{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0001","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-subset-closure；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"subset\":[0,2,4]}。請依 [n,subset,closed] 欄序回答。","answer":{"kind":"expression","value":"(6, [0, 2, 4], 是)","explanation_zh":"現代橋梁核對=(6, [0, 2, 4], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-subset-closure","seed":202610233001,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],"subset":[0,2,4],"domain":"Complete_finite_group_morphisms","operation":"subgroup-subset-closure","semantic_template":"qf-mod23-subgroup-subset-closure/{\"n\":6,\"subset\":[0,2,4],\"table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0002","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-subset-closure；資料={\"n\":6,\"table\":[[2,0,3,4,5,1],[0,1,2,3,4,5],[3,2,4,5,1,0],[4,3,5,1,0,2],[5,4,1,0,2,3],[1,5,0,2,3,4]],\"subset\":[1,2,4]}。請依 [n,subset,closed] 欄序回答。","answer":{"kind":"expression","value":"(6, [1, 2, 4], 是)","explanation_zh":"現代橋梁核對=(6, [1, 2, 4], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-subset-closure","seed":202610233002,"parameters":{"n":6,"table":[[2,0,3,4,5,1],[0,1,2,3,4,5],[3,2,4,5,1,0],[4,3,5,1,0,2],[5,4,1,0,2,3],[1,5,0,2,3,4]],"subset":[1,2,4],"domain":"Complete_finite_group_morphisms","operation":"subgroup-subset-closure","semantic_template":"qf-mod23-subgroup-subset-closure/{\"n\":6,\"subset\":[1,2,4],\"table\":[[2,0,3,4,5,1],[0,1,2,3,4,5],[3,2,4,5,1,0],[4,3,5,1,0,2],[5,4,1,0,2,3],[1,5,0,2,3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0003","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-subset-closure；資料={\"n\":6,\"table\":[[1,3,0,4,5,2],[3,4,1,5,2,0],[0,1,2,3,4,5],[4,5,3,2,0,1],[5,2,4,0,1,3],[2,0,5,1,3,4]],\"subset\":[1,2,4]}。請依 [n,subset,closed] 欄序回答。","answer":{"kind":"expression","value":"(6, [1, 2, 4], 是)","explanation_zh":"現代橋梁核對=(6, [1, 2, 4], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-subset-closure","seed":202610233003,"parameters":{"n":6,"table":[[1,3,0,4,5,2],[3,4,1,5,2,0],[0,1,2,3,4,5],[4,5,3,2,0,1],[5,2,4,0,1,3],[2,0,5,1,3,4]],"subset":[1,2,4],"domain":"Complete_finite_group_morphisms","operation":"subgroup-subset-closure","semantic_template":"qf-mod23-subgroup-subset-closure/{\"n\":6,\"subset\":[1,2,4],\"table\":[[1,3,0,4,5,2],[3,4,1,5,2,0],[0,1,2,3,4,5],[4,5,3,2,0,1],[5,2,4,0,1,3],[2,0,5,1,3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0004","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-subset-closure；資料={\"n\":6,\"table\":[[1,2,4,0,5,3],[2,4,5,1,3,0],[4,5,3,2,0,1],[0,1,2,3,4,5],[5,3,0,4,1,2],[3,0,1,5,2,4]],\"subset\":[1,3,4]}。請依 [n,subset,closed] 欄序回答。","answer":{"kind":"expression","value":"(6, [1, 3, 4], 是)","explanation_zh":"現代橋梁核對=(6, [1, 3, 4], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-subset-closure","seed":202610233004,"parameters":{"n":6,"table":[[1,2,4,0,5,3],[2,4,5,1,3,0],[4,5,3,2,0,1],[0,1,2,3,4,5],[5,3,0,4,1,2],[3,0,1,5,2,4]],"subset":[1,3,4],"domain":"Complete_finite_group_morphisms","operation":"subgroup-subset-closure","semantic_template":"qf-mod23-subgroup-subset-closure/{\"n\":6,\"subset\":[1,3,4],\"table\":[[1,2,4,0,5,3],[2,4,5,1,3,0],[4,5,3,2,0,1],[0,1,2,3,4,5],[5,3,0,4,1,2],[3,0,1,5,2,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0005","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-subset-closure；資料={\"n\":6,\"table\":[[1,2,3,5,0,4],[2,3,5,4,1,0],[3,5,4,0,2,1],[5,4,0,1,3,2],[0,1,2,3,4,5],[4,0,1,2,5,3]],\"subset\":[1,3,4]}。請依 [n,subset,closed] 欄序回答。","answer":{"kind":"expression","value":"(6, [1, 3, 4], 是)","explanation_zh":"現代橋梁核對=(6, [1, 3, 4], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-subset-closure","seed":202610233005,"parameters":{"n":6,"table":[[1,2,3,5,0,4],[2,3,5,4,1,0],[3,5,4,0,2,1],[5,4,0,1,3,2],[0,1,2,3,4,5],[4,0,1,2,5,3]],"subset":[1,3,4],"domain":"Complete_finite_group_morphisms","operation":"subgroup-subset-closure","semantic_template":"qf-mod23-subgroup-subset-closure/{\"n\":6,\"subset\":[1,3,4],\"table\":[[1,2,3,5,0,4],[2,3,5,4,1,0],[3,5,4,0,2,1],[5,4,0,1,3,2],[0,1,2,3,4,5],[4,0,1,2,5,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0006","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-subset-closure；資料={\"n\":6,\"table\":[[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5]],\"subset\":[1,3,5]}。請依 [n,subset,closed] 欄序回答。","answer":{"kind":"expression","value":"(6, [1, 3, 5], 是)","explanation_zh":"現代橋梁核對=(6, [1, 3, 5], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-subset-closure","seed":202610233006,"parameters":{"n":6,"table":[[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5]],"subset":[1,3,5],"domain":"Complete_finite_group_morphisms","operation":"subgroup-subset-closure","semantic_template":"qf-mod23-subgroup-subset-closure/{\"n\":6,\"subset\":[1,3,5],\"table\":[[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0007","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-subset-closure；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,4,3,5,0,2],[2,3,1,4,5,0],[3,5,4,0,2,1],[4,0,5,2,1,3],[5,2,0,1,3,4]],\"subset\":[0,1,4]}。請依 [n,subset,closed] 欄序回答。","answer":{"kind":"expression","value":"(6, [0, 1, 4], 是)","explanation_zh":"現代橋梁核對=(6, [0, 1, 4], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-subset-closure","seed":202610233007,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,4,3,5,0,2],[2,3,1,4,5,0],[3,5,4,0,2,1],[4,0,5,2,1,3],[5,2,0,1,3,4]],"subset":[0,1,4],"domain":"Complete_finite_group_morphisms","operation":"subgroup-subset-closure","semantic_template":"qf-mod23-subgroup-subset-closure/{\"n\":6,\"subset\":[0,1,4],\"table\":[[0,1,2,3,4,5],[1,4,3,5,0,2],[2,3,1,4,5,0],[3,5,4,0,2,1],[4,0,5,2,1,3],[5,2,0,1,3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0008","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-subset-closure；資料={\"n\":6,\"table\":[[4,0,3,5,1,2],[0,1,2,3,4,5],[3,2,0,4,5,1],[5,3,4,1,2,0],[1,4,5,2,0,3],[2,5,1,0,3,4]],\"subset\":[0,1,4]}。請依 [n,subset,closed] 欄序回答。","answer":{"kind":"expression","value":"(6, [0, 1, 4], 是)","explanation_zh":"現代橋梁核對=(6, [0, 1, 4], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-subset-closure","seed":202610233008,"parameters":{"n":6,"table":[[4,0,3,5,1,2],[0,1,2,3,4,5],[3,2,0,4,5,1],[5,3,4,1,2,0],[1,4,5,2,0,3],[2,5,1,0,3,4]],"subset":[0,1,4],"domain":"Complete_finite_group_morphisms","operation":"subgroup-subset-closure","semantic_template":"qf-mod23-subgroup-subset-closure/{\"n\":6,\"subset\":[0,1,4],\"table\":[[4,0,3,5,1,2],[0,1,2,3,4,5],[3,2,0,4,5,1],[5,3,4,1,2,0],[1,4,5,2,0,3],[2,5,1,0,3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0009","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-subset-closure；資料={\"n\":6,\"table\":[[4,3,0,5,2,1],[3,0,1,4,5,2],[0,1,2,3,4,5],[5,4,3,2,1,0],[2,5,4,1,0,3],[1,2,5,0,3,4]],\"subset\":[0,2,4]}。請依 [n,subset,closed] 欄序回答。","answer":{"kind":"expression","value":"(6, [0, 2, 4], 是)","explanation_zh":"現代橋梁核對=(6, [0, 2, 4], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-subset-closure","seed":202610233009,"parameters":{"n":6,"table":[[4,3,0,5,2,1],[3,0,1,4,5,2],[0,1,2,3,4,5],[5,4,3,2,1,0],[2,5,4,1,0,3],[1,2,5,0,3,4]],"subset":[0,2,4],"domain":"Complete_finite_group_morphisms","operation":"subgroup-subset-closure","semantic_template":"qf-mod23-subgroup-subset-closure/{\"n\":6,\"subset\":[0,2,4],\"table\":[[4,3,0,5,2,1],[3,0,1,4,5,2],[0,1,2,3,4,5],[5,4,3,2,1,0],[2,5,4,1,0,3],[1,2,5,0,3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0010","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-subset-closure；資料={\"n\":6,\"table\":[[4,2,5,0,3,1],[2,0,4,1,5,3],[5,4,3,2,1,0],[0,1,2,3,4,5],[3,5,1,4,0,2],[1,3,0,5,2,4]],\"subset\":[0,3,4]}。請依 [n,subset,closed] 欄序回答。","answer":{"kind":"expression","value":"(6, [0, 3, 4], 是)","explanation_zh":"現代橋梁核對=(6, [0, 3, 4], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-subset-closure","seed":202610233010,"parameters":{"n":6,"table":[[4,2,5,0,3,1],[2,0,4,1,5,3],[5,4,3,2,1,0],[0,1,2,3,4,5],[3,5,1,4,0,2],[1,3,0,5,2,4]],"subset":[0,3,4],"domain":"Complete_finite_group_morphisms","operation":"subgroup-subset-closure","semantic_template":"qf-mod23-subgroup-subset-closure/{\"n\":6,\"subset\":[0,3,4],\"table\":[[4,2,5,0,3,1],[2,0,4,1,5,3],[5,4,3,2,1,0],[0,1,2,3,4,5],[3,5,1,4,0,2],[1,3,0,5,2,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0011","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-subset-closure；資料={\"n\":6,\"table\":[[3,2,5,4,0,1],[2,0,3,5,1,4],[5,3,4,1,2,0],[4,5,1,0,3,2],[0,1,2,3,4,5],[1,4,0,2,5,3]],\"subset\":[0,3,4]}。請依 [n,subset,closed] 欄序回答。","answer":{"kind":"expression","value":"(6, [0, 3, 4], 是)","explanation_zh":"現代橋梁核對=(6, [0, 3, 4], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-subset-closure","seed":202610233011,"parameters":{"n":6,"table":[[3,2,5,4,0,1],[2,0,3,5,1,4],[5,3,4,1,2,0],[4,5,1,0,3,2],[0,1,2,3,4,5],[1,4,0,2,5,3]],"subset":[0,3,4],"domain":"Complete_finite_group_morphisms","operation":"subgroup-subset-closure","semantic_template":"qf-mod23-subgroup-subset-closure/{\"n\":6,\"subset\":[0,3,4],\"table\":[[3,2,5,4,0,1],[2,0,3,5,1,4],[5,3,4,1,2,0],[4,5,1,0,3,2],[0,1,2,3,4,5],[1,4,0,2,5,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0012","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-subset-closure；資料={\"n\":6,\"table\":[[3,2,4,5,1,0],[2,0,3,4,5,1],[4,3,5,1,0,2],[5,4,1,0,2,3],[1,5,0,2,3,4],[0,1,2,3,4,5]],\"subset\":[0,3,5]}。請依 [n,subset,closed] 欄序回答。","answer":{"kind":"expression","value":"(6, [0, 3, 5], 是)","explanation_zh":"現代橋梁核對=(6, [0, 3, 5], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-subset-closure","seed":202610233012,"parameters":{"n":6,"table":[[3,2,4,5,1,0],[2,0,3,4,5,1],[4,3,5,1,0,2],[5,4,1,0,2,3],[1,5,0,2,3,4],[0,1,2,3,4,5]],"subset":[0,3,5],"domain":"Complete_finite_group_morphisms","operation":"subgroup-subset-closure","semantic_template":"qf-mod23-subgroup-subset-closure/{\"n\":6,\"subset\":[0,3,5],\"table\":[[3,2,4,5,1,0],[2,0,3,4,5,1],[4,3,5,1,0,2],[5,4,1,0,2,3],[1,5,0,2,3,4],[0,1,2,3,4,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0013","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-subset-closure；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,4,5,2,0,3],[2,5,0,4,3,1],[3,2,4,1,5,0],[4,0,3,5,1,2],[5,3,1,0,2,4]],\"subset\":[0,1,4]}。請依 [n,subset,closed] 欄序回答。","answer":{"kind":"expression","value":"(6, [0, 1, 4], 是)","explanation_zh":"現代橋梁核對=(6, [0, 1, 4], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-subset-closure","seed":202610233013,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,4,5,2,0,3],[2,5,0,4,3,1],[3,2,4,1,5,0],[4,0,3,5,1,2],[5,3,1,0,2,4]],"subset":[0,1,4],"domain":"Complete_finite_group_morphisms","operation":"subgroup-subset-closure","semantic_template":"qf-mod23-subgroup-subset-closure/{\"n\":6,\"subset\":[0,1,4],\"table\":[[0,1,2,3,4,5],[1,4,5,2,0,3],[2,5,0,4,3,1],[3,2,4,1,5,0],[4,0,3,5,1,2],[5,3,1,0,2,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0014","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-subset-closure；資料={\"n\":6,\"table\":[[4,0,5,2,1,3],[0,1,2,3,4,5],[5,2,1,4,3,0],[2,3,4,0,5,1],[1,4,3,5,0,2],[3,5,0,1,2,4]],\"subset\":[0,1,4]}。請依 [n,subset,closed] 欄序回答。","answer":{"kind":"expression","value":"(6, [0, 1, 4], 是)","explanation_zh":"現代橋梁核對=(6, [0, 1, 4], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-subset-closure","seed":202610233014,"parameters":{"n":6,"table":[[4,0,5,2,1,3],[0,1,2,3,4,5],[5,2,1,4,3,0],[2,3,4,0,5,1],[1,4,3,5,0,2],[3,5,0,1,2,4]],"subset":[0,1,4],"domain":"Complete_finite_group_morphisms","operation":"subgroup-subset-closure","semantic_template":"qf-mod23-subgroup-subset-closure/{\"n\":6,\"subset\":[0,1,4],\"table\":[[4,0,5,2,1,3],[0,1,2,3,4,5],[5,2,1,4,3,0],[2,3,4,0,5,1],[1,4,3,5,0,2],[3,5,0,1,2,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0015","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-subset-closure；資料={\"n\":6,\"table\":[[4,5,0,1,2,3],[5,2,1,4,3,0],[0,1,2,3,4,5],[1,4,3,0,5,2],[2,3,4,5,0,1],[3,0,5,2,1,4]],\"subset\":[0,2,4]}。請依 [n,subset,closed] 欄序回答。","answer":{"kind":"expression","value":"(6, [0, 2, 4], 是)","explanation_zh":"現代橋梁核對=(6, [0, 2, 4], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-subset-closure","seed":202610233015,"parameters":{"n":6,"table":[[4,5,0,1,2,3],[5,2,1,4,3,0],[0,1,2,3,4,5],[1,4,3,0,5,2],[2,3,4,5,0,1],[3,0,5,2,1,4]],"subset":[0,2,4],"domain":"Complete_finite_group_morphisms","operation":"subgroup-subset-closure","semantic_template":"qf-mod23-subgroup-subset-closure/{\"n\":6,\"subset\":[0,2,4],\"table\":[[4,5,0,1,2,3],[5,2,1,4,3,0],[0,1,2,3,4,5],[1,4,3,0,5,2],[2,3,4,5,0,1],[3,0,5,2,1,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0016","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-subset-closure；資料={\"n\":6,\"table\":[[4,5,1,0,3,2],[5,3,4,1,2,0],[1,4,0,2,5,3],[0,1,2,3,4,5],[3,2,5,4,0,1],[2,0,3,5,1,4]],\"subset\":[0,3,4]}。