[2,2],[2,4],[2,5],[3,3],[4,0],[4,1],[4,2],[4,4],[4,5],[5,0],[5,1],[5,2],[5,4],[5,5]],"domain":"Complete_finite_quotient_set","operation":"quotient-partition","semantic_template":"qf-mod30-quotient-partition/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[0,4],[0,5],[1,0],[1,1],[1,2],[1,4],[1,5],[2,0],[2,1],[2,2],[2,4],[2,5],[3,3],[4,0],[4,1],[4,2],[4,4],[4,5],[5,0],[5,1],[5,2],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0066","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-partition；資料={\"n\":6,\"pairs\":[[0,0],[0,3],[1,1],[1,2],[1,4],[1,5],[2,1],[2,2],[2,4],[2,5],[3,0],[3,3],[4,1],[4,2],[4,4],[4,5],[5,1],[5,2],[5,4],[5,5]]}。請依 [classes,covers,disjoint] 欄序回答。","answer":{"kind":"expression","value":"([[0, 3], [1, 2, 4, 5]], 是, 是)","explanation_zh":"Wave4第四層核對=([[0, 3], [1, 2, 4, 5]], 是, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-partition","seed":202610304066,"parameters":{"n":6,"pairs":[[0,0],[0,3],[1,1],[1,2],[1,4],[1,5],[2,1],[2,2],[2,4],[2,5],[3,0],[3,3],[4,1],[4,2],[4,4],[4,5],[5,1],[5,2],[5,4],[5,5]],"domain":"Complete_finite_quotient_set","operation":"quotient-partition","semantic_template":"qf-mod30-quotient-partition/{\"n\":6,\"pairs\":[[0,0],[0,3],[1,1],[1,2],[1,4],[1,5],[2,1],[2,2],[2,4],[2,5],[3,0],[3,3],[4,1],[4,2],[4,4],[4,5],[5,1],[5,2],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0067","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-partition；資料={\"n\":6,\"pairs\":[[0,0],[1,1],[1,2],[1,4],[1,5],[2,1],[2,2],[2,4],[2,5],[3,3],[4,1],[4,2],[4,4],[4,5],[5,1],[5,2],[5,4],[5,5]]}。請依 [classes,covers,disjoint] 欄序回答。","answer":{"kind":"expression","value":"([[0], [1, 2, 4, 5], [3]], 是, 是)","explanation_zh":"Wave4第四層核對=([[0], [1, 2, 4, 5], [3]], 是, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-partition","seed":202610304067,"parameters":{"n":6,"pairs":[[0,0],[1,1],[1,2],[1,4],[1,5],[2,1],[2,2],[2,4],[2,5],[3,3],[4,1],[4,2],[4,4],[4,5],[5,1],[5,2],[5,4],[5,5]],"domain":"Complete_finite_quotient_set","operation":"quotient-partition","semantic_template":"qf-mod30-quotient-partition/{\"n\":6,\"pairs\":[[0,0],[1,1],[1,2],[1,4],[1,5],[2,1],[2,2],[2,4],[2,5],[3,3],[4,1],[4,2],[4,4],[4,5],[5,1],[5,2],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0068","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-partition；資料={\"n\":6,\"pairs\":[[0,0],[0,2],[0,4],[0,5],[1,1],[1,3],[2,0],[2,2],[2,4],[2,5],[3,1],[3,3],[4,0],[4,2],[4,4],[4,5],[5,0],[5,2],[5,4],[5,5]]}。請依 [classes,covers,disjoint] 欄序回答。","answer":{"kind":"expression","value":"([[0, 2, 4, 5], [1, 3]], 是, 是)","explanation_zh":"Wave4第四層核對=([[0, 2, 4, 5], [1, 3]], 是, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-partition","seed":202610304068,"parameters":{"n":6,"pairs":[[0,0],[0,2],[0,4],[0,5],[1,1],[1,3],[2,0],[2,2],[2,4],[2,5],[3,1],[3,3],[4,0],[4,2],[4,4],[4,5],[5,0],[5,2],[5,4],[5,5]],"domain":"Complete_finite_quotient_set","operation":"quotient-partition","semantic_template":"qf-mod30-quotient-partition/{\"n\":6,\"pairs\":[[0,0],[0,2],[0,4],[0,5],[1,1],[1,3],[2,0],[2,2],[2,4],[2,5],[3,1],[3,3],[4,0],[4,2],[4,4],[4,5],[5,0],[5,2],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0069","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-partition；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[0,3],[1,0],[1,1],[1,3],[2,2],[2,4],[2,5],[3,0],[3,1],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]]}。請依 [classes,covers,disjoint] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 3], [2, 4, 5]], 是, 是)","explanation_zh":"Wave4第四層核對=([[0, 1, 3], [2, 4, 5]], 是, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-partition","seed":202610304069,"parameters":{"n":6,"pairs":[[0,0],[0,1],[0,3],[1,0],[1,1],[1,3],[2,2],[2,4],[2,5],[3,0],[3,1],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]],"domain":"Complete_finite_quotient_set","operation":"quotient-partition","semantic_template":"qf-mod30-quotient-partition/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,3],[1,0],[1,1],[1,3],[2,2],[2,4],[2,5],[3,0],[3,1],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0070","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-partition；資料={\"n\":6,\"pairs\":[[0,0],[1,1],[1,3],[2,2],[2,4],[2,5],[3,1],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]]}。