請依 [n,subset,closed] 欄序回答。","answer":{"kind":"expression","value":"(6, [0, 3, 4], 是)","explanation_zh":"現代橋梁核對=(6, [0, 3, 4], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-subset-closure","seed":202610233016,"parameters":{"n":6,"table":[[4,5,1,0,3,2],[5,3,4,1,2,0],[1,4,0,2,5,3],[0,1,2,3,4,5],[3,2,5,4,0,1],[2,0,3,5,1,4]],"subset":[0,3,4],"domain":"Complete_finite_group_morphisms","operation":"subgroup-subset-closure","semantic_template":"qf-mod23-subgroup-subset-closure/{\"n\":6,\"subset\":[0,3,4],\"table\":[[4,5,1,0,3,2],[5,3,4,1,2,0],[1,4,0,2,5,3],[0,1,2,3,4,5],[3,2,5,4,0,1],[2,0,3,5,1,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0017","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-subset-closure；資料={\"n\":6,\"table\":[[3,5,1,4,0,2],[5,4,3,2,1,0],[1,3,0,5,2,4],[4,2,5,0,3,1],[0,1,2,3,4,5],[2,0,4,1,5,3]],\"subset\":[0,3,4]}。請依 [n,subset,closed] 欄序回答。","answer":{"kind":"expression","value":"(6, [0, 3, 4], 是)","explanation_zh":"現代橋梁核對=(6, [0, 3, 4], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-subset-closure","seed":202610233017,"parameters":{"n":6,"table":[[3,5,1,4,0,2],[5,4,3,2,1,0],[1,3,0,5,2,4],[4,2,5,0,3,1],[0,1,2,3,4,5],[2,0,4,1,5,3]],"subset":[0,3,4],"domain":"Complete_finite_group_morphisms","operation":"subgroup-subset-closure","semantic_template":"qf-mod23-subgroup-subset-closure/{\"n\":6,\"subset\":[0,3,4],\"table\":[[3,5,1,4,0,2],[5,4,3,2,1,0],[1,3,0,5,2,4],[4,2,5,0,3,1],[0,1,2,3,4,5],[2,0,4,1,5,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0018","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-subset-closure；資料={\"n\":6,\"table\":[[3,4,1,5,2,0],[4,5,3,2,0,1],[1,3,0,4,5,2],[5,2,4,0,1,3],[2,0,5,1,3,4],[0,1,2,3,4,5]],\"subset\":[0,3,5]}。請依 [n,subset,closed] 欄序回答。","answer":{"kind":"expression","value":"(6, [0, 3, 5], 是)","explanation_zh":"現代橋梁核對=(6, [0, 3, 5], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-subset-closure","seed":202610233018,"parameters":{"n":6,"table":[[3,4,1,5,2,0],[4,5,3,2,0,1],[1,3,0,4,5,2],[5,2,4,0,1,3],[2,0,5,1,3,4],[0,1,2,3,4,5]],"subset":[0,3,5],"domain":"Complete_finite_group_morphisms","operation":"subgroup-subset-closure","semantic_template":"qf-mod23-subgroup-subset-closure/{\"n\":6,\"subset\":[0,3,5],\"table\":[[3,4,1,5,2,0],[4,5,3,2,0,1],[1,3,0,4,5,2],[5,2,4,0,1,3],[2,0,5,1,3,4],[0,1,2,3,4,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0019","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-subset-closure；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,3,5,0,2,4],[2,5,0,4,3,1],[3,0,4,1,5,2],[4,2,3,5,1,0],[5,4,1,2,0,3]],\"subset\":[0,1,3]}。請依 [n,subset,closed] 欄序回答。","answer":{"kind":"expression","value":"(6, [0, 1, 3], 是)","explanation_zh":"現代橋梁核對=(6, [0, 1, 3], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-subset-closure","seed":202610233019,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,3,5,0,2,4],[2,5,0,4,3,1],[3,0,4,1,5,2],[4,2,3,5,1,0],[5,4,1,2,0,3]],"subset":[0,1,3],"domain":"Complete_finite_group_morphisms","operation":"subgroup-subset-closure","semantic_template":"qf-mod23-subgroup-subset-closure/{\"n\":6,\"subset\":[0,1,3],\"table\":[[0,1,2,3,4,5],[1,3,5,0,2,4],[2,5,0,4,3,1],[3,0,4,1,5,2],[4,2,3,5,1,0],[5,4,1,2,0,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0020","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-subset-closure；資料={\"n\":6,\"table\":[[3,0,5,1,2,4],[0,1,2,3,4,5],[5,2,1,4,3,0],[1,3,4,0,5,2],[2,4,3,5,0,1],[4,5,0,2,1,3]],\"subset\":[0,1,3]}。請依 [n,subset,closed] 欄序回答。","answer":{"kind":"expression","value":"(6, [0, 1, 3], 是)","explanation_zh":"現代橋梁核對=(6, [0, 1, 3], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-subset-closure","seed":202610233020,"parameters":{"n":6,"table":[[3,0,5,1,2,4],[0,1,2,3,4,5],[5,2,1,4,3,0],[1,3,4,0,5,2],[2,4,3,5,0,1],[4,5,0,2,1,3]],"subset":[0,1,3],"domain":"Complete_finite_group_morphisms","operation":"subgroup-subset-closure","semantic_template":"qf-mod23-subgroup-subset-closure/{\"n\":6,\"subset\":[0,1,3],\"table\":[[3,0,5,1,2,4],[0,1,2,3,4,5],[5,2,1,4,3,0],[1,3,4,0,5,2],[2,4,3,5,0,1],[4,5,0,2,1,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0021","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-subset-closure；資料={\"n\":6,\"table\":[[3,5,0,2,1,4],[5,2,1,4,3,0],[0,1,2,3,4,5],[2,4,3,0,5,1],[1,3,4,5,0,2],[4,0,5,1,2,3]],\"subset\":[0,2,3]}。請依 [n,subset,closed] 欄序回答。","answer":{"kind":"expression","value":"(6, [0, 2, 3], 是)","explanation_zh":"現代橋梁核對=(6, [0, 2, 3], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-subset-closure","seed":202610233021,"parameters":{"n":6,"table":[[3,5,0,2,1,4],[5,2,1,4,3,0],[0,1,2,3,4,5],[2,4,3,0,5,1],[1,3,4,5,0,2],[4,0,5,1,2,3]],"subset":[0,2,3],"domain":"Complete_finite_group_morphisms","operation":"subgroup-subset-closure","semantic_template":"qf-mod23-subgroup-subset-closure/{\"n\":6,\"subset\":[0,2,3],\"table\":[[3,5,0,2,1,4],[5,2,1,4,3,0],[0,1,2,3,4,5],[2,4,3,0,5,1],[1,3,4,5,0,2],[4,0,5,1,2,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0022","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-subset-closure；資料={\"n\":6,\"table\":[[2,5,3,0,1,4],[5,3,4,1,2,0],[3,4,0,2,5,1],[0,1,2,3,4,5],[1,2,5,4,0,3],[4,0,1,5,3,2]],\"subset\":[0,2,3]}。請依 [n,subset,closed] 欄序回答。","answer":{"kind":"expression","value":"(6, [0, 2, 3], 是)","explanation_zh":"現代橋梁核對=(6, [0, 2, 3], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-subset-closure","seed":202610233022,"parameters":{"n":6,"table":[[2,5,3,0,1,4],[5,3,4,1,2,0],[3,4,0,2,5,1],[0,1,2,3,4,5],[1,2,5,4,0,3],[4,0,1,5,3,2]],"subset":[0,2,3],"domain":"Complete_finite_group_morphisms","operation":"subgroup-subset-closure","semantic_template":"qf-mod23-subgroup-subset-closure/{\"n\":6,\"subset\":[0,2,3],\"table\":[[2,5,3,0,1,4],[5,3,4,1,2,0],[3,4,0,2,5,1],[0,1,2,3,4,5],[1,2,5,4,0,3],[4,0,1,5,3,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0023","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-subset-closure；資料={\"n\":6,\"table\":[[2,5,4,1,0,3],[5,4,3,2,1,0],[4,3,0,5,2,1],[1,2,5,0,3,4],[0,1,2,3,4,5],[3,0,1,4,5,2]],\"subset\":[0,2,4]}。請依 [n,subset,closed] 欄序回答。","answer":{"kind":"expression","value":"(6, [0, 2, 4], 是)","explanation_zh":"現代橋梁核對=(6, [0, 2, 4], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-subset-closure","seed":202610233023,"parameters":{"n":6,"table":[[2,5,4,1,0,3],[5,4,3,2,1,0],[4,3,0,5,2,1],[1,2,5,0,3,4],[0,1,2,3,4,5],[3,0,1,4,5,2]],"subset":[0,2,4],"domain":"Complete_finite_group_morphisms","operation":"subgroup-subset-closure","semantic_template":"qf-mod23-subgroup-subset-closure/{\"n\":6,\"subset\":[0,2,4],\"table\":[[2,5,4,1,0,3],[5,4,3,2,1,0],[4,3,0,5,2,1],[1,2,5,0,3,4],[0,1,2,3,4,5],[3,0,1,4,5,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0024","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-subset-closure；資料={\"n\":6,\"table\":[[2,4,5,1,3,0],[4,5,3,2,0,1],[5,3,0,4,1,2],[1,2,4,0,5,3],[3,0,1,5,2,4],[0,1,2,3,4,5]],\"subset\":[0,2,5]}。請依 [n,subset,closed] 欄序回答。","answer":{"kind":"expression","value":"(6, [0, 2, 5], 是)","explanation_zh":"現代橋梁核對=(6, [0, 2, 5], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-subset-closure","seed":202610233024,"parameters":{"n":6,"table":[[2,4,5,1,3,0],[4,5,3,2,0,1],[5,3,0,4,1,2],[1,2,4,0,5,3],[3,0,1,5,2,4],[0,1,2,3,4,5]],"subset":[0,2,5],"domain":"Complete_finite_group_morphisms","operation":"subgroup-subset-closure","semantic_template":"qf-mod23-subgroup-subset-closure/{\"n\":6,\"subset\":[0,2,5],\"table\":[[2,4,5,1,3,0],[4,5,3,2,0,1],[5,3,0,4,1,2],[1,2,4,0,5,3],[3,0,1,5,2,4],[0,1,2,3,4,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0025","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-subset-closure；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,3,4,0,5,2],[2,4,0,5,1,3],[3,0,5,1,2,4],[4,5,1,2,3,0],[5,2,3,4,0,1]],\"subset\":[0,1,3]}。請依 [n,subset,closed] 欄序回答。","answer":{"kind":"expression","value":"(6, [0, 1, 3], 是)","explanation_zh":"現代橋梁核對=(6, [0, 1, 3], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-subset-closure","seed":202610233025,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,3,4,0,5,2],[2,4,0,5,1,3],[3,0,5,1,2,4],[4,5,1,2,3,0],[5,2,3,4,0,1]],"subset":[0,1,3],"domain":"Complete_finite_group_morphisms","operation":"subgroup-subset-closure","semantic_template":"qf-mod23-subgroup-subset-closure/{\"n\":6,\"subset\":[0,1,3],\"table\":[[0,1,2,3,4,5],[1,3,4,0,5,2],[2,4,0,5,1,3],[3,0,5,1,2,4],[4,5,1,2,3,0],[5,2,3,4,0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0026","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-subgroup-audit；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"subset\":[0,2,4]}。請依 [n,subset,identity,closed,has_identity,inverse_closed,subgroup] 欄序回答。","answer":{"kind":"expression","value":"(6, [0, 2, 4], 0, 是, 是, 是, 是)","explanation_zh":"現代橋梁核對=(6, [0, 2, 4], 0, 是, 是, 是, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-subgroup-audit","seed":202610233026,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],"subset":[0,2,4],"domain":"Complete_finite_group_morphisms","operation":"subgroup-subgroup-audit","semantic_template":"qf-mod23-subgroup-subgroup-audit/{\"n\":6,\"subset\":[0,2,4],\"table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0027","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-subgroup-audit；資料={\"n\":6,\"table\":[[2,0,3,4,5,1],[0,1,2,3,4,5],[3,2,4,5,1,0],[4,3,5,1,0,2],[5,4,1,0,2,3],[1,5,0,2,3,4]],\"subset\":[1,2,4]}。請依 [n,subset,identity,closed,has_identity,inverse_closed,subgroup] 欄序回答。","answer":{"kind":"expression","value":"(6, [1, 2, 4], 1, 是, 是, 是, 是)","explanation_zh":"現代橋梁核對=(6, [1, 2, 4], 1, 是, 是, 是, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-subgroup-audit","seed":202610233027,"parameters":{"n":6,"table":[[2,0,3,4,5,1],[0,1,2,3,4,5],[3,2,4,5,1,0],[4,3,5,1,0,2],[5,4,1,0,2,3],[1,5,0,2,3,4]],"subset":[1,2,4],"domain":"Complete_finite_group_morphisms","operation":"subgroup-subgroup-audit","semantic_template":"qf-mod23-subgroup-subgroup-audit/{\"n\":6,\"subset\":[1,2,4],\"table\":[[2,0,3,4,5,1],[0,1,2,3,4,5],[3,2,4,5,1,0],[4,3,5,1,0,2],[5,4,1,0,2,3],[1,5,0,2,3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0028","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-subgroup-audit；資料={\"n\":6,\"table\":[[1,3,0,4,5,2],[3,4,1,5,2,0],[0,1,2,3,4,5],[4,5,3,2,0,1],[5,2,4,0,1,3],[2,0,5,1,3,4]],\"subset\":[1,2,4]}。請依 [n,subset,identity,closed,has_identity,inverse_closed,subgroup] 欄序回答。","answer":{"kind":"expression","value":"(6, [1, 2, 4], 2, 是, 是, 是, 是)","explanation_zh":"現代橋梁核對=(6, [1, 2, 4], 2, 是, 是, 是, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-subgroup-audit","seed":202610233028,"parameters":{"n":6,"table":[[1,3,0,4,5,2],[3,4,1,5,2,0],[0,1,2,3,4,5],[4,5,3,2,0,1],[5,2,4,0,1,3],[2,0,5,1,3,4]],"subset":[1,2,4],"domain":"Complete_finite_group_morphisms","operation":"subgroup-subgroup-audit","semantic_template":"qf-mod23-subgroup-subgroup-audit/{\"n\":6,\"subset\":[1,2,4],\"table\":[[1,3,0,4,5,2],[3,4,1,5,2,0],[0,1,2,3,4,5],[4,5,3,2,0,1],[5,2,4,0,1,3],[2,0,5,1,3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0029","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-subgroup-audit；資料={\"n\":6,\"table\":[[1,2,4,0,5,3],[2,4,5,1,3,0],[4,5,3,2,0,1],[0,1,2,3,4,5],[5,3,0,4,1,2],[3,0,1,5,2,4]],\"subset\":[1,3,4]}。請依 [n,subset,identity,closed,has_identity,inverse_closed,subgroup] 欄序回答。","answer":{"kind":"expression","value":"(6, [1, 3, 4], 3, 是, 是, 是, 是)","explanation_zh":"現代橋梁核對=(6, [1, 3, 4], 3, 是, 是, 是, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-subgroup-audit","seed":202610233029,"parameters":{"n":6,"table":[[1,2,4,0,5,3],[2,4,5,1,3,0],[4,5,3,2,0,1],[0,1,2,3,4,5],[5,3,0,4,1,2],[3,0,1,5,2,4]],"subset":[1,3,4],"domain":"Complete_finite_group_morphisms","operation":"subgroup-subgroup-audit","semantic_template":"qf-mod23-subgroup-subgroup-audit/{\"n\":6,\"subset\":[1,3,4],\"table\":[[1,2,4,0,5,3],[2,4,5,1,3,0],[4,5,3,2,0,1],[0,1,2,3,4,5],[5,3,0,4,1,2],[3,0,1,5,2,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0030","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-subgroup-audit；資料={\"n\":6,\"table\":[[1,2,3,5,0,4],[2,3,5,4,1,0],[3,5,4,0,2,1],[5,4,0,1,3,2],[0,1,2,3,4,5],[4,0,1,2,5,3]],\"subset\":[1,3,4]}。請依 [n,subset,identity,closed,has_identity,inverse_closed,subgroup] 欄序回答。","answer":{"kind":"expression","value":"(6, [1, 3, 4], 4, 是, 是, 是, 是)","explanation_zh":"現代橋梁核對=(6, [1, 3, 4], 4, 是, 是, 是, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-subgroup-audit","seed":202610233030,"parameters":{"n":6,"table":[[1,2,3,5,0,4],[2,3,5,4,1,0],[3,5,4,0,2,1],[5,4,0,1,3,2],[0,1,2,3,4,5],[4,0,1,2,5,3]],"subset":[1,3,4],"domain":"Complete_finite_group_morphisms","operation":"subgroup-subgroup-audit","semantic_template":"qf-mod23-subgroup-subgroup-audit/{\"n\":6,\"subset\":[1,3,4],\"table\":[[1,2,3,5,0,4],[2,3,5,4,1,0],[3,5,4,0,2,1],[5,4,0,1,3,2],[0,1,2,3,4,5],[4,0,1,2,5,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0031","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-subgroup-audit；資料={\"n\":6,\"table\":[[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5]],\"subset\":[1,3,5]}。