請依 [classes,covers,disjoint] 欄序回答。","answer":{"kind":"expression","value":"([[0], [1, 3], [2, 4, 5]], 是, 是)","explanation_zh":"Wave4第四層核對=([[0], [1, 3], [2, 4, 5]], 是, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-partition","seed":202610304070,"parameters":{"n":6,"pairs":[[0,0],[1,1],[1,3],[2,2],[2,4],[2,5],[3,1],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]],"domain":"Complete_finite_quotient_set","operation":"quotient-partition","semantic_template":"qf-mod30-quotient-partition/{\"n\":6,\"pairs\":[[0,0],[1,1],[1,3],[2,2],[2,4],[2,5],[3,1],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0071","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-partition；資料={\"n\":6,\"pairs\":[[0,0],[0,2],[0,4],[0,5],[1,1],[2,0],[2,2],[2,4],[2,5],[3,3],[4,0],[4,2],[4,4],[4,5],[5,0],[5,2],[5,4],[5,5]]}。請依 [classes,covers,disjoint] 欄序回答。","answer":{"kind":"expression","value":"([[0, 2, 4, 5], [1], [3]], 是, 是)","explanation_zh":"Wave4第四層核對=([[0, 2, 4, 5], [1], [3]], 是, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-partition","seed":202610304071,"parameters":{"n":6,"pairs":[[0,0],[0,2],[0,4],[0,5],[1,1],[2,0],[2,2],[2,4],[2,5],[3,3],[4,0],[4,2],[4,4],[4,5],[5,0],[5,2],[5,4],[5,5]],"domain":"Complete_finite_quotient_set","operation":"quotient-partition","semantic_template":"qf-mod30-quotient-partition/{\"n\":6,\"pairs\":[[0,0],[0,2],[0,4],[0,5],[1,1],[2,0],[2,2],[2,4],[2,5],[3,3],[4,0],[4,2],[4,4],[4,5],[5,0],[5,2],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0072","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-partition；資料={\"n\":6,\"pairs\":[[0,0],[0,3],[1,1],[2,2],[2,4],[2,5],[3,0],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]]}。請依 [classes,covers,disjoint] 欄序回答。","answer":{"kind":"expression","value":"([[0, 3], [1], [2, 4, 5]], 是, 是)","explanation_zh":"Wave4第四層核對=([[0, 3], [1], [2, 4, 5]], 是, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-partition","seed":202610304072,"parameters":{"n":6,"pairs":[[0,0],[0,3],[1,1],[2,2],[2,4],[2,5],[3,0],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]],"domain":"Complete_finite_quotient_set","operation":"quotient-partition","semantic_template":"qf-mod30-quotient-partition/{\"n\":6,\"pairs\":[[0,0],[0,3],[1,1],[2,2],[2,4],[2,5],[3,0],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0073","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-partition；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[2,4],[2,5],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]]}。請依 [classes,covers,disjoint] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1], [2, 4, 5], [3]], 是, 是)","explanation_zh":"Wave4第四層核對=([[0, 1], [2, 4, 5], [3]], 是, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-partition","seed":202610304073,"parameters":{"n":6,"pairs":[[0,0],[0,1],[1,0],[1,1],[2,2],[2,4],[2,5],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]],"domain":"Complete_finite_quotient_set","operation":"quotient-partition","semantic_template":"qf-mod30-quotient-partition/{\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[2,4],[2,5],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0074","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-partition；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[0,4],[0,5],[1,0],[1,1],[1,4],[1,5],[2,2],[2,3],[3,2],[3,3],[4,0],[4,1],[4,4],[4,5],[5,0],[5,1],[5,4],[5,5]]}。請依 [classes,covers,disjoint] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 4, 5], [2, 3]], 是, 是)","explanation_zh":"Wave4第四層核對=([[0, 1, 4, 5], [2, 3]], 是, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-partition","seed":202610304074,"parameters":{"n":6,"pairs":[[0,0],[0,1],[0,4],[0,5],[1,0],[1,1],[1,4],[1,5],[2,2],[2,3],[3,2],[3,3],[4,0],[4,1],[4,4],[4,5],[5,0],[5,1],[5,4],[5,5]],"domain":"Complete_finite_quotient_set","operation":"quotient-partition","semantic_template":"qf-mod30-quotient-partition/{\"n\":6,\"pairs\":[[0,0],[0,1],[0,4],[0,5],[1,0],[1,1],[1,4],[1,5],[2,2],[2,3],[3,2],[3,3],[4,0],[4,1],[4,4],[4,5],[5,0],[5,1],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0075","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-partition；資料={\"n\":6,\"pairs\":[[0,0],[0,2],[0,3],[1,1],[1,4],[1,5],[2,0],[2,2],[2,3],[3,0],[3,2],[3,3],[4,1],[4,4],[4,5],[5,1],[5,4],[5,5]]}。