請依 [n,subset,identity,closed,has_identity,inverse_closed,subgroup] 欄序回答。","answer":{"kind":"expression","value":"(6, [1, 3, 5], 5, 是, 是, 是, 是)","explanation_zh":"現代橋梁核對=(6, [1, 3, 5], 5, 是, 是, 是, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-subgroup-audit","seed":202610233031,"parameters":{"n":6,"table":[[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5]],"subset":[1,3,5],"domain":"Complete_finite_group_morphisms","operation":"subgroup-subgroup-audit","semantic_template":"qf-mod23-subgroup-subgroup-audit/{\"n\":6,\"subset\":[1,3,5],\"table\":[[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0032","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-subgroup-audit；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,4,3,5,0,2],[2,3,1,4,5,0],[3,5,4,0,2,1],[4,0,5,2,1,3],[5,2,0,1,3,4]],\"subset\":[0,1,4]}。請依 [n,subset,identity,closed,has_identity,inverse_closed,subgroup] 欄序回答。","answer":{"kind":"expression","value":"(6, [0, 1, 4], 0, 是, 是, 是, 是)","explanation_zh":"現代橋梁核對=(6, [0, 1, 4], 0, 是, 是, 是, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-subgroup-audit","seed":202610233032,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,4,3,5,0,2],[2,3,1,4,5,0],[3,5,4,0,2,1],[4,0,5,2,1,3],[5,2,0,1,3,4]],"subset":[0,1,4],"domain":"Complete_finite_group_morphisms","operation":"subgroup-subgroup-audit","semantic_template":"qf-mod23-subgroup-subgroup-audit/{\"n\":6,\"subset\":[0,1,4],\"table\":[[0,1,2,3,4,5],[1,4,3,5,0,2],[2,3,1,4,5,0],[3,5,4,0,2,1],[4,0,5,2,1,3],[5,2,0,1,3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0033","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-subgroup-audit；資料={\"n\":6,\"table\":[[4,0,3,5,1,2],[0,1,2,3,4,5],[3,2,0,4,5,1],[5,3,4,1,2,0],[1,4,5,2,0,3],[2,5,1,0,3,4]],\"subset\":[0,1,4]}。請依 [n,subset,identity,closed,has_identity,inverse_closed,subgroup] 欄序回答。","answer":{"kind":"expression","value":"(6, [0, 1, 4], 1, 是, 是, 是, 是)","explanation_zh":"現代橋梁核對=(6, [0, 1, 4], 1, 是, 是, 是, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-subgroup-audit","seed":202610233033,"parameters":{"n":6,"table":[[4,0,3,5,1,2],[0,1,2,3,4,5],[3,2,0,4,5,1],[5,3,4,1,2,0],[1,4,5,2,0,3],[2,5,1,0,3,4]],"subset":[0,1,4],"domain":"Complete_finite_group_morphisms","operation":"subgroup-subgroup-audit","semantic_template":"qf-mod23-subgroup-subgroup-audit/{\"n\":6,\"subset\":[0,1,4],\"table\":[[4,0,3,5,1,2],[0,1,2,3,4,5],[3,2,0,4,5,1],[5,3,4,1,2,0],[1,4,5,2,0,3],[2,5,1,0,3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0034","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-subgroup-audit；資料={\"n\":6,\"table\":[[4,3,0,5,2,1],[3,0,1,4,5,2],[0,1,2,3,4,5],[5,4,3,2,1,0],[2,5,4,1,0,3],[1,2,5,0,3,4]],\"subset\":[0,2,4]}。請依 [n,subset,identity,closed,has_identity,inverse_closed,subgroup] 欄序回答。","answer":{"kind":"expression","value":"(6, [0, 2, 4], 2, 是, 是, 是, 是)","explanation_zh":"現代橋梁核對=(6, [0, 2, 4], 2, 是, 是, 是, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-subgroup-audit","seed":202610233034,"parameters":{"n":6,"table":[[4,3,0,5,2,1],[3,0,1,4,5,2],[0,1,2,3,4,5],[5,4,3,2,1,0],[2,5,4,1,0,3],[1,2,5,0,3,4]],"subset":[0,2,4],"domain":"Complete_finite_group_morphisms","operation":"subgroup-subgroup-audit","semantic_template":"qf-mod23-subgroup-subgroup-audit/{\"n\":6,\"subset\":[0,2,4],\"table\":[[4,3,0,5,2,1],[3,0,1,4,5,2],[0,1,2,3,4,5],[5,4,3,2,1,0],[2,5,4,1,0,3],[1,2,5,0,3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0035","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-subgroup-audit；資料={\"n\":6,\"table\":[[4,2,5,0,3,1],[2,0,4,1,5,3],[5,4,3,2,1,0],[0,1,2,3,4,5],[3,5,1,4,0,2],[1,3,0,5,2,4]],\"subset\":[0,3,4]}。請依 [n,subset,identity,closed,has_identity,inverse_closed,subgroup] 欄序回答。","answer":{"kind":"expression","value":"(6, [0, 3, 4], 3, 是, 是, 是, 是)","explanation_zh":"現代橋梁核對=(6, [0, 3, 4], 3, 是, 是, 是, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-subgroup-audit","seed":202610233035,"parameters":{"n":6,"table":[[4,2,5,0,3,1],[2,0,4,1,5,3],[5,4,3,2,1,0],[0,1,2,3,4,5],[3,5,1,4,0,2],[1,3,0,5,2,4]],"subset":[0,3,4],"domain":"Complete_finite_group_morphisms","operation":"subgroup-subgroup-audit","semantic_template":"qf-mod23-subgroup-subgroup-audit/{\"n\":6,\"subset\":[0,3,4],\"table\":[[4,2,5,0,3,1],[2,0,4,1,5,3],[5,4,3,2,1,0],[0,1,2,3,4,5],[3,5,1,4,0,2],[1,3,0,5,2,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0036","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-subgroup-audit；資料={\"n\":6,\"table\":[[3,2,5,4,0,1],[2,0,3,5,1,4],[5,3,4,1,2,0],[4,5,1,0,3,2],[0,1,2,3,4,5],[1,4,0,2,5,3]],\"subset\":[0,3,4]}。請依 [n,subset,identity,closed,has_identity,inverse_closed,subgroup] 欄序回答。","answer":{"kind":"expression","value":"(6, [0, 3, 4], 4, 是, 是, 是, 是)","explanation_zh":"現代橋梁核對=(6, [0, 3, 4], 4, 是, 是, 是, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-subgroup-audit","seed":202610233036,"parameters":{"n":6,"table":[[3,2,5,4,0,1],[2,0,3,5,1,4],[5,3,4,1,2,0],[4,5,1,0,3,2],[0,1,2,3,4,5],[1,4,0,2,5,3]],"subset":[0,3,4],"domain":"Complete_finite_group_morphisms","operation":"subgroup-subgroup-audit","semantic_template":"qf-mod23-subgroup-subgroup-audit/{\"n\":6,\"subset\":[0,3,4],\"table\":[[3,2,5,4,0,1],[2,0,3,5,1,4],[5,3,4,1,2,0],[4,5,1,0,3,2],[0,1,2,3,4,5],[1,4,0,2,5,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0037","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-subgroup-audit；資料={\"n\":6,\"table\":[[3,2,4,5,1,0],[2,0,3,4,5,1],[4,3,5,1,0,2],[5,4,1,0,2,3],[1,5,0,2,3,4],[0,1,2,3,4,5]],\"subset\":[0,3,5]}。請依 [n,subset,identity,closed,has_identity,inverse_closed,subgroup] 欄序回答。","answer":{"kind":"expression","value":"(6, [0, 3, 5], 5, 是, 是, 是, 是)","explanation_zh":"現代橋梁核對=(6, [0, 3, 5], 5, 是, 是, 是, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-subgroup-audit","seed":202610233037,"parameters":{"n":6,"table":[[3,2,4,5,1,0],[2,0,3,4,5,1],[4,3,5,1,0,2],[5,4,1,0,2,3],[1,5,0,2,3,4],[0,1,2,3,4,5]],"subset":[0,3,5],"domain":"Complete_finite_group_morphisms","operation":"subgroup-subgroup-audit","semantic_template":"qf-mod23-subgroup-subgroup-audit/{\"n\":6,\"subset\":[0,3,5],\"table\":[[3,2,4,5,1,0],[2,0,3,4,5,1],[4,3,5,1,0,2],[5,4,1,0,2,3],[1,5,0,2,3,4],[0,1,2,3,4,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0038","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-subgroup-audit；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,4,5,2,0,3],[2,5,0,4,3,1],[3,2,4,1,5,0],[4,0,3,5,1,2],[5,3,1,0,2,4]],\"subset\":[0,1,4]}。請依 [n,subset,identity,closed,has_identity,inverse_closed,subgroup] 欄序回答。","answer":{"kind":"expression","value":"(6, [0, 1, 4], 0, 是, 是, 是, 是)","explanation_zh":"現代橋梁核對=(6, [0, 1, 4], 0, 是, 是, 是, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-subgroup-audit","seed":202610233038,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,4,5,2,0,3],[2,5,0,4,3,1],[3,2,4,1,5,0],[4,0,3,5,1,2],[5,3,1,0,2,4]],"subset":[0,1,4],"domain":"Complete_finite_group_morphisms","operation":"subgroup-subgroup-audit","semantic_template":"qf-mod23-subgroup-subgroup-audit/{\"n\":6,\"subset\":[0,1,4],\"table\":[[0,1,2,3,4,5],[1,4,5,2,0,3],[2,5,0,4,3,1],[3,2,4,1,5,0],[4,0,3,5,1,2],[5,3,1,0,2,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0039","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-subgroup-audit；資料={\"n\":6,\"table\":[[4,0,5,2,1,3],[0,1,2,3,4,5],[5,2,1,4,3,0],[2,3,4,0,5,1],[1,4,3,5,0,2],[3,5,0,1,2,4]],\"subset\":[0,1,4]}。請依 [n,subset,identity,closed,has_identity,inverse_closed,subgroup] 欄序回答。","answer":{"kind":"expression","value":"(6, [0, 1, 4], 1, 是, 是, 是, 是)","explanation_zh":"現代橋梁核對=(6, [0, 1, 4], 1, 是, 是, 是, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-subgroup-audit","seed":202610233039,"parameters":{"n":6,"table":[[4,0,5,2,1,3],[0,1,2,3,4,5],[5,2,1,4,3,0],[2,3,4,0,5,1],[1,4,3,5,0,2],[3,5,0,1,2,4]],"subset":[0,1,4],"domain":"Complete_finite_group_morphisms","operation":"subgroup-subgroup-audit","semantic_template":"qf-mod23-subgroup-subgroup-audit/{\"n\":6,\"subset\":[0,1,4],\"table\":[[4,0,5,2,1,3],[0,1,2,3,4,5],[5,2,1,4,3,0],[2,3,4,0,5,1],[1,4,3,5,0,2],[3,5,0,1,2,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0040","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-subgroup-audit；資料={\"n\":6,\"table\":[[4,5,0,1,2,3],[5,2,1,4,3,0],[0,1,2,3,4,5],[1,4,3,0,5,2],[2,3,4,5,0,1],[3,0,5,2,1,4]],\"subset\":[0,2,4]}。請依 [n,subset,identity,closed,has_identity,inverse_closed,subgroup] 欄序回答。","answer":{"kind":"expression","value":"(6, [0, 2, 4], 2, 是, 是, 是, 是)","explanation_zh":"現代橋梁核對=(6, [0, 2, 4], 2, 是, 是, 是, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-subgroup-audit","seed":202610233040,"parameters":{"n":6,"table":[[4,5,0,1,2,3],[5,2,1,4,3,0],[0,1,2,3,4,5],[1,4,3,0,5,2],[2,3,4,5,0,1],[3,0,5,2,1,4]],"subset":[0,2,4],"domain":"Complete_finite_group_morphisms","operation":"subgroup-subgroup-audit","semantic_template":"qf-mod23-subgroup-subgroup-audit/{\"n\":6,\"subset\":[0,2,4],\"table\":[[4,5,0,1,2,3],[5,2,1,4,3,0],[0,1,2,3,4,5],[1,4,3,0,5,2],[2,3,4,5,0,1],[3,0,5,2,1,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0041","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-subgroup-audit；資料={\"n\":6,\"table\":[[4,5,1,0,3,2],[5,3,4,1,2,0],[1,4,0,2,5,3],[0,1,2,3,4,5],[3,2,5,4,0,1],[2,0,3,5,1,4]],\"subset\":[0,3,4]}。請依 [n,subset,identity,closed,has_identity,inverse_closed,subgroup] 欄序回答。","answer":{"kind":"expression","value":"(6, [0, 3, 4], 3, 是, 是, 是, 是)","explanation_zh":"現代橋梁核對=(6, [0, 3, 4], 3, 是, 是, 是, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-subgroup-audit","seed":202610233041,"parameters":{"n":6,"table":[[4,5,1,0,3,2],[5,3,4,1,2,0],[1,4,0,2,5,3],[0,1,2,3,4,5],[3,2,5,4,0,1],[2,0,3,5,1,4]],"subset":[0,3,4],"domain":"Complete_finite_group_morphisms","operation":"subgroup-subgroup-audit","semantic_template":"qf-mod23-subgroup-subgroup-audit/{\"n\":6,\"subset\":[0,3,4],\"table\":[[4,5,1,0,3,2],[5,3,4,1,2,0],[1,4,0,2,5,3],[0,1,2,3,4,5],[3,2,5,4,0,1],[2,0,3,5,1,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0042","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-subgroup-audit；資料={\"n\":6,\"table\":[[3,5,1,4,0,2],[5,4,3,2,1,0],[1,3,0,5,2,4],[4,2,5,0,3,1],[0,1,2,3,4,5],[2,0,4,1,5,3]],\"subset\":[0,3,4]}。請依 [n,subset,identity,closed,has_identity,inverse_closed,subgroup] 欄序回答。","answer":{"kind":"expression","value":"(6, [0, 3, 4], 4, 是, 是, 是, 是)","explanation_zh":"現代橋梁核對=(6, [0, 3, 4], 4, 是, 是, 是, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-subgroup-audit","seed":202610233042,"parameters":{"n":6,"table":[[3,5,1,4,0,2],[5,4,3,2,1,0],[1,3,0,5,2,4],[4,2,5,0,3,1],[0,1,2,3,4,5],[2,0,4,1,5,3]],"subset":[0,3,4],"domain":"Complete_finite_group_morphisms","operation":"subgroup-subgroup-audit","semantic_template":"qf-mod23-subgroup-subgroup-audit/{\"n\":6,\"subset\":[0,3,4],\"table\":[[3,5,1,4,0,2],[5,4,3,2,1,0],[1,3,0,5,2,4],[4,2,5,0,3,1],[0,1,2,3,4,5],[2,0,4,1,5,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0043","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-subgroup-audit；資料={\"n\":6,\"table\":[[3,4,1,5,2,0],[4,5,3,2,0,1],[1,3,0,4,5,2],[5,2,4,0,1,3],[2,0,5,1,3,4],[0,1,2,3,4,5]],\"subset\":[0,3,5]}。請依 [n,subset,identity,closed,has_identity,inverse_closed,subgroup] 欄序回答。","answer":{"kind":"expression","value":"(6, [0, 3, 5], 5, 是, 是, 是, 是)","explanation_zh":"現代橋梁核對=(6, [0, 3, 5], 5, 是, 是, 是, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-subgroup-audit","seed":202610233043,"parameters":{"n":6,"table":[[3,4,1,5,2,0],[4,5,3,2,0,1],[1,3,0,4,5,2],[5,2,4,0,1,3],[2,0,5,1,3,4],[0,1,2,3,4,5]],"subset":[0,3,5],"domain":"Complete_finite_group_morphisms","operation":"subgroup-subgroup-audit","semantic_template":"qf-mod23-subgroup-subgroup-audit/{\"n\":6,\"subset\":[0,3,5],\"table\":[[3,4,1,5,2,0],[4,5,3,2,0,1],[1,3,0,4,5,2],[5,2,4,0,1,3],[2,0,5,1,3,4],[0,1,2,3,4,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0044","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-subgroup-audit；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,3,5,0,2,4],[2,5,0,4,3,1],[3,0,4,1,5,2],[4,2,3,5,1,0],[5,4,1,2,0,3]],\"subset\":[0,1,3]}。請依 [n,subset,identity,closed,has_identity,inverse_closed,subgroup] 欄序回答。","answer":{"kind":"expression","value":"(6, [0, 1, 3], 0, 是, 是, 是, 是)","explanation_zh":"現代橋梁核對=(6, [0, 1, 3], 0, 是, 是, 是, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-subgroup-audit","seed":202610233044,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,3,5,0,2,4],[2,5,0,4,3,1],[3,0,4,1,5,2],[4,2,3,5,1,0],[5,4,1,2,0,3]],"subset":[0,1,3],"domain":"Complete_finite_group_morphisms","operation":"subgroup-subgroup-audit","semantic_template":"qf-mod23-subgroup-subgroup-audit/{\"n\":6,\"subset\":[0,1,3],\"table\":[[0,1,2,3,4,5],[1,3,5,0,2,4],[2,5,0,4,3,1],[3,0,4,1,5,2],[4,2,3,5,1,0],[5,4,1,2,0,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0045","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-subgroup-audit；資料={\"n\":6,\"table\":[[3,0,5,1,2,4],[0,1,2,3,4,5],[5,2,1,4,3,0],[1,3,4,0,5,2],[2,4,3,5,0,1],[4,5,0,2,1,3]],\"subset\":[0,1,3]}。