請依 [classes,covers,disjoint] 欄序回答。","answer":{"kind":"expression","value":"([[0, 2, 3], [1, 4, 5]], 是, 是)","explanation_zh":"Wave4第四層核對=([[0, 2, 3], [1, 4, 5]], 是, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-partition","seed":202610304075,"parameters":{"n":6,"pairs":[[0,0],[0,2],[0,3],[1,1],[1,4],[1,5],[2,0],[2,2],[2,3],[3,0],[3,2],[3,3],[4,1],[4,4],[4,5],[5,1],[5,4],[5,5]],"domain":"Complete_finite_quotient_set","operation":"quotient-partition","semantic_template":"qf-mod30-quotient-partition/{\"n\":6,\"pairs\":[[0,0],[0,2],[0,3],[1,1],[1,4],[1,5],[2,0],[2,2],[2,3],[3,0],[3,2],[3,3],[4,1],[4,4],[4,5],[5,1],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0076","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-representative；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[0,3],[0,4],[0,5],[1,0],[1,1],[1,2],[1,3],[1,4],[1,5],[2,0],[2,1],[2,2],[2,3],[2,4],[2,5],[3,0],[3,1],[3,2],[3,3],[3,4],[3,5],[4,0],[4,1],[4,2],[4,3],[4,4],[4,5],[5,0],[5,1],[5,2],[5,3],[5,4],[5,5]],\"a\":0,\"b\":1}。請依 [a,b,related,class_a,class_b,invariance,boundary] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 是, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是, no_quotient_operation_claim)","explanation_zh":"Wave4第四層核對=(0, 1, 是, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是, no_quotient_operation_claim)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-representative","seed":202610304076,"parameters":{"n":6,"pairs":[[0,0],[0,1],[0,2],[0,3],[0,4],[0,5],[1,0],[1,1],[1,2],[1,3],[1,4],[1,5],[2,0],[2,1],[2,2],[2,3],[2,4],[2,5],[3,0],[3,1],[3,2],[3,3],[3,4],[3,5],[4,0],[4,1],[4,2],[4,3],[4,4],[4,5],[5,0],[5,1],[5,2],[5,3],[5,4],[5,5]],"a":0,"b":1,"domain":"Complete_finite_quotient_set","operation":"quotient-representative","semantic_template":"qf-mod30-quotient-representative/{\"a\":0,\"b\":1,\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[0,3],[0,4],[0,5],[1,0],[1,1],[1,2],[1,3],[1,4],[1,5],[2,0],[2,1],[2,2],[2,3],[2,4],[2,5],[3,0],[3,1],[3,2],[3,3],[3,4],[3,5],[4,0],[4,1],[4,2],[4,3],[4,4],[4,5],[5,0],[5,1],[5,2],[5,3],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0077","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-representative；資料={\"n\":6,\"pairs\":[[0,0],[1,1],[1,2],[1,3],[1,4],[1,5],[2,1],[2,2],[2,3],[2,4],[2,5],[3,1],[3,2],[3,3],[3,4],[3,5],[4,1],[4,2],[4,3],[4,4],[4,5],[5,1],[5,2],[5,3],[5,4],[5,5]],\"a\":1,\"b\":4}。請依 [a,b,related,class_a,class_b,invariance,boundary] 欄序回答。","answer":{"kind":"expression","value":"(1, 4, 是, [1, 2, 3, 4, 5], [1, 2, 3, 4, 5], 是, no_quotient_operation_claim)","explanation_zh":"Wave4第四層核對=(1, 4, 是, [1, 2, 3, 4, 5], [1, 2, 3, 4, 5], 是, no_quotient_operation_claim)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-representative","seed":202610304077,"parameters":{"n":6,"pairs":[[0,0],[1,1],[1,2],[1,3],[1,4],[1,5],[2,1],[2,2],[2,3],[2,4],[2,5],[3,1],[3,2],[3,3],[3,4],[3,5],[4,1],[4,2],[4,3],[4,4],[4,5],[5,1],[5,2],[5,3],[5,4],[5,5]],"a":1,"b":4,"domain":"Complete_finite_quotient_set","operation":"quotient-representative","semantic_template":"qf-mod30-quotient-representative/{\"a\":1,\"b\":4,\"n\":6,\"pairs\":[[0,0],[1,1],[1,2],[1,3],[1,4],[1,5],[2,1],[2,2],[2,3],[2,4],[2,5],[3,1],[3,2],[3,3],[3,4],[3,5],[4,1],[4,2],[4,3],[4,4],[4,5],[5,1],[5,2],[5,3],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0078","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-representative；資料={\"n\":6,\"pairs\":[[0,0],[0,2],[0,3],[0,4],[0,5],[1,1],[2,0],[2,2],[2,3],[2,4],[2,5],[3,0],[3,2],[3,3],[3,4],[3,5],[4,0],[4,2],[4,3],[4,4],[4,5],[5,0],[5,2],[5,3],[5,4],[5,5]],\"a\":2,\"b\":1}。請依 [a,b,related,class_a,class_b,invariance,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, 1, 否, [0, 2, 3, 4, 5], [1], 是, no_quotient_operation_claim)","explanation_zh":"Wave4第四層核對=(2, 1, 否, [0, 2, 3, 4, 5], [1], 是, no_quotient_operation_claim)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-representative","seed":202610304078,"parameters":{"n":6,"pairs":[[0,0],[0,2],[0,3],[0,4],[0,5],[1,1],[2,0],[2,2],[2,3],[2,4],[2,5],[3,0],[3,2],[3,3],[3,4],[3,5],[4,0],[4,2],[4,3],[4,4],[4,5],[5,0],[5,2],[5,3],[5,4],[5,5]],"a":2,"b":1,"domain":"Complete_finite_quotient_set","operation":"quotient-representative","semantic_template":"qf-mod30-quotient-representative/{\"a\":2,\"b\":1,\"n\":6,\"pairs\":[[0,0],[0,2],[0,3],[0,4],[0,5],[1,1],[2,0],[2,2],[2,3],[2,4],[2,5],[3,0],[3,2],[3,3],[3,4],[3,5],[4,0],[4,2],[4,3],[4,4],[4,5],[5,0],[5,2],[5,3],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0079","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-representative；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[2,3],[2,4],[2,5],[3,2],[3,3],[3,4],[3,5],[4,2],[4,3],[4,4],[4,5],[5,2],[5,3],[5,4],[5,5]],\"a\":3,\"b\":4}。請依 [a,b,related,class_a,class_b,invariance,boundary] 欄序回答。","answer":{"kind":"expression","value":"(3, 4, 是, [2, 3, 4, 5], [2, 3, 4, 5], 是, no_quotient_operation_claim)","explanation_zh":"Wave4第四層核對=(3, 4, 是, [2, 3, 4, 5], [2, 3, 4, 5], 是, no_quotient_operation_claim)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-representative","seed":202610304079,"parameters":{"n":6,"pairs":[[0,0],[0,1],[1,0],[1,1],[2,2],[2,3],[2,4],[2,5],[3,2],[3,3],[3,4],[3,5],[4,2],[4,3],[4,4],[4,5],[5,2],[5,3],[5,4],[5,5]],"a":3,"b":4,"domain":"Complete_finite_quotient_set","operation":"quotient-representative","semantic_template":"qf-mod30-quotient-representative/{\"a\":3,\"b\":4,\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[2,3],[2,4],[2,5],[3,2],[3,3],[3,4],[3,5],[4,2],[4,3],[4,4],[4,5],[5,2],[5,3],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0080","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-representative；資料={\"n\":6,\"pairs\":[[0,0],[1,1],[2,2],[2,3],[2,4],[2,5],[3,2],[3,3],[3,4],[3,5],[4,2],[4,3],[4,4],[4,5],[5,2],[5,3],[5,4],[5,5]],\"a\":4,\"b\":1}。請依 [a,b,related,class_a,class_b,invariance,boundary] 欄序回答。","answer":{"kind":"expression","value":"(4, 1, 否, [2, 3, 4, 5], [1], 是, no_quotient_operation_claim)","explanation_zh":"Wave4第四層核對=(4, 1, 否, [2, 3, 4, 5], [1], 是, no_quotient_operation_claim)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-representative","seed":202610304080,"parameters":{"n":6,"pairs":[[0,0],[1,1],[2,2],[2,3],[2,4],[2,5],[3,2],[3,3],[3,4],[3,5],[4,2],[4,3],[4,4],[4,5],[5,2],[5,3],[5,4],[5,5]],"a":4,"b":1,"domain":"Complete_finite_quotient_set","operation":"quotient-representative","semantic_template":"qf-mod30-quotient-representative/{\"a\":4,\"b\":1,\"n\":6,\"pairs\":[[0,0],[1,1],[2,2],[2,3],[2,4],[2,5],[3,2],[3,3],[3,4],[3,5],[4,2],[4,3],[4,4],[4,5],[5,2],[5,3],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0081","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-representative；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[0,3],[0,4],[0,5],[1,0],[1,1],[1,3],[1,4],[1,5],[2,2],[3,0],[3,1],[3,3],[3,4],[3,5],[4,0],[4,1],[4,3],[4,4],[4,5],[5,0],[5,1],[5,3],[5,4],[5,5]],\"a\":5,\"b\":4}。請依 [a,b,related,class_a,class_b,invariance,boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, 4, 是, [0, 1, 3, 4, 5], [0, 1, 3, 4, 5], 是, no_quotient_operation_claim)","explanation_zh":"Wave4第四層核對=(5, 4, 是, [0, 1, 3, 4, 5], [0, 1, 3, 4, 5], 是, no_quotient_operation_claim)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-representative","seed":202610304081,"parameters":{"n":6,"pairs":[[0,0],[0,1],[0,3],[0,4],[0,5],[1,0],[1,1],[1,3],[1,4],[1,5],[2,2],[3,0],[3,1],[3,3],[3,4],[3,5],[4,0],[4,1],[4,3],[4,4],[4,5],[5,0],[5,1],[5,3],[5,4],[5,5]],"a":5,"b":4,"domain":"Complete_finite_quotient_set","operation":"quotient-representative","semantic_template":"qf-mod30-quotient-representative/{\"a\":5,\"b\":4,\"n\":6,\"pairs\":[[0,0],[0,1],[0,3],[0,4],[0,5],[1,0],[1,1],[1,3],[1,4],[1,5],[2,2],[3,0],[3,1],[3,3],[3,4],[3,5],[4,0],[4,1],[4,3],[4,4],[4,5],[5,0],[5,1],[5,3],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0082","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-representative；資料={\"n\":6,\"pairs\":[[0,0],[0,2],[1,1],[1,3],[1,4],[1,5],[2,0],[2,2],[3,1],[3,3],[3,4],[3,5],[4,1],[4,3],[4,4],[4,5],[5,1],[5,3],[5,4],[5,5]],\"a\":0,\"b\":1}。請依 [a,b,related,class_a,class_b,invariance,boundary] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 否, [0, 2], [1, 3, 4, 5], 是, no_quotient_operation_claim)","explanation_zh":"Wave4第四層核對=(0, 1, 否, [0, 2], [1, 3, 4, 5], 是, no_quotient_operation_claim)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-representative","seed":202610304082,"parameters":{"n":6,"pairs":[[0,0],[0,2],[1,1],[1,3],[1,4],[1,5],[2,0],[2,2],[3,1],[3,3],[3,4],[3,5],[4,1],[4,3],[4,4],[4,5],[5,1],[5,3],[5,4],[5,5]],"a":0,"b":1,"domain":"Complete_finite_quotient_set","operation":"quotient-representative","semantic_template":"qf-mod30-quotient-representative/{\"a\":0,\"b\":1,\"n\":6,\"pairs\":[[0,0],[0,2],[1,1],[1,3],[1,4],[1,5],[2,0],[2,2],[3,1],[3,3],[3,4],[3,5],[4,1],[4,3],[4,4],[4,5],[5,1],[5,3],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0083","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-representative；資料={\"n\":6,\"pairs\":[[0,0],[1,1],[1,3],[1,4],[1,5],[2,2],[3,1],[3,3],[3,4],[3,5],[4,1],[4,3],[4,4],[4,5],[5,1],[5,3],[5,4],[5,5]],\"a\":1,\"b\":4}。