請依 [n,subset,identity,closed,has_identity,inverse_closed,subgroup] 欄序回答。","answer":{"kind":"expression","value":"(6, [0, 1, 3], 1, 是, 是, 是, 是)","explanation_zh":"現代橋梁核對=(6, [0, 1, 3], 1, 是, 是, 是, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-subgroup-audit","seed":202610233045,"parameters":{"n":6,"table":[[3,0,5,1,2,4],[0,1,2,3,4,5],[5,2,1,4,3,0],[1,3,4,0,5,2],[2,4,3,5,0,1],[4,5,0,2,1,3]],"subset":[0,1,3],"domain":"Complete_finite_group_morphisms","operation":"subgroup-subgroup-audit","semantic_template":"qf-mod23-subgroup-subgroup-audit/{\"n\":6,\"subset\":[0,1,3],\"table\":[[3,0,5,1,2,4],[0,1,2,3,4,5],[5,2,1,4,3,0],[1,3,4,0,5,2],[2,4,3,5,0,1],[4,5,0,2,1,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0046","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-subgroup-audit；資料={\"n\":6,\"table\":[[3,5,0,2,1,4],[5,2,1,4,3,0],[0,1,2,3,4,5],[2,4,3,0,5,1],[1,3,4,5,0,2],[4,0,5,1,2,3]],\"subset\":[0,2,3]}。請依 [n,subset,identity,closed,has_identity,inverse_closed,subgroup] 欄序回答。","answer":{"kind":"expression","value":"(6, [0, 2, 3], 2, 是, 是, 是, 是)","explanation_zh":"現代橋梁核對=(6, [0, 2, 3], 2, 是, 是, 是, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-subgroup-audit","seed":202610233046,"parameters":{"n":6,"table":[[3,5,0,2,1,4],[5,2,1,4,3,0],[0,1,2,3,4,5],[2,4,3,0,5,1],[1,3,4,5,0,2],[4,0,5,1,2,3]],"subset":[0,2,3],"domain":"Complete_finite_group_morphisms","operation":"subgroup-subgroup-audit","semantic_template":"qf-mod23-subgroup-subgroup-audit/{\"n\":6,\"subset\":[0,2,3],\"table\":[[3,5,0,2,1,4],[5,2,1,4,3,0],[0,1,2,3,4,5],[2,4,3,0,5,1],[1,3,4,5,0,2],[4,0,5,1,2,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0047","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-subgroup-audit；資料={\"n\":6,\"table\":[[2,5,3,0,1,4],[5,3,4,1,2,0],[3,4,0,2,5,1],[0,1,2,3,4,5],[1,2,5,4,0,3],[4,0,1,5,3,2]],\"subset\":[0,2,3]}。請依 [n,subset,identity,closed,has_identity,inverse_closed,subgroup] 欄序回答。","answer":{"kind":"expression","value":"(6, [0, 2, 3], 3, 是, 是, 是, 是)","explanation_zh":"現代橋梁核對=(6, [0, 2, 3], 3, 是, 是, 是, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-subgroup-audit","seed":202610233047,"parameters":{"n":6,"table":[[2,5,3,0,1,4],[5,3,4,1,2,0],[3,4,0,2,5,1],[0,1,2,3,4,5],[1,2,5,4,0,3],[4,0,1,5,3,2]],"subset":[0,2,3],"domain":"Complete_finite_group_morphisms","operation":"subgroup-subgroup-audit","semantic_template":"qf-mod23-subgroup-subgroup-audit/{\"n\":6,\"subset\":[0,2,3],\"table\":[[2,5,3,0,1,4],[5,3,4,1,2,0],[3,4,0,2,5,1],[0,1,2,3,4,5],[1,2,5,4,0,3],[4,0,1,5,3,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0048","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-subgroup-audit；資料={\"n\":6,\"table\":[[2,5,4,1,0,3],[5,4,3,2,1,0],[4,3,0,5,2,1],[1,2,5,0,3,4],[0,1,2,3,4,5],[3,0,1,4,5,2]],\"subset\":[0,2,4]}。請依 [n,subset,identity,closed,has_identity,inverse_closed,subgroup] 欄序回答。","answer":{"kind":"expression","value":"(6, [0, 2, 4], 4, 是, 是, 是, 是)","explanation_zh":"現代橋梁核對=(6, [0, 2, 4], 4, 是, 是, 是, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-subgroup-audit","seed":202610233048,"parameters":{"n":6,"table":[[2,5,4,1,0,3],[5,4,3,2,1,0],[4,3,0,5,2,1],[1,2,5,0,3,4],[0,1,2,3,4,5],[3,0,1,4,5,2]],"subset":[0,2,4],"domain":"Complete_finite_group_morphisms","operation":"subgroup-subgroup-audit","semantic_template":"qf-mod23-subgroup-subgroup-audit/{\"n\":6,\"subset\":[0,2,4],\"table\":[[2,5,4,1,0,3],[5,4,3,2,1,0],[4,3,0,5,2,1],[1,2,5,0,3,4],[0,1,2,3,4,5],[3,0,1,4,5,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0049","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-subgroup-audit；資料={\"n\":6,\"table\":[[2,4,5,1,3,0],[4,5,3,2,0,1],[5,3,0,4,1,2],[1,2,4,0,5,3],[3,0,1,5,2,4],[0,1,2,3,4,5]],\"subset\":[0,2,5]}。請依 [n,subset,identity,closed,has_identity,inverse_closed,subgroup] 欄序回答。","answer":{"kind":"expression","value":"(6, [0, 2, 5], 5, 是, 是, 是, 是)","explanation_zh":"現代橋梁核對=(6, [0, 2, 5], 5, 是, 是, 是, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-subgroup-audit","seed":202610233049,"parameters":{"n":6,"table":[[2,4,5,1,3,0],[4,5,3,2,0,1],[5,3,0,4,1,2],[1,2,4,0,5,3],[3,0,1,5,2,4],[0,1,2,3,4,5]],"subset":[0,2,5],"domain":"Complete_finite_group_morphisms","operation":"subgroup-subgroup-audit","semantic_template":"qf-mod23-subgroup-subgroup-audit/{\"n\":6,\"subset\":[0,2,5],\"table\":[[2,4,5,1,3,0],[4,5,3,2,0,1],[5,3,0,4,1,2],[1,2,4,0,5,3],[3,0,1,5,2,4],[0,1,2,3,4,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0050","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-subgroup-audit；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,3,4,0,5,2],[2,4,0,5,1,3],[3,0,5,1,2,4],[4,5,1,2,3,0],[5,2,3,4,0,1]],\"subset\":[0,1,3]}。請依 [n,subset,identity,closed,has_identity,inverse_closed,subgroup] 欄序回答。","answer":{"kind":"expression","value":"(6, [0, 1, 3], 0, 是, 是, 是, 是)","explanation_zh":"現代橋梁核對=(6, [0, 1, 3], 0, 是, 是, 是, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-subgroup-audit","seed":202610233050,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,3,4,0,5,2],[2,4,0,5,1,3],[3,0,5,1,2,4],[4,5,1,2,3,0],[5,2,3,4,0,1]],"subset":[0,1,3],"domain":"Complete_finite_group_morphisms","operation":"subgroup-subgroup-audit","semantic_template":"qf-mod23-subgroup-subgroup-audit/{\"n\":6,\"subset\":[0,1,3],\"table\":[[0,1,2,3,4,5],[1,3,4,0,5,2],[2,4,0,5,1,3],[3,0,5,1,2,4],[4,5,1,2,3,0],[5,2,3,4,0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0051","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-homomorphism-audit；資料={\"source_n\":5,\"source_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"target_n\":5,\"target_table\":[[0,1,2,3,4],[1,3,0,4,2],[2,0,4,1,3],[3,4,1,2,0],[4,2,3,0,1]],\"mapping\":[0,3,2,1,4]}。請依 [source_n,target_n,mapping,bad_pairs,homomorphism] 欄序回答。","answer":{"kind":"expression","value":"(5, 5, [0, 3, 2, 1, 4], [], 是)","explanation_zh":"現代橋梁核對=(5, 5, [0, 3, 2, 1, 4], [], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-homomorphism-audit","seed":202610233051,"parameters":{"source_n":5,"source_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"target_n":5,"target_table":[[0,1,2,3,4],[1,3,0,4,2],[2,0,4,1,3],[3,4,1,2,0],[4,2,3,0,1]],"mapping":[0,3,2,1,4],"domain":"Complete_finite_group_morphisms","operation":"subgroup-homomorphism-audit","semantic_template":"qf-mod23-subgroup-homomorphism-audit/{\"mapping\":[0,3,2,1,4],\"source_n\":5,\"source_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"target_n\":5,\"target_table\":[[0,1,2,3,4],[1,3,0,4,2],[2,0,4,1,3],[3,4,1,2,0],[4,2,3,0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0052","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-homomorphism-audit；資料={\"source_n\":6,\"source_table\":[[2,0,3,4,5,1],[0,1,2,3,4,5],[3,2,4,5,1,0],[4,3,5,1,0,2],[5,4,1,0,2,3],[1,5,0,2,3,4]],\"target_n\":3,\"target_table\":[[2,0,1],[0,1,2],[1,2,0]],\"mapping\":[0,1,2,1,0,2]}。請依 [source_n,target_n,mapping,bad_pairs,homomorphism] 欄序回答。","answer":{"kind":"expression","value":"(6, 3, [0, 1, 2, 1, 0, 2], [], 是)","explanation_zh":"現代橋梁核對=(6, 3, [0, 1, 2, 1, 0, 2], [], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-homomorphism-audit","seed":202610233052,"parameters":{"source_n":6,"source_table":[[2,0,3,4,5,1],[0,1,2,3,4,5],[3,2,4,5,1,0],[4,3,5,1,0,2],[5,4,1,0,2,3],[1,5,0,2,3,4]],"target_n":3,"target_table":[[2,0,1],[0,1,2],[1,2,0]],"mapping":[0,1,2,1,0,2],"domain":"Complete_finite_group_morphisms","operation":"subgroup-homomorphism-audit","semantic_template":"qf-mod23-subgroup-homomorphism-audit/{\"mapping\":[0,1,2,1,0,2],\"source_n\":6,\"source_table\":[[2,0,3,4,5,1],[0,1,2,3,4,5],[3,2,4,5,1,0],[4,3,5,1,0,2],[5,4,1,0,2,3],[1,5,0,2,3,4]],\"target_n\":3,\"target_table\":[[2,0,1],[0,1,2],[1,2,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0053","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-homomorphism-audit；資料={\"source_n\":5,\"source_table\":[[1,3,0,4,2],[3,4,1,2,0],[0,1,2,3,4],[4,2,3,0,1],[2,0,4,1,3]],\"target_n\":5,\"target_table\":[[3,2,0,4,1],[2,4,1,0,3],[0,1,2,3,4],[4,0,3,1,2],[1,3,4,2,0]],\"mapping\":[3,1,2,0,4]}。請依 [source_n,target_n,mapping,bad_pairs,homomorphism] 欄序回答。","answer":{"kind":"expression","value":"(5, 5, [3, 1, 2, 0, 4], [], 是)","explanation_zh":"現代橋梁核對=(5, 5, [3, 1, 2, 0, 4], [], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-homomorphism-audit","seed":202610233053,"parameters":{"source_n":5,"source_table":[[1,3,0,4,2],[3,4,1,2,0],[0,1,2,3,4],[4,2,3,0,1],[2,0,4,1,3]],"target_n":5,"target_table":[[3,2,0,4,1],[2,4,1,0,3],[0,1,2,3,4],[4,0,3,1,2],[1,3,4,2,0]],"mapping":[3,1,2,0,4],"domain":"Complete_finite_group_morphisms","operation":"subgroup-homomorphism-audit","semantic_template":"qf-mod23-subgroup-homomorphism-audit/{\"mapping\":[3,1,2,0,4],\"source_n\":5,\"source_table\":[[1,3,0,4,2],[3,4,1,2,0],[0,1,2,3,4],[4,2,3,0,1],[2,0,4,1,3]],\"target_n\":5,\"target_table\":[[3,2,0,4,1],[2,4,1,0,3],[0,1,2,3,4],[4,0,3,1,2],[1,3,4,2,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0054","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-homomorphism-audit；資料={\"source_n\":6,\"source_table\":[[1,2,4,0,5,3],[2,4,5,1,3,0],[4,5,3,2,0,1],[0,1,2,3,4,5],[5,3,0,4,1,2],[3,0,1,5,2,4]],\"target_n\":3,\"target_table\":[[0,1,2],[1,2,0],[2,0,1]],\"mapping\":[2,1,0,0,2,1]}。請依 [source_n,target_n,mapping,bad_pairs,homomorphism] 欄序回答。","answer":{"kind":"expression","value":"(6, 3, [2, 1, 0, 0, 2, 1], [], 是)","explanation_zh":"現代橋梁核對=(6, 3, [2, 1, 0, 0, 2, 1], [], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-homomorphism-audit","seed":202610233054,"parameters":{"source_n":6,"source_table":[[1,2,4,0,5,3],[2,4,5,1,3,0],[4,5,3,2,0,1],[0,1,2,3,4,5],[5,3,0,4,1,2],[3,0,1,5,2,4]],"target_n":3,"target_table":[[0,1,2],[1,2,0],[2,0,1]],"mapping":[2,1,0,0,2,1],"domain":"Complete_finite_group_morphisms","operation":"subgroup-homomorphism-audit","semantic_template":"qf-mod23-subgroup-homomorphism-audit/{\"mapping\":[2,1,0,0,2,1],\"source_n\":6,\"source_table\":[[1,2,4,0,5,3],[2,4,5,1,3,0],[4,5,3,2,0,1],[0,1,2,3,4,5],[5,3,0,4,1,2],[3,0,1,5,2,4]],\"target_n\":3,\"target_table\":[[0,1,2],[1,2,0],[2,0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0055","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-homomorphism-audit；資料={\"source_n\":5,\"source_table\":[[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3],[0,1,2,3,4]],\"target_n\":5,\"target_table\":[[2,4,3,1,0],[4,3,0,2,1],[3,0,1,4,2],[1,2,4,0,3],[0,1,2,3,4]],\"mapping\":[2,1,0,3,4]}。請依 [source_n,target_n,mapping,bad_pairs,homomorphism] 欄序回答。","answer":{"kind":"expression","value":"(5, 5, [2, 1, 0, 3, 4], [], 是)","explanation_zh":"現代橋梁核對=(5, 5, [2, 1, 0, 3, 4], [], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-homomorphism-audit","seed":202610233055,"parameters":{"source_n":5,"source_table":[[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3],[0,1,2,3,4]],"target_n":5,"target_table":[[2,4,3,1,0],[4,3,0,2,1],[3,0,1,4,2],[1,2,4,0,3],[0,1,2,3,4]],"mapping":[2,1,0,3,4],"domain":"Complete_finite_group_morphisms","operation":"subgroup-homomorphism-audit","semantic_template":"qf-mod23-subgroup-homomorphism-audit/{\"mapping\":[2,1,0,3,4],\"source_n\":5,\"source_table\":[[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3],[0,1,2,3,4]],\"target_n\":5,\"target_table\":[[2,4,3,1,0],[4,3,0,2,1],[3,0,1,4,2],[1,2,4,0,3],[0,1,2,3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0056","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-homomorphism-audit；資料={\"source_n\":6,\"source_table\":[[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5]],\"target_n\":3,\"target_table\":[[1,2,0],[2,0,1],[0,1,2]],\"mapping\":[1,0,2,1,0,2]}。請依 [source_n,target_n,mapping,bad_pairs,homomorphism] 欄序回答。","answer":{"kind":"expression","value":"(6, 3, [1, 0, 2, 1, 0, 2], [], 是)","explanation_zh":"現代橋梁核對=(6, 3, [1, 0, 2, 1, 0, 2], [], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-homomorphism-audit","seed":202610233056,"parameters":{"source_n":6,"source_table":[[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5]],"target_n":3,"target_table":[[1,2,0],[2,0,1],[0,1,2]],"mapping":[1,0,2,1,0,2],"domain":"Complete_finite_group_morphisms","operation":"subgroup-homomorphism-audit","semantic_template":"qf-mod23-subgroup-homomorphism-audit/{\"mapping\":[1,0,2,1,0,2],\"source_n\":6,\"source_table\":[[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5]],\"target_n\":3,\"target_table\":[[1,2,0],[2,0,1],[0,1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0057","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-homomorphism-audit；資料={\"source_n\":5,\"source_table\":[[4,0,3,1,2],[0,1,2,3,4],[3,2,0,4,1],[1,3,4,2,0],[2,4,1,0,3]],\"target_n\":5,\"target_table\":[[4,0,1,2,3],[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2]],\"mapping\":[2,1,4,0,3]}。請依 [source_n,target_n,mapping,bad_pairs,homomorphism] 欄序回答。","answer":{"kind":"expression","value":"(5, 5, [2, 1, 4, 0, 3], [], 是)","explanation_zh":"現代橋梁核對=(5, 5, [2, 1, 4, 0, 3], [], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-homomorphism-audit","seed":202610233057,"parameters":{"source_n":5,"source_table":[[4,0,3,1,2],[0,1,2,3,4],[3,2,0,4,1],[1,3,4,2,0],[2,4,1,0,3]],"target_n":5,"target_table":[[4,0,1,2,3],[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2]],"mapping":[2,1,4,0,3],"domain":"Complete_finite_group_morphisms","operation":"subgroup-homomorphism-audit","semantic_template":"qf-mod23-subgroup-homomorphism-audit/{\"mapping\":[2,1,4,0,3],\"source_n\":5,\"source_table\":[[4,0,3,1,2],[0,1,2,3,4],[3,2,0,4,1],[1,3,4,2,0],[2,4,1,0,3]],\"target_n\":5,\"target_table\":[[4,0,1,2,3],[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0058","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-homomorphism-audit；資料={\"source_n\":6,\"source_table\":[[4,0,3,5,1,2],[0,1,2,3,4,5],[3,2,0,4,5,1],[5,3,4,1,2,0],[1,4,5,2,0,3],[2,5,1,0,3,4]],\"target_n\":3,\"target_table\":[[2,0,1],[0,1,2],[1,2,0]],\"mapping\":[2,1,0,1,0,2]}。請依 [source_n,target_n,mapping,bad_pairs,homomorphism] 欄序回答。","answer":{"kind":"expression","value":"(6, 3, [2, 1, 0, 1, 0, 2], [], 是)","explanation_zh":"現代橋梁核對=(6, 3, [2, 1, 0, 1, 0, 2], [], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-homomorphism-audit","seed":202610233058,"parameters":{"source_n":6,"source_table":[[4,0,3,5,1,2],[0,1,2,3,4,5],[3,2,0,4,5,1],[5,3,4,1,2,0],[1,4,5,2,0,3],[2,5,1,0,3,4]],"target_n":3,"target_table":[[2,0,1],[0,1,2],[1,2,0]],"mapping":[2,1,0,1,0,2],"domain":"Complete_finite_group_morphisms","operation":"subgroup-homomorphism-audit","semantic_template":"qf-mod23-subgroup-homomorphism-audit/{\"mapping\":[2,1,0,1,0,2],\"source_n\":6,\"source_table\":[[4,0,3,5,1,2],[0,1,2,3,4,5],[3,2,0,4,5,1],[5,3,4,1,2,0],[1,4,5,2,0,3],[2,5,1,0,3,4]],\"target_n\":3,\"target_table\":[[2,0,1],[0,1,2],[1,2,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0059","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-homomorphism-audit；資料={\"source_n\":5,\"source_table\":[[4,2,3,0,1],[2,0,4,1,3],[3,4,1,2,0],[0,1,2,3,4],[1,3,0,4,2]],\"target_n\":5,\"target_table\":[[4,3,1,0,2],[3,2,4,1,0],[1,4,0,2,3],[0,1,2,3,4],[2,0,3,4,1]],\"mapping\":[1,4,0,3,2]}。請依 [source_n,target_n,mapping,bad_pairs,homomorphism] 欄序回答。","answer":{"kind":"expression","value":"(5, 5, [1, 4, 0, 3, 2], [], 是)","explanation_zh":"現代橋梁核對=(5, 5, [1, 4, 0, 3, 2], [], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-homomorphism-audit","seed":202610233059,"parameters":{"source_n":5,"source_table":[[4,2,3,0,1],[2,0,4,1,3],[3,4,1,2,0],[0,1,2,3,4],[1,3,0,4,2]],"target_n":5,"target_table":[[4,3,1,0,2],[3,2,4,1,0],[1,4,0,2,3],[0,1,2,3,4],[2,0,3,4,1]],"mapping":[1,4,0,3,2],"domain":"Complete_finite_group_morphisms","operation":"subgroup-homomorphism-audit","semantic_template":"qf-mod23-subgroup-homomorphism-audit/{\"mapping\":[1,4,0,3,2],\"source_n\":5,\"source_table\":[[4,2,3,0,1],[2,0,4,1,3],[3,4,1,2,0],[0,1,2,3,4],[1,3,0,4,2]],\"target_n\":5,\"target_table\":[[4,3,1,0,2],[3,2,4,1,0],[1,4,0,2,3],[0,1,2,3,4],[2,0,3,4,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0060","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-homomorphism-audit；資料={\"source_n\":6,\"source_table\":[[4,2,5,0,3,1],[2,0,4,1,5,3],[5,4,3,2,1,0],[0,1,2,3,4,5],[3,5,1,4,0,2],[1,3,0,5,2,4]],\"target_n\":3,\"target_table\":[[0,1,2],[1,2,0],[2,0,1]],\"mapping\":[1,2,0,0,2,1]}。請依 [source_n,target_n,mapping,bad_pairs,homomorphism] 欄序回答。","answer":{"kind":"expression","value":"(6, 3, [1, 2, 0, 0, 2, 1], [], 是)","explanation_zh":"現代橋梁核對=(6, 3, [1, 2, 0, 0, 2, 1], [], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-homomorphism-audit","seed":202610233060,"parameters":{"source_n":6,"source_table":[[4,2,5,0,3,1],[2,0,4,1,5,3],[5,4,3,2,1,0],[0,1,2,3,4,5],[3,5,1,4,0,2],[1,3,0,5,2,4]],"target_n":3,"target_table":[[0,1,2],[1,2,0],[2,0,1]],"mapping":[1,2,0,0,2,1],"domain":"Complete_finite_group_morphisms","operation":"subgroup-homomorphism-audit","semantic_template":"qf-mod23-subgroup-homomorphism-audit/{\"mapping\":[1,2,0,0,2,1],\"source_n\":6,\"source_table\":[[4,2,5,0,3,1],[2,0,4,1,5,3],[5,4,3,2,1,0],[0,1,2,3,4,5],[3,5,1,4,0,2],[1,3,0,5,2,4]],\"target_n\":3,\"target_table\":[[0,1,2],[1,2,0],[2,0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0061","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-homomorphism-audit；資料={\"source_n\":5,\"source_table\":[[0,1,2,3,4],[1,4,0,2,3],[2,0,3,4,1],[3,2,4,1,0],[4,3,1,0,2]],\"target_n\":5,\"target_table\":[[0,1,2,3,4],[1,4,3,0,2],[2,3,1,4,0],[3,0,4,2,1],[4,2,0,1,3]],\"mapping\":[0,4,2,1,3]}。請依 [source_n,target_n,mapping,bad_pairs,homomorphism] 欄序回答。","answer":{"kind":"expression","value":"(5, 5, [0, 4, 2, 1, 3], [], 是)","explanation_zh":"現代橋梁核對=(5, 5, [0, 4, 2, 1, 3], [], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-homomorphism-audit","seed":202610233061,"parameters":{"source_n":5,"source_table":[[0,1,2,3,4],[1,4,0,2,3],[2,0,3,4,1],[3,2,4,1,0],[4,3,1,0,2]],"target_n":5,"target_table":[[0,1,2,3,4],[1,4,3,0,2],[2,3,1,4,0],[3,0,4,2,1],[4,2,0,1,3]],"mapping":[0,4,2,1,3],"domain":"Complete_finite_group_morphisms","operation":"subgroup-homomorphism-audit","semantic_template":"qf-mod23-subgroup-homomorphism-audit/{\"mapping\":[0,4,2,1,3],\"source_n\":5,\"source_table\":[[0,1,2,3,4],[1,4,0,2,3],[2,0,3,4,1],[3,2,4,1,0],[4,3,1,0,2]],\"target_n\":5,\"target_table\":[[0,1,2,3,4],[1,4,3,0,2],[2,3,1,4,0],[3,0,4,2,1],[4,2,0,1,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0062","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-homomorphism-audit；資料={\"source_n\":6,\"source_table\":[[3,2,4,5,1,0],[2,0,3,4,5,1],[4,3,5,1,0,2],[5,4,1,0,2,3],[1,5,0,2,3,4],[0,1,2,3,4,5]],\"target_n\":3,\"target_table\":[[1,2,0],[2,0,1],[0,1,2]],\"mapping\":[0,1,2,1,0,2]}。請依 [source_n,target_n,mapping,bad_pairs,homomorphism] 欄序回答。","answer":{"kind":"expression","value":"(6, 3, [0, 1, 2, 1, 0, 2], [], 是)","explanation_zh":"現代橋梁核對=(6, 3, [0, 1, 2, 1, 0, 2], [], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-homomorphism-audit","seed":202610233062,"parameters":{"source_n":6,"source_table":[[3,2,4,5,1,0],[2,0,3,4,5,1],[4,3,5,1,0,2],[5,4,1,0,2,3],[1,5,0,2,3,4],[0,1,2,3,4,5]],"target_n":3,"target_table":[[1,2,0],[2,0,1],[0,1,2]],"mapping":[0,1,2,1,0,2],"domain":"Complete_finite_group_morphisms","operation":"subgroup-homomorphism-audit","semantic_template":"qf-mod23-subgroup-homomorphism-audit/{\"mapping\":[0,1,2,1,0,2],\"source_n\":6,\"source_table\":[[3,2,4,5,1,0],[2,0,3,4,5,1],[4,3,5,1,0,2],[5,4,1,0,2,3],[1,5,0,2,3,4],[0,1,2,3,4,5]],\"target_n\":3,\"target_table\":[[1,2,0],[2,0,1],[0,1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0063","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-homomorphism-audit；資料={\"source_n\":5,\"source_table\":[[4,2,0,1,3],[2,3,1,4,0],[0,1,2,3,4],[1,4,3,0,2],[3,0,4,2,1]],\"target_n\":5,\"target_table\":[[4,3,0,2,1],[3,0,1,4,2],[0,1,2,3,4],[2,4,3,1,0],[1,2,4,0,3]],\"mapping\":[4,1,2,0,3]}。請依 [source_n,target_n,mapping,bad_pairs,homomorphism] 欄序回答。","answer":{"kind":"expression","value":"(5, 5, [4, 1, 2, 0, 3], [], 是)","explanation_zh":"現代橋梁核對=(5, 5, [4, 1, 2, 0, 3], [], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-homomorphism-audit","seed":202610233063,"parameters":{"source_n":5,"source_table":[[4,2,0,1,3],[2,3,1,4,0],[0,1,2,3,4],[1,4,3,0,2],[3,0,4,2,1]],"target_n":5,"target_table":[[4,3,0,2,1],[3,0,1,4,2],[0,1,2,3,4],[2,4,3,1,0],[1,2,4,0,3]],"mapping":[4,1,2,0,3],"domain":"Complete_finite_group_morphisms","operation":"subgroup-homomorphism-audit","semantic_template":"qf-mod23-subgroup-homomorphism-audit/{\"mapping\":[4,1,2,0,3],\"source_n\":5,\"source_table\":[[4,2,0,1,3],[2,3,1,4,0],[0,1,2,3,4],[1,4,3,0,2],[3,0,4,2,1]],\"target_n\":5,\"target_table\":[[4,3,0,2,1],[3,0,1,4,2],[0,1,2,3,4],[2,4,3,1,0],[1,2,4,0,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0064","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-homomorphism-audit；資料={\"source_n\":6,\"source_table\":[[4,0,5,2,1,3],[0,1,2,3,4,5],[5,2,1,4,3,0],[2,3,4,0,5,1],[1,4,3,5,0,2],[3,5,0,1,2,4]],\"target_n\":3,\"target_table\":[[2,0,1],[0,1,2],[1,2,0]],\"mapping\":[2,1,1,0,0,2]}。請依 [source_n,target_n,mapping,bad_pairs,homomorphism] 欄序回答。","answer":{"kind":"expression","value":"(6, 3, [2, 1, 1, 0, 0, 2], [], 是)","explanation_zh":"現代橋梁核對=(6, 3, [2, 1, 1, 0, 0, 2], [], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-homomorphism-audit","seed":202610233064,"parameters":{"source_n":6,"source_table":[[4,0,5,2,1,3],[0,1,2,3,4,5],[5,2,1,4,3,0],[2,3,4,0,5,1],[1,4,3,5,0,2],[3,5,0,1,2,4]],"target_n":3,"target_table":[[2,0,1],[0,1,2],[1,2,0]],"mapping":[2,1,1,0,0,2],"domain":"Complete_finite_group_morphisms","operation":"subgroup-homomorphism-audit","semantic_template":"qf-mod23-subgroup-homomorphism-audit/{\"mapping\":[2,1,1,0,0,2],\"source_n\":6,\"source_table\":[[4,0,5,2,1,3],[0,1,2,3,4,5],[5,2,1,4,3,0],[2,3,4,0,5,1],[1,4,3,5,0,2],[3,5,0,1,2,4]],\"target_n\":3,\"target_table\":[[2,0,1],[0,1,2],[1,2,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0065","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-homomorphism-audit；資料={\"source_n\":5,\"source_table\":[[3,4,1,2,0],[4,2,3,0,1],[1,3,0,4,2],[2,0,4,1,3],[0,1,2,3,4]],\"target_n\":5,\"target_table\":[[3,2,4,1,0],[2,0,3,4,1],[4,3,1,0,2],[1,4,0,2,3],[0,1,2,3,4]],\"mapping\":[3,1,0,2,4]}。請依 [source_n,target_n,mapping,bad_pairs,homomorphism] 欄序回答。","answer":{"kind":"expression","value":"(5, 5, [3, 1, 0, 2, 4], [], 是)","explanation_zh":"現代橋梁核對=(5, 5, [3, 1, 0, 2, 4], [], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-homomorphism-audit","seed":202610233065,"parameters":{"source_n":5,"source_table":[[3,4,1,2,0],[4,2,3,0,1],[1,3,0,4,2],[2,0,4,1,3],[0,1,2,3,4]],"target_n":5,"target_table":[[3,2,4,1,0],[2,0,3,4,1],[4,3,1,0,2],[1,4,0,2,3],[0,1,2,3,4]],"mapping":[3,1,0,2,4],"domain":"Complete_finite_group_morphisms","operation":"subgroup-homomorphism-audit","semantic_template":"qf-mod23-subgroup-homomorphism-audit/{\"mapping\":[3,1,0,2,4],\"source_n\":5,\"source_table\":[[3,4,1,2,0],[4,2,3,0,1],[1,3,0,4,2],[2,0,4,1,3],[0,1,2,3,4]],\"target_n\":5,\"target_table\":[[3,2,4,1,0],[2,0,3,4,1],[4,3,1,0,2],[1,4,0,2,3],[0,1,2,3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0066","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-homomorphism-audit；資料={\"source_n\":6,\"source_table\":[[4,5,1,0,3,2],[5,3,4,1,2,0],[1,4,0,2,5,3],[0,1,2,3,4,5],[3,2,5,4,0,1],[2,0,3,5,1,4]],\"target_n\":3,\"target_table\":[[0,1,2],[1,2,0],[2,0,1]],\"mapping\":[1,0,2,0,2,1]}。請依 [source_n,target_n,mapping,bad_pairs,homomorphism] 欄序回答。","answer":{"kind":"expression","value":"(6, 3, [1, 0, 2, 0, 2, 1], [], 是)","explanation_zh":"現代橋梁核對=(6, 3, [1, 0, 2, 0, 2, 1], [], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-homomorphism-audit","seed":202610233066,"parameters":{"source_n":6,"source_table":[[4,5,1,0,3,2],[5,3,4,1,2,0],[1,4,0,2,5,3],[0,1,2,3,4,5],[3,2,5,4,0,1],[2,0,3,5,1,4]],"target_n":3,"target_table":[[0,1,2],[1,2,0],[2,0,1]],"mapping":[1,0,2,0,2,1],"domain":"Complete_finite_group_morphisms","operation":"subgroup-homomorphism-audit","semantic_template":"qf-mod23-subgroup-homomorphism-audit/{\"mapping\":[1,0,2,0,2,1],\"source_n\":6,\"source_table\":[[4,5,1,0,3,2],[5,3,4,1,2,0],[1,4,0,2,5,3],[0,1,2,3,4,5],[3,2,5,4,0,1],[2,0,3,5,1,4]],\"target_n\":3,\"target_table\":[[0,1,2],[1,2,0],[2,0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0067","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-homomorphism-audit；資料={\"source_n\":5,\"source_table\":[[3,0,1,4,2],[0,1,2,3,4],[1,2,4,0,3],[4,3,0,2,1],[2,4,3,1,0]],\"target_n\":5,\"target_table\":[[2,0,3,4,1],[0,1,2,3,4],[3,2,4,1,0],[4,3,1,0,2],[1,4,0,2,3]],\"mapping\":[4,1,0,3,2]}。請依 [source_n,target_n,mapping,bad_pairs,homomorphism] 欄序回答。","answer":{"kind":"expression","value":"(5, 5, [4, 1, 0, 3, 2], [], 是)","explanation_zh":"現代橋梁核對=(5, 5, [4, 1, 0, 3, 2], [], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-homomorphism-audit","seed":202610233067,"parameters":{"source_n":5,"source_table":[[3,0,1,4,2],[0,1,2,3,4],[1,2,4,0,3],[4,3,0,2,1],[2,4,3,1,0]],"target_n":5,"target_table":[[2,0,3,4,1],[0,1,2,3,4],[3,2,4,1,0],[4,3,1,0,2],[1,4,0,2,3]],"mapping":[4,1,0,3,2],"domain":"Complete_finite_group_morphisms","operation":"subgroup-homomorphism-audit","semantic_template":"qf-mod23-subgroup-homomorphism-audit/{\"mapping\":[4,1,0,3,2],\"source_n\":5,\"source_table\":[[3,0,1,4,2],[0,1,2,3,4],[1,2,4,0,3],[4,3,0,2,1],[2,4,3,1,0]],\"target_n\":5,\"target_table\":[[2,0,3,4,1],[0,1,2,3,4],[3,2,4,1,0],[4,3,1,0,2],[1,4,0,2,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0068","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-homomorphism-audit；資料={\"source_n\":6,\"source_table\":[[3,4,1,5,2,0],[4,5,3,2,0,1],[1,3,0,4,5,2],[5,2,4,0,1,3],[2,0,5,1,3,4],[0,1,2,3,4,5]],\"target_n\":3,\"target_table\":[[1,2,0],[2,0,1],[0,1,2]],\"mapping\":[0,2,1,1,0,2]}。