請依 [a,b,related,class_a,class_b,invariance,boundary] 欄序回答。","answer":{"kind":"expression","value":"(1, 4, 是, [1, 3, 4, 5], [1, 3, 4, 5], 是, no_quotient_operation_claim)","explanation_zh":"Wave4第四層核對=(1, 4, 是, [1, 3, 4, 5], [1, 3, 4, 5], 是, no_quotient_operation_claim)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-representative","seed":202610304083,"parameters":{"n":6,"pairs":[[0,0],[1,1],[1,3],[1,4],[1,5],[2,2],[3,1],[3,3],[3,4],[3,5],[4,1],[4,3],[4,4],[4,5],[5,1],[5,3],[5,4],[5,5]],"a":1,"b":4,"domain":"Complete_finite_quotient_set","operation":"quotient-representative","semantic_template":"qf-mod30-quotient-representative/{\"a\":1,\"b\":4,\"n\":6,\"pairs\":[[0,0],[1,1],[1,3],[1,4],[1,5],[2,2],[3,1],[3,3],[3,4],[3,5],[4,1],[4,3],[4,4],[4,5],[5,1],[5,3],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0084","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-representative；資料={\"n\":6,\"pairs\":[[0,0],[0,3],[0,4],[0,5],[1,1],[1,2],[2,1],[2,2],[3,0],[3,3],[3,4],[3,5],[4,0],[4,3],[4,4],[4,5],[5,0],[5,3],[5,4],[5,5]],\"a\":2,\"b\":1}。請依 [a,b,related,class_a,class_b,invariance,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, 1, 是, [1, 2], [1, 2], 是, no_quotient_operation_claim)","explanation_zh":"Wave4第四層核對=(2, 1, 是, [1, 2], [1, 2], 是, no_quotient_operation_claim)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-representative","seed":202610304084,"parameters":{"n":6,"pairs":[[0,0],[0,3],[0,4],[0,5],[1,1],[1,2],[2,1],[2,2],[3,0],[3,3],[3,4],[3,5],[4,0],[4,3],[4,4],[4,5],[5,0],[5,3],[5,4],[5,5]],"a":2,"b":1,"domain":"Complete_finite_quotient_set","operation":"quotient-representative","semantic_template":"qf-mod30-quotient-representative/{\"a\":2,\"b\":1,\"n\":6,\"pairs\":[[0,0],[0,3],[0,4],[0,5],[1,1],[1,2],[2,1],[2,2],[3,0],[3,3],[3,4],[3,5],[4,0],[4,3],[4,4],[4,5],[5,0],[5,3],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0085","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-representative；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[1,0],[1,1],[1,2],[2,0],[2,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]],\"a\":3,\"b\":4}。請依 [a,b,related,class_a,class_b,invariance,boundary] 欄序回答。","answer":{"kind":"expression","value":"(3, 4, 是, [3, 4, 5], [3, 4, 5], 是, no_quotient_operation_claim)","explanation_zh":"Wave4第四層核對=(3, 4, 是, [3, 4, 5], [3, 4, 5], 是, no_quotient_operation_claim)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-representative","seed":202610304085,"parameters":{"n":6,"pairs":[[0,0],[0,1],[0,2],[1,0],[1,1],[1,2],[2,0],[2,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]],"a":3,"b":4,"domain":"Complete_finite_quotient_set","operation":"quotient-representative","semantic_template":"qf-mod30-quotient-representative/{\"a\":3,\"b\":4,\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[1,0],[1,1],[1,2],[2,0],[2,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0086","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-representative；資料={\"n\":6,\"pairs\":[[0,0],[1,1],[1,2],[2,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]],\"a\":4,\"b\":1}。請依 [a,b,related,class_a,class_b,invariance,boundary] 欄序回答。","answer":{"kind":"expression","value":"(4, 1, 否, [3, 4, 5], [1, 2], 是, no_quotient_operation_claim)","explanation_zh":"Wave4第四層核對=(4, 1, 否, [3, 4, 5], [1, 2], 是, no_quotient_operation_claim)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-representative","seed":202610304086,"parameters":{"n":6,"pairs":[[0,0],[1,1],[1,2],[2,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]],"a":4,"b":1,"domain":"Complete_finite_quotient_set","operation":"quotient-representative","semantic_template":"qf-mod30-quotient-representative/{\"a\":4,\"b\":1,\"n\":6,\"pairs\":[[0,0],[1,1],[1,2],[2,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0087","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-representative；資料={\"n\":6,\"pairs\":[[0,0],[0,3],[0,4],[0,5],[1,1],[2,2],[3,0],[3,3],[3,4],[3,5],[4,0],[4,3],[4,4],[4,5],[5,0],[5,3],[5,4],[5,5]],\"a\":5,\"b\":4}。請依 [a,b,related,class_a,class_b,invariance,boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, 4, 是, [0, 3, 4, 5], [0, 3, 4, 5], 是, no_quotient_operation_claim)","explanation_zh":"Wave4第四層核對=(5, 4, 是, [0, 3, 4, 5], [0, 3, 4, 5], 是, no_quotient_operation_claim)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-representative","seed":202610304087,"parameters":{"n":6,"pairs":[[0,0],[0,3],[0,4],[0,5],[1,1],[2,2],[3,0],[3,3],[3,4],[3,5],[4,0],[4,3],[4,4],[4,5],[5,0],[5,3],[5,4],[5,5]],"a":5,"b":4,"domain":"Complete_finite_quotient_set","operation":"quotient-representative","semantic_template":"qf-mod30-quotient-representative/{\"a\":5,\"b\":4,\"n\":6,\"pairs\":[[0,0],[0,3],[0,4],[0,5],[1,1],[2,2],[3,0],[3,3],[3,4],[3,5],[4,0],[4,3],[4,4],[4,5],[5,0],[5,3],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0088","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-representative；資料={\"n\":6,\"pairs\":[[0,0],[0,2],[1,1],[2,0],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]],\"a\":0,\"b\":1}。請依 [a,b,related,class_a,class_b,invariance,boundary] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 否, [0, 2], [1], 是, no_quotient_operation_claim)","explanation_zh":"Wave4第四層核對=(0, 1, 否, [0, 2], [1], 是, no_quotient_operation_claim)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-representative","seed":202610304088,"parameters":{"n":6,"pairs":[[0,0],[0,2],[1,1],[2,0],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]],"a":0,"b":1,"domain":"Complete_finite_quotient_set","operation":"quotient-representative","semantic_template":"qf-mod30-quotient-representative/{\"a\":0,\"b\":1,\"n\":6,\"pairs\":[[0,0],[0,2],[1,1],[2,0],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0089","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-representative；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]],\"a\":1,\"b\":4}。請依 [a,b,related,class_a,class_b,invariance,boundary] 欄序回答。","answer":{"kind":"expression","value":"(1, 4, 否, [0, 1], [3, 4, 5], 是, no_quotient_operation_claim)","explanation_zh":"Wave4第四層核對=(1, 4, 否, [0, 1], [3, 4, 5], 是, no_quotient_operation_claim)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-representative","seed":202610304089,"parameters":{"n":6,"pairs":[[0,0],[0,1],[1,0],[1,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]],"a":1,"b":4,"domain":"Complete_finite_quotient_set","operation":"quotient-representative","semantic_template":"qf-mod30-quotient-representative/{\"a\":1,\"b\":4,\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[3,3],[3,4],[3,5],[4,3],[4,4],[4,5],[5,3],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0090","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-representative；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[0,4],[0,5],[1,0],[1,1],[1,2],[1,4],[1,5],[2,0],[2,1],[2,2],[2,4],[2,5],[3,3],[4,0],[4,1],[4,2],[4,4],[4,5],[5,0],[5,1],[5,2],[5,4],[5,5]],\"a\":2,\"b\":1}。請依 [a,b,related,class_a,class_b,invariance,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, 1, 是, [0, 1, 2, 4, 5], [0, 1, 2, 4, 5], 是, no_quotient_operation_claim)","explanation_zh":"Wave4第四層核對=(2, 1, 是, [0, 1, 2, 4, 5], [0, 1, 2, 4, 5], 是, no_quotient_operation_claim)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-representative","seed":202610304090,"parameters":{"n":6,"pairs":[[0,0],[0,1],[0,2],[0,4],[0,5],[1,0],[1,1],[1,2],[1,4],[1,5],[2,0],[2,1],[2,2],[2,4],[2,5],[3,3],[4,0],[4,1],[4,2],[4,4],[4,5],[5,0],[5,1],[5,2],[5,4],[5,5]],"a":2,"b":1,"domain":"Complete_finite_quotient_set","operation":"quotient-representative","semantic_template":"qf-mod30-quotient-representative/{\"a\":2,\"b\":1,\"n\":6,\"pairs\":[[0,0],[0,1],[0,2],[0,4],[0,5],[1,0],[1,1],[1,2],[1,4],[1,5],[2,0],[2,1],[2,2],[2,4],[2,5],[3,3],[4,0],[4,1],[4,2],[4,4],[4,5],[5,0],[5,1],[5,2],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0091","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-representative；資料={\"n\":6,\"pairs\":[[0,0],[0,3],[1,1],[1,2],[1,4],[1,5],[2,1],[2,2],[2,4],[2,5],[3,0],[3,3],[4,1],[4,2],[4,4],[4,5],[5,1],[5,2],[5,4],[5,5]],\"a\":3,\"b\":4}。請依 [a,b,related,class_a,class_b,invariance,boundary] 欄序回答。","answer":{"kind":"expression","value":"(3, 4, 否, [0, 3], [1, 2, 4, 5], 是, no_quotient_operation_claim)","explanation_zh":"Wave4第四層核對=(3, 4, 否, [0, 3], [1, 2, 4, 5], 是, no_quotient_operation_claim)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-representative","seed":202610304091,"parameters":{"n":6,"pairs":[[0,0],[0,3],[1,1],[1,2],[1,4],[1,5],[2,1],[2,2],[2,4],[2,5],[3,0],[3,3],[4,1],[4,2],[4,4],[4,5],[5,1],[5,2],[5,4],[5,5]],"a":3,"b":4,"domain":"Complete_finite_quotient_set","operation":"quotient-representative","semantic_template":"qf-mod30-quotient-representative/{\"a\":3,\"b\":4,\"n\":6,\"pairs\":[[0,0],[0,3],[1,1],[1,2],[1,4],[1,5],[2,1],[2,2],[2,4],[2,5],[3,0],[3,3],[4,1],[4,2],[4,4],[4,5],[5,1],[5,2],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0092","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-representative；資料={\"n\":6,\"pairs\":[[0,0],[1,1],[1,2],[1,4],[1,5],[2,1],[2,2],[2,4],[2,5],[3,3],[4,1],[4,2],[4,4],[4,5],[5,1],[5,2],[5,4],[5,5]],\"a\":4,\"b\":1}。