請依 [source_n,target_n,mapping,bad_pairs,homomorphism] 欄序回答。","answer":{"kind":"expression","value":"(6, 3, [0, 2, 1, 1, 0, 2], [], 是)","explanation_zh":"現代橋梁核對=(6, 3, [0, 2, 1, 1, 0, 2], [], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-homomorphism-audit","seed":202610233068,"parameters":{"source_n":6,"source_table":[[3,4,1,5,2,0],[4,5,3,2,0,1],[1,3,0,4,5,2],[5,2,4,0,1,3],[2,0,5,1,3,4],[0,1,2,3,4,5]],"target_n":3,"target_table":[[1,2,0],[2,0,1],[0,1,2]],"mapping":[0,2,1,1,0,2],"domain":"Complete_finite_group_morphisms","operation":"subgroup-homomorphism-audit","semantic_template":"qf-mod23-subgroup-homomorphism-audit/{\"mapping\":[0,2,1,1,0,2],\"source_n\":6,\"source_table\":[[3,4,1,5,2,0],[4,5,3,2,0,1],[1,3,0,4,5,2],[5,2,4,0,1,3],[2,0,5,1,3,4],[0,1,2,3,4,5]],\"target_n\":3,\"target_table\":[[1,2,0],[2,0,1],[0,1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0069","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-homomorphism-audit；資料={\"source_n\":5,\"source_table\":[[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3],[0,1,2,3,4],[1,2,3,4,0]],\"target_n\":5,\"target_table\":[[1,2,4,0,3],[2,4,3,1,0],[4,3,0,2,1],[0,1,2,3,4],[3,0,1,4,2]],\"mapping\":[4,0,2,3,1]}。請依 [source_n,target_n,mapping,bad_pairs,homomorphism] 欄序回答。","answer":{"kind":"expression","value":"(5, 5, [4, 0, 2, 3, 1], [], 是)","explanation_zh":"現代橋梁核對=(5, 5, [4, 0, 2, 3, 1], [], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-homomorphism-audit","seed":202610233069,"parameters":{"source_n":5,"source_table":[[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3],[0,1,2,3,4],[1,2,3,4,0]],"target_n":5,"target_table":[[1,2,4,0,3],[2,4,3,1,0],[4,3,0,2,1],[0,1,2,3,4],[3,0,1,4,2]],"mapping":[4,0,2,3,1],"domain":"Complete_finite_group_morphisms","operation":"subgroup-homomorphism-audit","semantic_template":"qf-mod23-subgroup-homomorphism-audit/{\"mapping\":[4,0,2,3,1],\"source_n\":5,\"source_table\":[[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3],[0,1,2,3,4],[1,2,3,4,0]],\"target_n\":5,\"target_table\":[[1,2,4,0,3],[2,4,3,1,0],[4,3,0,2,1],[0,1,2,3,4],[3,0,1,4,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0070","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-homomorphism-audit；資料={\"source_n\":6,\"source_table\":[[3,0,5,1,2,4],[0,1,2,3,4,5],[5,2,1,4,3,0],[1,3,4,0,5,2],[2,4,3,5,0,1],[4,5,0,2,1,3]],\"target_n\":3,\"target_table\":[[2,0,1],[0,1,2],[1,2,0]],\"mapping\":[2,1,1,0,0,2]}。請依 [source_n,target_n,mapping,bad_pairs,homomorphism] 欄序回答。","answer":{"kind":"expression","value":"(6, 3, [2, 1, 1, 0, 0, 2], [], 是)","explanation_zh":"現代橋梁核對=(6, 3, [2, 1, 1, 0, 0, 2], [], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-homomorphism-audit","seed":202610233070,"parameters":{"source_n":6,"source_table":[[3,0,5,1,2,4],[0,1,2,3,4,5],[5,2,1,4,3,0],[1,3,4,0,5,2],[2,4,3,5,0,1],[4,5,0,2,1,3]],"target_n":3,"target_table":[[2,0,1],[0,1,2],[1,2,0]],"mapping":[2,1,1,0,0,2],"domain":"Complete_finite_group_morphisms","operation":"subgroup-homomorphism-audit","semantic_template":"qf-mod23-subgroup-homomorphism-audit/{\"mapping\":[2,1,1,0,0,2],\"source_n\":6,\"source_table\":[[3,0,5,1,2,4],[0,1,2,3,4,5],[5,2,1,4,3,0],[1,3,4,0,5,2],[2,4,3,5,0,1],[4,5,0,2,1,3]],\"target_n\":3,\"target_table\":[[2,0,1],[0,1,2],[1,2,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0071","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-homomorphism-audit；資料={\"source_n\":5,\"source_table\":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],\"target_n\":5,\"target_table\":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],\"mapping\":[0,3,1,4,2]}。請依 [source_n,target_n,mapping,bad_pairs,homomorphism] 欄序回答。","answer":{"kind":"expression","value":"(5, 5, [0, 3, 1, 4, 2], [], 是)","explanation_zh":"現代橋梁核對=(5, 5, [0, 3, 1, 4, 2], [], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-homomorphism-audit","seed":202610233071,"parameters":{"source_n":5,"source_table":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],"target_n":5,"target_table":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],"mapping":[0,3,1,4,2],"domain":"Complete_finite_group_morphisms","operation":"subgroup-homomorphism-audit","semantic_template":"qf-mod23-subgroup-homomorphism-audit/{\"mapping\":[0,3,1,4,2],\"source_n\":5,\"source_table\":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],\"target_n\":5,\"target_table\":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0072","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-homomorphism-audit；資料={\"source_n\":6,\"source_table\":[[2,5,3,0,1,4],[5,3,4,1,2,0],[3,4,0,2,5,1],[0,1,2,3,4,5],[1,2,5,4,0,3],[4,0,1,5,3,2]],\"target_n\":3,\"target_table\":[[0,1,2],[1,2,0],[2,0,1]],\"mapping\":[1,0,2,0,2,1]}。請依 [source_n,target_n,mapping,bad_pairs,homomorphism] 欄序回答。","answer":{"kind":"expression","value":"(6, 3, [1, 0, 2, 0, 2, 1], [], 是)","explanation_zh":"現代橋梁核對=(6, 3, [1, 0, 2, 0, 2, 1], [], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-homomorphism-audit","seed":202610233072,"parameters":{"source_n":6,"source_table":[[2,5,3,0,1,4],[5,3,4,1,2,0],[3,4,0,2,5,1],[0,1,2,3,4,5],[1,2,5,4,0,3],[4,0,1,5,3,2]],"target_n":3,"target_table":[[0,1,2],[1,2,0],[2,0,1]],"mapping":[1,0,2,0,2,1],"domain":"Complete_finite_group_morphisms","operation":"subgroup-homomorphism-audit","semantic_template":"qf-mod23-subgroup-homomorphism-audit/{\"mapping\":[1,0,2,0,2,1],\"source_n\":6,\"source_table\":[[2,5,3,0,1,4],[5,3,4,1,2,0],[3,4,0,2,5,1],[0,1,2,3,4,5],[1,2,5,4,0,3],[4,0,1,5,3,2]],\"target_n\":3,\"target_table\":[[0,1,2],[1,2,0],[2,0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0073","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-homomorphism-audit；資料={\"source_n\":5,\"source_table\":[[3,4,0,1,2],[4,0,1,2,3],[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1]],\"target_n\":5,\"target_table\":[[3,4,0,1,2],[4,0,1,2,3],[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1]],\"mapping\":[3,0,2,4,1]}。請依 [source_n,target_n,mapping,bad_pairs,homomorphism] 欄序回答。","answer":{"kind":"expression","value":"(5, 5, [3, 0, 2, 4, 1], [], 是)","explanation_zh":"現代橋梁核對=(5, 5, [3, 0, 2, 4, 1], [], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-homomorphism-audit","seed":202610233073,"parameters":{"source_n":5,"source_table":[[3,4,0,1,2],[4,0,1,2,3],[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1]],"target_n":5,"target_table":[[3,4,0,1,2],[4,0,1,2,3],[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1]],"mapping":[3,0,2,4,1],"domain":"Complete_finite_group_morphisms","operation":"subgroup-homomorphism-audit","semantic_template":"qf-mod23-subgroup-homomorphism-audit/{\"mapping\":[3,0,2,4,1],\"source_n\":5,\"source_table\":[[3,4,0,1,2],[4,0,1,2,3],[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1]],\"target_n\":5,\"target_table\":[[3,4,0,1,2],[4,0,1,2,3],[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0074","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-homomorphism-audit；資料={\"source_n\":6,\"source_table\":[[2,4,5,1,3,0],[4,5,3,2,0,1],[5,3,0,4,1,2],[1,2,4,0,5,3],[3,0,1,5,2,4],[0,1,2,3,4,5]],\"target_n\":3,\"target_table\":[[1,2,0],[2,0,1],[0,1,2]],\"mapping\":[0,2,1,1,0,2]}。請依 [source_n,target_n,mapping,bad_pairs,homomorphism] 欄序回答。","answer":{"kind":"expression","value":"(6, 3, [0, 2, 1, 1, 0, 2], [], 是)","explanation_zh":"現代橋梁核對=(6, 3, [0, 2, 1, 1, 0, 2], [], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-homomorphism-audit","seed":202610233074,"parameters":{"source_n":6,"source_table":[[2,4,5,1,3,0],[4,5,3,2,0,1],[5,3,0,4,1,2],[1,2,4,0,5,3],[3,0,1,5,2,4],[0,1,2,3,4,5]],"target_n":3,"target_table":[[1,2,0],[2,0,1],[0,1,2]],"mapping":[0,2,1,1,0,2],"domain":"Complete_finite_group_morphisms","operation":"subgroup-homomorphism-audit","semantic_template":"qf-mod23-subgroup-homomorphism-audit/{\"mapping\":[0,2,1,1,0,2],\"source_n\":6,\"source_table\":[[2,4,5,1,3,0],[4,5,3,2,0,1],[5,3,0,4,1,2],[1,2,4,0,5,3],[3,0,1,5,2,4],[0,1,2,3,4,5]],\"target_n\":3,\"target_table\":[[1,2,0],[2,0,1],[0,1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0075","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-homomorphism-audit；資料={\"source_n\":5,\"source_table\":[[2,3,1,4,0],[3,0,4,2,1],[1,4,3,0,2],[4,2,0,1,3],[0,1,2,3,4]],\"target_n\":5,\"target_table\":[[2,3,1,4,0],[3,0,4,2,1],[1,4,3,0,2],[4,2,0,1,3],[0,1,2,3,4]],\"mapping\":[2,0,3,1,4]}。請依 [source_n,target_n,mapping,bad_pairs,homomorphism] 欄序回答。","answer":{"kind":"expression","value":"(5, 5, [2, 0, 3, 1, 4], [], 是)","explanation_zh":"現代橋梁核對=(5, 5, [2, 0, 3, 1, 4], [], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-homomorphism-audit","seed":202610233075,"parameters":{"source_n":5,"source_table":[[2,3,1,4,0],[3,0,4,2,1],[1,4,3,0,2],[4,2,0,1,3],[0,1,2,3,4]],"target_n":5,"target_table":[[2,3,1,4,0],[3,0,4,2,1],[1,4,3,0,2],[4,2,0,1,3],[0,1,2,3,4]],"mapping":[2,0,3,1,4],"domain":"Complete_finite_group_morphisms","operation":"subgroup-homomorphism-audit","semantic_template":"qf-mod23-subgroup-homomorphism-audit/{\"mapping\":[2,0,3,1,4],\"source_n\":5,\"source_table\":[[2,3,1,4,0],[3,0,4,2,1],[1,4,3,0,2],[4,2,0,1,3],[0,1,2,3,4]],\"target_n\":5,\"target_table\":[[2,3,1,4,0],[3,0,4,2,1],[1,4,3,0,2],[4,2,0,1,3],[0,1,2,3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0076","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-kernel-image；資料={\"source_n\":5,\"source_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"target_n\":5,\"target_table\":[[0,1,2,3,4],[1,3,0,4,2],[2,0,4,1,3],[3,4,1,2,0],[4,2,3,0,1]],\"mapping\":[0,3,2,1,4]}。請依 [source_n,target_n,mapping,kernel,image,injective] 欄序回答。","answer":{"kind":"expression","value":"(5, 5, [0, 3, 2, 1, 4], [0], [0, 1, 2, 3, 4], 是)","explanation_zh":"現代橋梁核對=(5, 5, [0, 3, 2, 1, 4], [0], [0, 1, 2, 3, 4], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-kernel-image","seed":202610233076,"parameters":{"source_n":5,"source_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"target_n":5,"target_table":[[0,1,2,3,4],[1,3,0,4,2],[2,0,4,1,3],[3,4,1,2,0],[4,2,3,0,1]],"mapping":[0,3,2,1,4],"domain":"Complete_finite_group_morphisms","operation":"subgroup-kernel-image","semantic_template":"qf-mod23-subgroup-kernel-image/{\"mapping\":[0,3,2,1,4],\"source_n\":5,\"source_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"target_n\":5,\"target_table\":[[0,1,2,3,4],[1,3,0,4,2],[2,0,4,1,3],[3,4,1,2,0],[4,2,3,0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0077","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-kernel-image；資料={\"source_n\":6,\"source_table\":[[2,0,3,4,5,1],[0,1,2,3,4,5],[3,2,4,5,1,0],[4,3,5,1,0,2],[5,4,1,0,2,3],[1,5,0,2,3,4]],\"target_n\":3,\"target_table\":[[2,0,1],[0,1,2],[1,2,0]],\"mapping\":[0,1,2,1,0,2]}。請依 [source_n,target_n,mapping,kernel,image,injective] 欄序回答。","answer":{"kind":"expression","value":"(6, 3, [0, 1, 2, 1, 0, 2], [1, 3], [0, 1, 2], 否)","explanation_zh":"現代橋梁核對=(6, 3, [0, 1, 2, 1, 0, 2], [1, 3], [0, 1, 2], 否)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-kernel-image","seed":202610233077,"parameters":{"source_n":6,"source_table":[[2,0,3,4,5,1],[0,1,2,3,4,5],[3,2,4,5,1,0],[4,3,5,1,0,2],[5,4,1,0,2,3],[1,5,0,2,3,4]],"target_n":3,"target_table":[[2,0,1],[0,1,2],[1,2,0]],"mapping":[0,1,2,1,0,2],"domain":"Complete_finite_group_morphisms","operation":"subgroup-kernel-image","semantic_template":"qf-mod23-subgroup-kernel-image/{\"mapping\":[0,1,2,1,0,2],\"source_n\":6,\"source_table\":[[2,0,3,4,5,1],[0,1,2,3,4,5],[3,2,4,5,1,0],[4,3,5,1,0,2],[5,4,1,0,2,3],[1,5,0,2,3,4]],\"target_n\":3,\"target_table\":[[2,0,1],[0,1,2],[1,2,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0078","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-kernel-image；資料={\"source_n\":5,\"source_table\":[[1,3,0,4,2],[3,4,1,2,0],[0,1,2,3,4],[4,2,3,0,1],[2,0,4,1,3]],\"target_n\":5,\"target_table\":[[3,2,0,4,1],[2,4,1,0,3],[0,1,2,3,4],[4,0,3,1,2],[1,3,4,2,0]],\"mapping\":[3,1,2,0,4]}。請依 [source_n,target_n,mapping,kernel,image,injective] 欄序回答。","answer":{"kind":"expression","value":"(5, 5, [3, 1, 2, 0, 4], [2], [0, 1, 2, 3, 4], 是)","explanation_zh":"現代橋梁核對=(5, 5, [3, 1, 2, 0, 4], [2], [0, 1, 2, 3, 4], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-kernel-image","seed":202610233078,"parameters":{"source_n":5,"source_table":[[1,3,0,4,2],[3,4,1,2,0],[0,1,2,3,4],[4,2,3,0,1],[2,0,4,1,3]],"target_n":5,"target_table":[[3,2,0,4,1],[2,4,1,0,3],[0,1,2,3,4],[4,0,3,1,2],[1,3,4,2,0]],"mapping":[3,1,2,0,4],"domain":"Complete_finite_group_morphisms","operation":"subgroup-kernel-image","semantic_template":"qf-mod23-subgroup-kernel-image/{\"mapping\":[3,1,2,0,4],\"source_n\":5,\"source_table\":[[1,3,0,4,2],[3,4,1,2,0],[0,1,2,3,4],[4,2,3,0,1],[2,0,4,1,3]],\"target_n\":5,\"target_table\":[[3,2,0,4,1],[2,4,1,0,3],[0,1,2,3,4],[4,0,3,1,2],[1,3,4,2,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0079","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-kernel-image；資料={\"source_n\":6,\"source_table\":[[1,2,4,0,5,3],[2,4,5,1,3,0],[4,5,3,2,0,1],[0,1,2,3,4,5],[5,3,0,4,1,2],[3,0,1,5,2,4]],\"target_n\":3,\"target_table\":[[0,1,2],[1,2,0],[2,0,1]],\"mapping\":[2,1,0,0,2,1]}。請依 [source_n,target_n,mapping,kernel,image,injective] 欄序回答。","answer":{"kind":"expression","value":"(6, 3, [2, 1, 0, 0, 2, 1], [2, 3], [0, 1, 2], 否)","explanation_zh":"現代橋梁核對=(6, 3, [2, 1, 0, 0, 2, 1], [2, 3], [0, 1, 2], 否)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-kernel-image","seed":202610233079,"parameters":{"source_n":6,"source_table":[[1,2,4,0,5,3],[2,4,5,1,3,0],[4,5,3,2,0,1],[0,1,2,3,4,5],[5,3,0,4,1,2],[3,0,1,5,2,4]],"target_n":3,"target_table":[[0,1,2],[1,2,0],[2,0,1]],"mapping":[2,1,0,0,2,1],"domain":"Complete_finite_group_morphisms","operation":"subgroup-kernel-image","semantic_template":"qf-mod23-subgroup-kernel-image/{\"mapping\":[2,1,0,0,2,1],\"source_n\":6,\"source_table\":[[1,2,4,0,5,3],[2,4,5,1,3,0],[4,5,3,2,0,1],[0,1,2,3,4,5],[5,3,0,4,1,2],[3,0,1,5,2,4]],\"target_n\":3,\"target_table\":[[0,1,2],[1,2,0],[2,0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0080","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-kernel-image；資料={\"source_n\":5,\"source_table\":[[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3],[0,1,2,3,4]],\"target_n\":5,\"target_table\":[[2,4,3,1,0],[4,3,0,2,1],[3,0,1,4,2],[1,2,4,0,3],[0,1,2,3,4]],\"mapping\":[2,1,0,3,4]}。請依 [source_n,target_n,mapping,kernel,image,injective] 欄序回答。","answer":{"kind":"expression","value":"(5, 5, [2, 1, 0, 3, 4], [4], [0, 1, 2, 3, 4], 是)","explanation_zh":"現代橋梁核對=(5, 5, [2, 1, 0, 3, 4], [4], [0, 1, 2, 3, 4], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-kernel-image","seed":202610233080,"parameters":{"source_n":5,"source_table":[[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3],[0,1,2,3,4]],"target_n":5,"target_table":[[2,4,3,1,0],[4,3,0,2,1],[3,0,1,4,2],[1,2,4,0,3],[0,1,2,3,4]],"mapping":[2,1,0,3,4],"domain":"Complete_finite_group_morphisms","operation":"subgroup-kernel-image","semantic_template":"qf-mod23-subgroup-kernel-image/{\"mapping\":[2,1,0,3,4],\"source_n\":5,\"source_table\":[[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3],[0,1,2,3,4]],\"target_n\":5,\"target_table\":[[2,4,3,1,0],[4,3,0,2,1],[3,0,1,4,2],[1,2,4,0,3],[0,1,2,3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0081","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-kernel-image；資料={\"source_n\":6,\"source_table\":[[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5]],\"target_n\":3,\"target_table\":[[1,2,0],[2,0,1],[0,1,2]],\"mapping\":[1,0,2,1,0,2]}。請依 [source_n,target_n,mapping,kernel,image,injective] 欄序回答。","answer":{"kind":"expression","value":"(6, 3, [1, 0, 2, 1, 0, 2], [2, 5], [0, 1, 2], 否)","explanation_zh":"現代橋梁核對=(6, 3, [1, 0, 2, 1, 0, 2], [2, 5], [0, 1, 2], 否)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-kernel-image","seed":202610233081,"parameters":{"source_n":6,"source_table":[[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5]],"target_n":3,"target_table":[[1,2,0],[2,0,1],[0,1,2]],"mapping":[1,0,2,1,0,2],"domain":"Complete_finite_group_morphisms","operation":"subgroup-kernel-image","semantic_template":"qf-mod23-subgroup-kernel-image/{\"mapping\":[1,0,2,1,0,2],\"source_n\":6,\"source_table\":[[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5]],\"target_n\":3,\"target_table\":[[1,2,0],[2,0,1],[0,1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0082","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-kernel-image；資料={\"source_n\":5,\"source_table\":[[4,0,3,1,2],[0,1,2,3,4],[3,2,0,4,1],[1,3,4,2,0],[2,4,1,0,3]],\"target_n\":5,\"target_table\":[[4,0,1,2,3],[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2]],\"mapping\":[2,1,4,0,3]}。請依 [source_n,target_n,mapping,kernel,image,injective] 欄序回答。","answer":{"kind":"expression","value":"(5, 5, [2, 1, 4, 0, 3], [1], [0, 1, 2, 3, 4], 是)","explanation_zh":"現代橋梁核對=(5, 5, [2, 1, 4, 0, 3], [1], [0, 1, 2, 3, 4], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-kernel-image","seed":202610233082,"parameters":{"source_n":5,"source_table":[[4,0,3,1,2],[0,1,2,3,4],[3,2,0,4,1],[1,3,4,2,0],[2,4,1,0,3]],"target_n":5,"target_table":[[4,0,1,2,3],[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2]],"mapping":[2,1,4,0,3],"domain":"Complete_finite_group_morphisms","operation":"subgroup-kernel-image","semantic_template":"qf-mod23-subgroup-kernel-image/{\"mapping\":[2,1,4,0,3],\"source_n\":5,\"source_table\":[[4,0,3,1,2],[0,1,2,3,4],[3,2,0,4,1],[1,3,4,2,0],[2,4,1,0,3]],\"target_n\":5,\"target_table\":[[4,0,1,2,3],[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0083","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-kernel-image；資料={\"source_n\":6,\"source_table\":[[4,0,3,5,1,2],[0,1,2,3,4,5],[3,2,0,4,5,1],[5,3,4,1,2,0],[1,4,5,2,0,3],[2,5,1,0,3,4]],\"target_n\":3,\"target_table\":[[2,0,1],[0,1,2],[1,2,0]],\"mapping\":[2,1,0,1,0,2]}。請依 [source_n,target_n,mapping,kernel,image,injective] 欄序回答。","answer":{"kind":"expression","value":"(6, 3, [2, 1, 0, 1, 0, 2], [1, 3], [0, 1, 2], 否)","explanation_zh":"現代橋梁核對=(6, 3, [2, 1, 0, 1, 0, 2], [1, 3], [0, 1, 2], 否)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-kernel-image","seed":202610233083,"parameters":{"source_n":6,"source_table":[[4,0,3,5,1,2],[0,1,2,3,4,5],[3,2,0,4,5,1],[5,3,4,1,2,0],[1,4,5,2,0,3],[2,5,1,0,3,4]],"target_n":3,"target_table":[[2,0,1],[0,1,2],[1,2,0]],"mapping":[2,1,0,1,0,2],"domain":"Complete_finite_group_morphisms","operation":"subgroup-kernel-image","semantic_template":"qf-mod23-subgroup-kernel-image/{\"mapping\":[2,1,0,1,0,2],\"source_n\":6,\"source_table\":[[4,0,3,5,1,2],[0,1,2,3,4,5],[3,2,0,4,5,1],[5,3,4,1,2,0],[1,4,5,2,0,3],[2,5,1,0,3,4]],\"target_n\":3,\"target_table\":[[2,0,1],[0,1,2],[1,2,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0084","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-kernel-image；資料={\"source_n\":5,\"source_table\":[[4,2,3,0,1],[2,0,4,1,3],[3,4,1,2,0],[0,1,2,3,4],[1,3,0,4,2]],\"target_n\":5,\"target_table\":[[4,3,1,0,2],[3,2,4,1,0],[1,4,0,2,3],[0,1,2,3,4],[2,0,3,4,1]],\"mapping\":[1,4,0,3,2]}。請依 [source_n,target_n,mapping,kernel,image,injective] 欄序回答。","answer":{"kind":"expression","value":"(5, 5, [1, 4, 0, 3, 2], [3], [0, 1, 2, 3, 4], 是)","explanation_zh":"現代橋梁核對=(5, 5, [1, 4, 0, 3, 2], [3], [0, 1, 2, 3, 4], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-kernel-image","seed":202610233084,"parameters":{"source_n":5,"source_table":[[4,2,3,0,1],[2,0,4,1,3],[3,4,1,2,0],[0,1,2,3,4],[1,3,0,4,2]],"target_n":5,"target_table":[[4,3,1,0,2],[3,2,4,1,0],[1,4,0,2,3],[0,1,2,3,4],[2,0,3,4,1]],"mapping":[1,4,0,3,2],"domain":"Complete_finite_group_morphisms","operation":"subgroup-kernel-image","semantic_template":"qf-mod23-subgroup-kernel-image/{\"mapping\":[1,4,0,3,2],\"source_n\":5,\"source_table\":[[4,2,3,0,1],[2,0,4,1,3],[3,4,1,2,0],[0,1,2,3,4],[1,3,0,4,2]],\"target_n\":5,\"target_table\":[[4,3,1,0,2],[3,2,4,1,0],[1,4,0,2,3],[0,1,2,3,4],[2,0,3,4,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0085","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-kernel-image；資料={\"source_n\":6,\"source_table\":[[4,2,5,0,3,1],[2,0,4,1,5,3],[5,4,3,2,1,0],[0,1,2,3,4,5],[3,5,1,4,0,2],[1,3,0,5,2,4]],\"target_n\":3,\"target_table\":[[0,1,2],[1,2,0],[2,0,1]],\"mapping\":[1,2,0,0,2,1]}。請依 [source_n,target_n,mapping,kernel,image,injective] 欄序回答。","answer":{"kind":"expression","value":"(6, 3, [1, 2, 0, 0, 2, 1], [2, 3], [0, 1, 2], 否)","explanation_zh":"現代橋梁核對=(6, 3, [1, 2, 0, 0, 2, 1], [2, 3], [0, 1, 2], 否)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-kernel-image","seed":202610233085,"parameters":{"source_n":6,"source_table":[[4,2,5,0,3,1],[2,0,4,1,5,3],[5,4,3,2,1,0],[0,1,2,3,4,5],[3,5,1,4,0,2],[1,3,0,5,2,4]],"target_n":3,"target_table":[[0,1,2],[1,2,0],[2,0,1]],"mapping":[1,2,0,0,2,1],"domain":"Complete_finite_group_morphisms","operation":"subgroup-kernel-image","semantic_template":"qf-mod23-subgroup-kernel-image/{\"mapping\":[1,2,0,0,2,1],\"source_n\":6,\"source_table\":[[4,2,5,0,3,1],[2,0,4,1,5,3],[5,4,3,2,1,0],[0,1,2,3,4,5],[3,5,1,4,0,2],[1,3,0,5,2,4]],\"target_n\":3,\"target_table\":[[0,1,2],[1,2,0],[2,0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0086","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-kernel-image；資料={\"source_n\":5,\"source_table\":[[0,1,2,3,4],[1,4,0,2,3],[2,0,3,4,1],[3,2,4,1,0],[4,3,1,0,2]],\"target_n\":5,\"target_table\":[[0,1,2,3,4],[1,4,3,0,2],[2,3,1,4,0],[3,0,4,2,1],[4,2,0,1,3]],\"mapping\":[0,4,2,1,3]}。請依 [source_n,target_n,mapping,kernel,image,injective] 欄序回答。","answer":{"kind":"expression","value":"(5, 5, [0, 4, 2, 1, 3], [0], [0, 1, 2, 3, 4], 是)","explanation_zh":"現代橋梁核對=(5, 5, [0, 4, 2, 1, 3], [0], [0, 1, 2, 3, 4], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-kernel-image","seed":202610233086,"parameters":{"source_n":5,"source_table":[[0,1,2,3,4],[1,4,0,2,3],[2,0,3,4,1],[3,2,4,1,0],[4,3,1,0,2]],"target_n":5,"target_table":[[0,1,2,3,4],[1,4,3,0,2],[2,3,1,4,0],[3,0,4,2,1],[4,2,0,1,3]],"mapping":[0,4,2,1,3],"domain":"Complete_finite_group_morphisms","operation":"subgroup-kernel-image","semantic_template":"qf-mod23-subgroup-kernel-image/{\"mapping\":[0,4,2,1,3],\"source_n\":5,\"source_table\":[[0,1,2,3,4],[1,4,0,2,3],[2,0,3,4,1],[3,2,4,1,0],[4,3,1,0,2]],\"target_n\":5,\"target_table\":[[0,1,2,3,4],[1,4,3,0,2],[2,3,1,4,0],[3,0,4,2,1],[4,2,0,1,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0087","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-kernel-image；資料={\"source_n\":6,\"source_table\":[[3,2,4,5,1,0],[2,0,3,4,5,1],[4,3,5,1,0,2],[5,4,1,0,2,3],[1,5,0,2,3,4],[0,1,2,3,4,5]],\"target_n\":3,\"target_table\":[[1,2,0],[2,0,1],[0,1,2]],\"mapping\":[0,1,2,1,0,2]}。請依 [source_n,target_n,mapping,kernel,image,injective] 欄序回答。","answer":{"kind":"expression","value":"(6, 3, [0, 1, 2, 1, 0, 2], [2, 5], [0, 1, 2], 否)","explanation_zh":"現代橋梁核對=(6, 3, [0, 1, 2, 1, 0, 2], [2, 5], [0, 1, 2], 否)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-kernel-image","seed":202610233087,"parameters":{"source_n":6,"source_table":[[3,2,4,5,1,0],[2,0,3,4,5,1],[4,3,5,1,0,2],[5,4,1,0,2,3],[1,5,0,2,3,4],[0,1,2,3,4,5]],"target_n":3,"target_table":[[1,2,0],[2,0,1],[0,1,2]],"mapping":[0,1,2,1,0,2],"domain":"Complete_finite_group_morphisms","operation":"subgroup-kernel-image","semantic_template":"qf-mod23-subgroup-kernel-image/{\"mapping\":[0,1,2,1,0,2],\"source_n\":6,\"source_table\":[[3,2,4,5,1,0],[2,0,3,4,5,1],[4,3,5,1,0,2],[5,4,1,0,2,3],[1,5,0,2,3,4],[0,1,2,3,4,5]],\"target_n\":3,\"target_table\":[[1,2,0],[2,0,1],[0,1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0088","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-kernel-image；資料={\"source_n\":5,\"source_table\":[[4,2,0,1,3],[2,3,1,4,0],[0,1,2,3,4],[1,4,3,0,2],[3,0,4,2,1]],\"target_n\":5,\"target_table\":[[4,3,0,2,1],[3,0,1,4,2],[0,1,2,3,4],[2,4,3,1,0],[1,2,4,0,3]],\"mapping\":[4,1,2,0,3]}。請依 [source_n,target_n,mapping,kernel,image,injective] 欄序回答。","answer":{"kind":"expression","value":"(5, 5, [4, 1, 2, 0, 3], [2], [0, 1, 2, 3, 4], 是)","explanation_zh":"現代橋梁核對=(5, 5, [4, 1, 2, 0, 3], [2], [0, 1, 2, 3, 4], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-kernel-image","seed":202610233088,"parameters":{"source_n":5,"source_table":[[4,2,0,1,3],[2,3,1,4,0],[0,1,2,3,4],[1,4,3,0,2],[3,0,4,2,1]],"target_n":5,"target_table":[[4,3,0,2,1],[3,0,1,4,2],[0,1,2,3,4],[2,4,3,1,0],[1,2,4,0,3]],"mapping":[4,1,2,0,3],"domain":"Complete_finite_group_morphisms","operation":"subgroup-kernel-image","semantic_template":"qf-mod23-subgroup-kernel-image/{\"mapping\":[4,1,2,0,3],\"source_n\":5,\"source_table\":[[4,2,0,1,3],[2,3,1,4,0],[0,1,2,3,4],[1,4,3,0,2],[3,0,4,2,1]],\"target_n\":5,\"target_table\":[[4,3,0,2,1],[3,0,1,4,2],[0,1,2,3,4],[2,4,3,1,0],[1,2,4,0,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0089","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-kernel-image；資料={\"source_n\":6,\"source_table\":[[4,0,5,2,1,3],[0,1,2,3,4,5],[5,2,1,4,3,0],[2,3,4,0,5,1],[1,4,3,5,0,2],[3,5,0,1,2,4]],\"target_n\":3,\"target_table\":[[2,0,1],[0,1,2],[1,2,0]],\"mapping\":[2,1,1,0,0,2]}。請依 [source_n,target_n,mapping,kernel,image,injective] 欄序回答。","answer":{"kind":"expression","value":"(6, 3, [2, 1, 1, 0, 0, 2], [1, 2], [0, 1, 2], 否)","explanation_zh":"現代橋梁核對=(6, 3, [2, 1, 1, 0, 0, 2], [1, 2], [0, 1, 2], 否)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-kernel-image","seed":202610233089,"parameters":{"source_n":6,"source_table":[[4,0,5,2,1,3],[0,1,2,3,4,5],[5,2,1,4,3,0],[2,3,4,0,5,1],[1,4,3,5,0,2],[3,5,0,1,2,4]],"target_n":3,"target_table":[[2,0,1],[0,1,2],[1,2,0]],"mapping":[2,1,1,0,0,2],"domain":"Complete_finite_group_morphisms","operation":"subgroup-kernel-image","semantic_template":"qf-mod23-subgroup-kernel-image/{\"mapping\":[2,1,1,0,0,2],\"source_n\":6,\"source_table\":[[4,0,5,2,1,3],[0,1,2,3,4,5],[5,2,1,4,3,0],[2,3,4,0,5,1],[1,4,3,5,0,2],[3,5,0,1,2,4]],\"target_n\":3,\"target_table\":[[2,0,1],[0,1,2],[1,2,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0090","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-kernel-image；資料={\"source_n\":5,\"source_table\":[[3,4,1,2,0],[4,2,3,0,1],[1,3,0,4,2],[2,0,4,1,3],[0,1,2,3,4]],\"target_n\":5,\"target_table\":[[3,2,4,1,0],[2,0,3,4,1],[4,3,1,0,2],[1,4,0,2,3],[0,1,2,3,4]],\"mapping\":[3,1,0,2,4]}。