請依 [a,b,related,class_a,class_b,invariance,boundary] 欄序回答。","answer":{"kind":"expression","value":"(4, 1, 是, [1, 2, 4, 5], [1, 2, 4, 5], 是, no_quotient_operation_claim)","explanation_zh":"Wave4第四層核對=(4, 1, 是, [1, 2, 4, 5], [1, 2, 4, 5], 是, no_quotient_operation_claim)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-representative","seed":202610304092,"parameters":{"n":6,"pairs":[[0,0],[1,1],[1,2],[1,4],[1,5],[2,1],[2,2],[2,4],[2,5],[3,3],[4,1],[4,2],[4,4],[4,5],[5,1],[5,2],[5,4],[5,5]],"a":4,"b":1,"domain":"Complete_finite_quotient_set","operation":"quotient-representative","semantic_template":"qf-mod30-quotient-representative/{\"a\":4,\"b\":1,\"n\":6,\"pairs\":[[0,0],[1,1],[1,2],[1,4],[1,5],[2,1],[2,2],[2,4],[2,5],[3,3],[4,1],[4,2],[4,4],[4,5],[5,1],[5,2],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0093","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-representative；資料={\"n\":6,\"pairs\":[[0,0],[0,2],[0,4],[0,5],[1,1],[1,3],[2,0],[2,2],[2,4],[2,5],[3,1],[3,3],[4,0],[4,2],[4,4],[4,5],[5,0],[5,2],[5,4],[5,5]],\"a\":5,\"b\":4}。請依 [a,b,related,class_a,class_b,invariance,boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, 4, 是, [0, 2, 4, 5], [0, 2, 4, 5], 是, no_quotient_operation_claim)","explanation_zh":"Wave4第四層核對=(5, 4, 是, [0, 2, 4, 5], [0, 2, 4, 5], 是, no_quotient_operation_claim)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-representative","seed":202610304093,"parameters":{"n":6,"pairs":[[0,0],[0,2],[0,4],[0,5],[1,1],[1,3],[2,0],[2,2],[2,4],[2,5],[3,1],[3,3],[4,0],[4,2],[4,4],[4,5],[5,0],[5,2],[5,4],[5,5]],"a":5,"b":4,"domain":"Complete_finite_quotient_set","operation":"quotient-representative","semantic_template":"qf-mod30-quotient-representative/{\"a\":5,\"b\":4,\"n\":6,\"pairs\":[[0,0],[0,2],[0,4],[0,5],[1,1],[1,3],[2,0],[2,2],[2,4],[2,5],[3,1],[3,3],[4,0],[4,2],[4,4],[4,5],[5,0],[5,2],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0094","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-representative；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[0,3],[1,0],[1,1],[1,3],[2,2],[2,4],[2,5],[3,0],[3,1],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]],\"a\":0,\"b\":1}。請依 [a,b,related,class_a,class_b,invariance,boundary] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 是, [0, 1, 3], [0, 1, 3], 是, no_quotient_operation_claim)","explanation_zh":"Wave4第四層核對=(0, 1, 是, [0, 1, 3], [0, 1, 3], 是, no_quotient_operation_claim)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-representative","seed":202610304094,"parameters":{"n":6,"pairs":[[0,0],[0,1],[0,3],[1,0],[1,1],[1,3],[2,2],[2,4],[2,5],[3,0],[3,1],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]],"a":0,"b":1,"domain":"Complete_finite_quotient_set","operation":"quotient-representative","semantic_template":"qf-mod30-quotient-representative/{\"a\":0,\"b\":1,\"n\":6,\"pairs\":[[0,0],[0,1],[0,3],[1,0],[1,1],[1,3],[2,2],[2,4],[2,5],[3,0],[3,1],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0095","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-representative；資料={\"n\":6,\"pairs\":[[0,0],[1,1],[1,3],[2,2],[2,4],[2,5],[3,1],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]],\"a\":1,\"b\":4}。請依 [a,b,related,class_a,class_b,invariance,boundary] 欄序回答。","answer":{"kind":"expression","value":"(1, 4, 否, [1, 3], [2, 4, 5], 是, no_quotient_operation_claim)","explanation_zh":"Wave4第四層核對=(1, 4, 否, [1, 3], [2, 4, 5], 是, no_quotient_operation_claim)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-representative","seed":202610304095,"parameters":{"n":6,"pairs":[[0,0],[1,1],[1,3],[2,2],[2,4],[2,5],[3,1],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]],"a":1,"b":4,"domain":"Complete_finite_quotient_set","operation":"quotient-representative","semantic_template":"qf-mod30-quotient-representative/{\"a\":1,\"b\":4,\"n\":6,\"pairs\":[[0,0],[1,1],[1,3],[2,2],[2,4],[2,5],[3,1],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0096","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-representative；資料={\"n\":6,\"pairs\":[[0,0],[0,2],[0,4],[0,5],[1,1],[2,0],[2,2],[2,4],[2,5],[3,3],[4,0],[4,2],[4,4],[4,5],[5,0],[5,2],[5,4],[5,5]],\"a\":2,\"b\":1}。