請依 [source_n,target_n,mapping,kernel,image,injective] 欄序回答。","answer":{"kind":"expression","value":"(5, 5, [3, 1, 0, 2, 4], [4], [0, 1, 2, 3, 4], 是)","explanation_zh":"現代橋梁核對=(5, 5, [3, 1, 0, 2, 4], [4], [0, 1, 2, 3, 4], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-kernel-image","seed":202610233090,"parameters":{"source_n":5,"source_table":[[3,4,1,2,0],[4,2,3,0,1],[1,3,0,4,2],[2,0,4,1,3],[0,1,2,3,4]],"target_n":5,"target_table":[[3,2,4,1,0],[2,0,3,4,1],[4,3,1,0,2],[1,4,0,2,3],[0,1,2,3,4]],"mapping":[3,1,0,2,4],"domain":"Complete_finite_group_morphisms","operation":"subgroup-kernel-image","semantic_template":"qf-mod23-subgroup-kernel-image/{\"mapping\":[3,1,0,2,4],\"source_n\":5,\"source_table\":[[3,4,1,2,0],[4,2,3,0,1],[1,3,0,4,2],[2,0,4,1,3],[0,1,2,3,4]],\"target_n\":5,\"target_table\":[[3,2,4,1,0],[2,0,3,4,1],[4,3,1,0,2],[1,4,0,2,3],[0,1,2,3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0091","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-kernel-image；資料={\"source_n\":6,\"source_table\":[[4,5,1,0,3,2],[5,3,4,1,2,0],[1,4,0,2,5,3],[0,1,2,3,4,5],[3,2,5,4,0,1],[2,0,3,5,1,4]],\"target_n\":3,\"target_table\":[[0,1,2],[1,2,0],[2,0,1]],\"mapping\":[1,0,2,0,2,1]}。請依 [source_n,target_n,mapping,kernel,image,injective] 欄序回答。","answer":{"kind":"expression","value":"(6, 3, [1, 0, 2, 0, 2, 1], [1, 3], [0, 1, 2], 否)","explanation_zh":"現代橋梁核對=(6, 3, [1, 0, 2, 0, 2, 1], [1, 3], [0, 1, 2], 否)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-kernel-image","seed":202610233091,"parameters":{"source_n":6,"source_table":[[4,5,1,0,3,2],[5,3,4,1,2,0],[1,4,0,2,5,3],[0,1,2,3,4,5],[3,2,5,4,0,1],[2,0,3,5,1,4]],"target_n":3,"target_table":[[0,1,2],[1,2,0],[2,0,1]],"mapping":[1,0,2,0,2,1],"domain":"Complete_finite_group_morphisms","operation":"subgroup-kernel-image","semantic_template":"qf-mod23-subgroup-kernel-image/{\"mapping\":[1,0,2,0,2,1],\"source_n\":6,\"source_table\":[[4,5,1,0,3,2],[5,3,4,1,2,0],[1,4,0,2,5,3],[0,1,2,3,4,5],[3,2,5,4,0,1],[2,0,3,5,1,4]],\"target_n\":3,\"target_table\":[[0,1,2],[1,2,0],[2,0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0092","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-kernel-image；資料={\"source_n\":5,\"source_table\":[[3,0,1,4,2],[0,1,2,3,4],[1,2,4,0,3],[4,3,0,2,1],[2,4,3,1,0]],\"target_n\":5,\"target_table\":[[2,0,3,4,1],[0,1,2,3,4],[3,2,4,1,0],[4,3,1,0,2],[1,4,0,2,3]],\"mapping\":[4,1,0,3,2]}。請依 [source_n,target_n,mapping,kernel,image,injective] 欄序回答。","answer":{"kind":"expression","value":"(5, 5, [4, 1, 0, 3, 2], [1], [0, 1, 2, 3, 4], 是)","explanation_zh":"現代橋梁核對=(5, 5, [4, 1, 0, 3, 2], [1], [0, 1, 2, 3, 4], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-kernel-image","seed":202610233092,"parameters":{"source_n":5,"source_table":[[3,0,1,4,2],[0,1,2,3,4],[1,2,4,0,3],[4,3,0,2,1],[2,4,3,1,0]],"target_n":5,"target_table":[[2,0,3,4,1],[0,1,2,3,4],[3,2,4,1,0],[4,3,1,0,2],[1,4,0,2,3]],"mapping":[4,1,0,3,2],"domain":"Complete_finite_group_morphisms","operation":"subgroup-kernel-image","semantic_template":"qf-mod23-subgroup-kernel-image/{\"mapping\":[4,1,0,3,2],\"source_n\":5,\"source_table\":[[3,0,1,4,2],[0,1,2,3,4],[1,2,4,0,3],[4,3,0,2,1],[2,4,3,1,0]],\"target_n\":5,\"target_table\":[[2,0,3,4,1],[0,1,2,3,4],[3,2,4,1,0],[4,3,1,0,2],[1,4,0,2,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0093","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-kernel-image；資料={\"source_n\":6,\"source_table\":[[3,4,1,5,2,0],[4,5,3,2,0,1],[1,3,0,4,5,2],[5,2,4,0,1,3],[2,0,5,1,3,4],[0,1,2,3,4,5]],\"target_n\":3,\"target_table\":[[1,2,0],[2,0,1],[0,1,2]],\"mapping\":[0,2,1,1,0,2]}。請依 [source_n,target_n,mapping,kernel,image,injective] 欄序回答。","answer":{"kind":"expression","value":"(6, 3, [0, 2, 1, 1, 0, 2], [1, 5], [0, 1, 2], 否)","explanation_zh":"現代橋梁核對=(6, 3, [0, 2, 1, 1, 0, 2], [1, 5], [0, 1, 2], 否)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-kernel-image","seed":202610233093,"parameters":{"source_n":6,"source_table":[[3,4,1,5,2,0],[4,5,3,2,0,1],[1,3,0,4,5,2],[5,2,4,0,1,3],[2,0,5,1,3,4],[0,1,2,3,4,5]],"target_n":3,"target_table":[[1,2,0],[2,0,1],[0,1,2]],"mapping":[0,2,1,1,0,2],"domain":"Complete_finite_group_morphisms","operation":"subgroup-kernel-image","semantic_template":"qf-mod23-subgroup-kernel-image/{\"mapping\":[0,2,1,1,0,2],\"source_n\":6,\"source_table\":[[3,4,1,5,2,0],[4,5,3,2,0,1],[1,3,0,4,5,2],[5,2,4,0,1,3],[2,0,5,1,3,4],[0,1,2,3,4,5]],\"target_n\":3,\"target_table\":[[1,2,0],[2,0,1],[0,1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0094","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-kernel-image；資料={\"source_n\":5,\"source_table\":[[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3],[0,1,2,3,4],[1,2,3,4,0]],\"target_n\":5,\"target_table\":[[1,2,4,0,3],[2,4,3,1,0],[4,3,0,2,1],[0,1,2,3,4],[3,0,1,4,2]],\"mapping\":[4,0,2,3,1]}。請依 [source_n,target_n,mapping,kernel,image,injective] 欄序回答。","answer":{"kind":"expression","value":"(5, 5, [4, 0, 2, 3, 1], [3], [0, 1, 2, 3, 4], 是)","explanation_zh":"現代橋梁核對=(5, 5, [4, 0, 2, 3, 1], [3], [0, 1, 2, 3, 4], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-kernel-image","seed":202610233094,"parameters":{"source_n":5,"source_table":[[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3],[0,1,2,3,4],[1,2,3,4,0]],"target_n":5,"target_table":[[1,2,4,0,3],[2,4,3,1,0],[4,3,0,2,1],[0,1,2,3,4],[3,0,1,4,2]],"mapping":[4,0,2,3,1],"domain":"Complete_finite_group_morphisms","operation":"subgroup-kernel-image","semantic_template":"qf-mod23-subgroup-kernel-image/{\"mapping\":[4,0,2,3,1],\"source_n\":5,\"source_table\":[[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3],[0,1,2,3,4],[1,2,3,4,0]],\"target_n\":5,\"target_table\":[[1,2,4,0,3],[2,4,3,1,0],[4,3,0,2,1],[0,1,2,3,4],[3,0,1,4,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0095","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-kernel-image；資料={\"source_n\":6,\"source_table\":[[3,0,5,1,2,4],[0,1,2,3,4,5],[5,2,1,4,3,0],[1,3,4,0,5,2],[2,4,3,5,0,1],[4,5,0,2,1,3]],\"target_n\":3,\"target_table\":[[2,0,1],[0,1,2],[1,2,0]],\"mapping\":[2,1,1,0,0,2]}。請依 [source_n,target_n,mapping,kernel,image,injective] 欄序回答。","answer":{"kind":"expression","value":"(6, 3, [2, 1, 1, 0, 0, 2], [1, 2], [0, 1, 2], 否)","explanation_zh":"現代橋梁核對=(6, 3, [2, 1, 1, 0, 0, 2], [1, 2], [0, 1, 2], 否)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-kernel-image","seed":202610233095,"parameters":{"source_n":6,"source_table":[[3,0,5,1,2,4],[0,1,2,3,4,5],[5,2,1,4,3,0],[1,3,4,0,5,2],[2,4,3,5,0,1],[4,5,0,2,1,3]],"target_n":3,"target_table":[[2,0,1],[0,1,2],[1,2,0]],"mapping":[2,1,1,0,0,2],"domain":"Complete_finite_group_morphisms","operation":"subgroup-kernel-image","semantic_template":"qf-mod23-subgroup-kernel-image/{\"mapping\":[2,1,1,0,0,2],\"source_n\":6,\"source_table\":[[3,0,5,1,2,4],[0,1,2,3,4,5],[5,2,1,4,3,0],[1,3,4,0,5,2],[2,4,3,5,0,1],[4,5,0,2,1,3]],\"target_n\":3,\"target_table\":[[2,0,1],[0,1,2],[1,2,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0096","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-kernel-image；資料={\"source_n\":5,\"source_table\":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],\"target_n\":5,\"target_table\":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],\"mapping\":[0,3,1,4,2]}。請依 [source_n,target_n,mapping,kernel,image,injective] 欄序回答。","answer":{"kind":"expression","value":"(5, 5, [0, 3, 1, 4, 2], [0], [0, 1, 2, 3, 4], 是)","explanation_zh":"現代橋梁核對=(5, 5, [0, 3, 1, 4, 2], [0], [0, 1, 2, 3, 4], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-kernel-image","seed":202610233096,"parameters":{"source_n":5,"source_table":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],"target_n":5,"target_table":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],"mapping":[0,3,1,4,2],"domain":"Complete_finite_group_morphisms","operation":"subgroup-kernel-image","semantic_template":"qf-mod23-subgroup-kernel-image/{\"mapping\":[0,3,1,4,2],\"source_n\":5,\"source_table\":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],\"target_n\":5,\"target_table\":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0097","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-kernel-image；資料={\"source_n\":6,\"source_table\":[[2,5,3,0,1,4],[5,3,4,1,2,0],[3,4,0,2,5,1],[0,1,2,3,4,5],[1,2,5,4,0,3],[4,0,1,5,3,2]],\"target_n\":3,\"target_table\":[[0,1,2],[1,2,0],[2,0,1]],\"mapping\":[1,0,2,0,2,1]}。請依 [source_n,target_n,mapping,kernel,image,injective] 欄序回答。","answer":{"kind":"expression","value":"(6, 3, [1, 0, 2, 0, 2, 1], [1, 3], [0, 1, 2], 否)","explanation_zh":"現代橋梁核對=(6, 3, [1, 0, 2, 0, 2, 1], [1, 3], [0, 1, 2], 否)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-kernel-image","seed":202610233097,"parameters":{"source_n":6,"source_table":[[2,5,3,0,1,4],[5,3,4,1,2,0],[3,4,0,2,5,1],[0,1,2,3,4,5],[1,2,5,4,0,3],[4,0,1,5,3,2]],"target_n":3,"target_table":[[0,1,2],[1,2,0],[2,0,1]],"mapping":[1,0,2,0,2,1],"domain":"Complete_finite_group_morphisms","operation":"subgroup-kernel-image","semantic_template":"qf-mod23-subgroup-kernel-image/{\"mapping\":[1,0,2,0,2,1],\"source_n\":6,\"source_table\":[[2,5,3,0,1,4],[5,3,4,1,2,0],[3,4,0,2,5,1],[0,1,2,3,4,5],[1,2,5,4,0,3],[4,0,1,5,3,2]],\"target_n\":3,\"target_table\":[[0,1,2],[1,2,0],[2,0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0098","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-kernel-image；資料={\"source_n\":5,\"source_table\":[[3,4,0,1,2],[4,0,1,2,3],[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1]],\"target_n\":5,\"target_table\":[[3,4,0,1,2],[4,0,1,2,3],[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1]],\"mapping\":[3,0,2,4,1]}。請依 [source_n,target_n,mapping,kernel,image,injective] 欄序回答。","answer":{"kind":"expression","value":"(5, 5, [3, 0, 2, 4, 1], [2], [0, 1, 2, 3, 4], 是)","explanation_zh":"現代橋梁核對=(5, 5, [3, 0, 2, 4, 1], [2], [0, 1, 2, 3, 4], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-kernel-image","seed":202610233098,"parameters":{"source_n":5,"source_table":[[3,4,0,1,2],[4,0,1,2,3],[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1]],"target_n":5,"target_table":[[3,4,0,1,2],[4,0,1,2,3],[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1]],"mapping":[3,0,2,4,1],"domain":"Complete_finite_group_morphisms","operation":"subgroup-kernel-image","semantic_template":"qf-mod23-subgroup-kernel-image/{\"mapping\":[3,0,2,4,1],\"source_n\":5,\"source_table\":[[3,4,0,1,2],[4,0,1,2,3],[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1]],\"target_n\":5,\"target_table\":[[3,4,0,1,2],[4,0,1,2,3],[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0099","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-kernel-image；資料={\"source_n\":6,\"source_table\":[[2,4,5,1,3,0],[4,5,3,2,0,1],[5,3,0,4,1,2],[1,2,4,0,5,3],[3,0,1,5,2,4],[0,1,2,3,4,5]],\"target_n\":3,\"target_table\":[[1,2,0],[2,0,1],[0,1,2]],\"mapping\":[0,2,1,1,0,2]}。請依 [source_n,target_n,mapping,kernel,image,injective] 欄序回答。","answer":{"kind":"expression","value":"(6, 3, [0, 2, 1, 1, 0, 2], [1, 5], [0, 1, 2], 否)","explanation_zh":"現代橋梁核對=(6, 3, [0, 2, 1, 1, 0, 2], [1, 5], [0, 1, 2], 否)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-kernel-image","seed":202610233099,"parameters":{"source_n":6,"source_table":[[2,4,5,1,3,0],[4,5,3,2,0,1],[5,3,0,4,1,2],[1,2,4,0,5,3],[3,0,1,5,2,4],[0,1,2,3,4,5]],"target_n":3,"target_table":[[1,2,0],[2,0,1],[0,1,2]],"mapping":[0,2,1,1,0,2],"domain":"Complete_finite_group_morphisms","operation":"subgroup-kernel-image","semantic_template":"qf-mod23-subgroup-kernel-image/{\"mapping\":[0,2,1,1,0,2],\"source_n\":6,\"source_table\":[[2,4,5,1,3,0],[4,5,3,2,0,1],[5,3,0,4,1,2],[1,2,4,0,5,3],[3,0,1,5,2,4],[0,1,2,3,4,5]],\"target_n\":3,\"target_table\":[[1,2,0],[2,0,1],[0,1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r03-subgroup-homomorphism-0100","batch_id":"batch-20261013-advance-20261023-r03-subgroup-homomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-subgroup-homomorphism"],"title_zh":"有限子群與同態","stem_zh":"完整有限群Cayley表。操作=subgroup-kernel-image；資料={\"source_n\":5,\"source_table\":[[2,3,1,4,0],[3,0,4,2,1],[1,4,3,0,2],[4,2,0,1,3],[0,1,2,3,4]],\"target_n\":5,\"target_table\":[[2,3,1,4,0],[3,0,4,2,1],[1,4,3,0,2],[4,2,0,1,3],[0,1,2,3,4]],\"mapping\":[2,0,3,1,4]}。請依 [source_n,target_n,mapping,kernel,image,injective] 欄序回答。","answer":{"kind":"expression","value":"(5, 5, [2, 0, 3, 1, 4], [4], [0, 1, 2, 3, 4], 是)","explanation_zh":"現代橋梁核對=(5, 5, [2, 0, 3, 1, 4], [4], [0, 1, 2, 3, 4], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；同態不自動是單射；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限子群與同態"]},"variant":{"family_id":"qf-mod23-subgroup-kernel-image","seed":202610233100,"parameters":{"source_n":5,"source_table":[[2,3,1,4,0],[3,0,4,2,1],[1,4,3,0,2],[4,2,0,1,3],[0,1,2,3,4]],"target_n":5,"target_table":[[2,3,1,4,0],[3,0,4,2,1],[1,4,3,0,2],[4,2,0,1,3],[0,1,2,3,4]],"mapping":[2,0,3,1,4],"domain":"Complete_finite_group_morphisms","operation":"subgroup-kernel-image","semantic_template":"qf-mod23-subgroup-kernel-image/{\"mapping\":[2,0,3,1,4],\"source_n\":5,\"source_table\":[[2,3,1,4,0],[3,0,4,2,1],[1,4,3,0,2],[4,2,0,1,3],[0,1,2,3,4]],\"target_n\":5,\"target_table\":[[2,3,1,4,0],[3,0,4,2,1],[1,4,3,0,2],[4,2,0,1,3],[0,1,2,3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}