請依 [a,b,related,class_a,class_b,invariance,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, 1, 否, [0, 2, 4, 5], [1], 是, no_quotient_operation_claim)","explanation_zh":"Wave4第四層核對=(2, 1, 否, [0, 2, 4, 5], [1], 是, no_quotient_operation_claim)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-representative","seed":202610304096,"parameters":{"n":6,"pairs":[[0,0],[0,2],[0,4],[0,5],[1,1],[2,0],[2,2],[2,4],[2,5],[3,3],[4,0],[4,2],[4,4],[4,5],[5,0],[5,2],[5,4],[5,5]],"a":2,"b":1,"domain":"Complete_finite_quotient_set","operation":"quotient-representative","semantic_template":"qf-mod30-quotient-representative/{\"a\":2,\"b\":1,\"n\":6,\"pairs\":[[0,0],[0,2],[0,4],[0,5],[1,1],[2,0],[2,2],[2,4],[2,5],[3,3],[4,0],[4,2],[4,4],[4,5],[5,0],[5,2],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0097","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-representative；資料={\"n\":6,\"pairs\":[[0,0],[0,3],[1,1],[2,2],[2,4],[2,5],[3,0],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]],\"a\":3,\"b\":4}。請依 [a,b,related,class_a,class_b,invariance,boundary] 欄序回答。","answer":{"kind":"expression","value":"(3, 4, 否, [0, 3], [2, 4, 5], 是, no_quotient_operation_claim)","explanation_zh":"Wave4第四層核對=(3, 4, 否, [0, 3], [2, 4, 5], 是, no_quotient_operation_claim)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-representative","seed":202610304097,"parameters":{"n":6,"pairs":[[0,0],[0,3],[1,1],[2,2],[2,4],[2,5],[3,0],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]],"a":3,"b":4,"domain":"Complete_finite_quotient_set","operation":"quotient-representative","semantic_template":"qf-mod30-quotient-representative/{\"a\":3,\"b\":4,\"n\":6,\"pairs\":[[0,0],[0,3],[1,1],[2,2],[2,4],[2,5],[3,0],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0098","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-representative；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[2,4],[2,5],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]],\"a\":4,\"b\":1}。請依 [a,b,related,class_a,class_b,invariance,boundary] 欄序回答。","answer":{"kind":"expression","value":"(4, 1, 否, [2, 4, 5], [0, 1], 是, no_quotient_operation_claim)","explanation_zh":"Wave4第四層核對=(4, 1, 否, [2, 4, 5], [0, 1], 是, no_quotient_operation_claim)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-representative","seed":202610304098,"parameters":{"n":6,"pairs":[[0,0],[0,1],[1,0],[1,1],[2,2],[2,4],[2,5],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]],"a":4,"b":1,"domain":"Complete_finite_quotient_set","operation":"quotient-representative","semantic_template":"qf-mod30-quotient-representative/{\"a\":4,\"b\":1,\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[2,4],[2,5],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0099","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-representative；資料={\"n\":6,\"pairs\":[[0,0],[0,1],[0,4],[0,5],[1,0],[1,1],[1,4],[1,5],[2,2],[2,3],[3,2],[3,3],[4,0],[4,1],[4,4],[4,5],[5,0],[5,1],[5,4],[5,5]],\"a\":5,\"b\":4}。請依 [a,b,related,class_a,class_b,invariance,boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, 4, 是, [0, 1, 4, 5], [0, 1, 4, 5], 是, no_quotient_operation_claim)","explanation_zh":"Wave4第四層核對=(5, 4, 是, [0, 1, 4, 5], [0, 1, 4, 5], 是, no_quotient_operation_claim)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-representative","seed":202610304099,"parameters":{"n":6,"pairs":[[0,0],[0,1],[0,4],[0,5],[1,0],[1,1],[1,4],[1,5],[2,2],[2,3],[3,2],[3,3],[4,0],[4,1],[4,4],[4,5],[5,0],[5,1],[5,4],[5,5]],"a":5,"b":4,"domain":"Complete_finite_quotient_set","operation":"quotient-representative","semantic_template":"qf-mod30-quotient-representative/{\"a\":5,\"b\":4,\"n\":6,\"pairs\":[[0,0],[0,1],[0,4],[0,5],[1,0],[1,1],[1,4],[1,5],[2,2],[2,3],[3,2],[3,3],[4,0],[4,1],[4,4],[4,5],[5,0],[5,1],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r04-quotient-set-0100","batch_id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-set"],"title_zh":"商集","stem_zh":"同一完整等價關係的互異classes。操作=quotient-representative；資料={\"n\":6,\"pairs\":[[0,0],[0,2],[0,3],[1,1],[1,4],[1,5],[2,0],[2,2],[2,3],[3,0],[3,2],[3,3],[4,1],[4,4],[4,5],[5,1],[5,4],[5,5]],\"a\":0,\"b\":1}。請依 [a,b,related,class_a,class_b,invariance,boundary] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 否, [0, 2, 3], [1, 4, 5], 是, no_quotient_operation_claim)","explanation_zh":"Wave4第四層核對=(0, 1, 否, [0, 2, 3], [1, 4, 5], 是, no_quotient_operation_claim)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商集"]},"variant":{"family_id":"qf-mod30-quotient-representative","seed":202610304100,"parameters":{"n":6,"pairs":[[0,0],[0,2],[0,3],[1,1],[1,4],[1,5],[2,0],[2,2],[2,3],[3,0],[3,2],[3,3],[4,1],[4,4],[4,5],[5,1],[5,4],[5,5]],"a":0,"b":1,"domain":"Complete_finite_quotient_set","operation":"quotient-representative","semantic_template":"qf-mod30-quotient-representative/{\"a\":0,\"b\":1,\"n\":6,\"pairs\":[[0,0],[0,2],[0,3],[1,1],[1,4],[1,5],[2,0],[2,2],[2,3],[3,0],[3,2],[3,3],[4,1],[4,4],[4,5],[5,1],[5,4],[5,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}