{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r01-finite-abelian-group-0001","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-commutativity-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]]}。請依 [n,identity,noncommuting_pairs,abelian] 欄序回答。","answer":{"kind":"expression","value":"(6, 0, [], 是)","explanation_zh":"結構深度核對=(6, 0, [], 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-commutativity-audit","seed":202610261001,"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]],"domain":"Complete_finite_abelian_group_table","operation":"abelian-commutativity-audit","semantic_template":"qf-mod26-abelian-commutativity-audit/{\"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]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0002","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-commutativity-audit；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,2,3,5,0,4],[2,3,5,4,1,0],[3,5,4,0,2,1],[4,0,1,2,5,3],[5,4,0,1,3,2]]}。請依 [n,identity,noncommuting_pairs,abelian] 欄序回答。","answer":{"kind":"expression","value":"(6, 0, [], 是)","explanation_zh":"結構深度核對=(6, 0, [], 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-commutativity-audit","seed":202610261002,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,2,3,5,0,4],[2,3,5,4,1,0],[3,5,4,0,2,1],[4,0,1,2,5,3],[5,4,0,1,3,2]],"domain":"Complete_finite_abelian_group_table","operation":"abelian-commutativity-audit","semantic_template":"qf-mod26-abelian-commutativity-audit/{\"n\":6,\"table\":[[0,1,2,3,4,5],[1,2,3,5,0,4],[2,3,5,4,1,0],[3,5,4,0,2,1],[4,0,1,2,5,3],[5,4,0,1,3,2]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0003","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-commutativity-audit；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,2,4,5,3,0],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,3,5,1,0,2],[5,0,1,4,2,3]]}。請依 [n,identity,noncommuting_pairs,abelian] 欄序回答。","answer":{"kind":"expression","value":"(6, 0, [], 是)","explanation_zh":"結構深度核對=(6, 0, [], 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-commutativity-audit","seed":202610261003,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,2,4,5,3,0],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,3,5,1,0,2],[5,0,1,4,2,3]],"domain":"Complete_finite_abelian_group_table","operation":"abelian-commutativity-audit","semantic_template":"qf-mod26-abelian-commutativity-audit/{\"n\":6,\"table\":[[0,1,2,3,4,5],[1,2,4,5,3,0],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,3,5,1,0,2],[5,0,1,4,2,3]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0004","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-commutativity-audit；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,2,4,0,5,3],[2,4,5,1,3,0],[3,0,1,5,2,4],[4,5,3,2,0,1],[5,3,0,4,1,2]]}。請依 [n,identity,noncommuting_pairs,abelian] 欄序回答。","answer":{"kind":"expression","value":"(6, 0, [], 是)","explanation_zh":"結構深度核對=(6, 0, [], 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-commutativity-audit","seed":202610261004,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,2,4,0,5,3],[2,4,5,1,3,0],[3,0,1,5,2,4],[4,5,3,2,0,1],[5,3,0,4,1,2]],"domain":"Complete_finite_abelian_group_table","operation":"abelian-commutativity-audit","semantic_template":"qf-mod26-abelian-commutativity-audit/{\"n\":6,\"table\":[[0,1,2,3,4,5],[1,2,4,0,5,3],[2,4,5,1,3,0],[3,0,1,5,2,4],[4,5,3,2,0,1],[5,3,0,4,1,2]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0005","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-commutativity-audit；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,2,5,4,0,3],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,0,1,5,3,2],[5,3,4,1,2,0]]}。請依 [n,identity,noncommuting_pairs,abelian] 欄序回答。","answer":{"kind":"expression","value":"(6, 0, [], 是)","explanation_zh":"結構深度核對=(6, 0, [], 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-commutativity-audit","seed":202610261005,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,2,5,4,0,3],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,0,1,5,3,2],[5,3,4,1,2,0]],"domain":"Complete_finite_abelian_group_table","operation":"abelian-commutativity-audit","semantic_template":"qf-mod26-abelian-commutativity-audit/{\"n\":6,\"table\":[[0,1,2,3,4,5],[1,2,5,4,0,3],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,0,1,5,3,2],[5,3,4,1,2,0]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0006","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-commutativity-audit；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,2,5,0,3,4],[2,5,4,1,0,3],[3,0,1,4,5,2],[4,3,0,5,2,1],[5,4,3,2,1,0]]}。請依 [n,identity,noncommuting_pairs,abelian] 欄序回答。","answer":{"kind":"expression","value":"(6, 0, [], 是)","explanation_zh":"結構深度核對=(6, 0, [], 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-commutativity-audit","seed":202610261006,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,2,5,0,3,4],[2,5,4,1,0,3],[3,0,1,4,5,2],[4,3,0,5,2,1],[5,4,3,2,1,0]],"domain":"Complete_finite_abelian_group_table","operation":"abelian-commutativity-audit","semantic_template":"qf-mod26-abelian-commutativity-audit/{\"n\":6,\"table\":[[0,1,2,3,4,5],[1,2,5,0,3,4],[2,5,4,1,0,3],[3,0,1,4,5,2],[4,3,0,5,2,1],[5,4,3,2,1,0]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0007","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-commutativity-audit；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,3,4,2,5,0],[2,4,0,5,1,3],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,0,3,1,2,4]]}。請依 [n,identity,noncommuting_pairs,abelian] 欄序回答。","answer":{"kind":"expression","value":"(6, 0, [], 是)","explanation_zh":"結構深度核對=(6, 0, [], 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-commutativity-audit","seed":202610261007,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,3,4,2,5,0],[2,4,0,5,1,3],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,0,3,1,2,4]],"domain":"Complete_finite_abelian_group_table","operation":"abelian-commutativity-audit","semantic_template":"qf-mod26-abelian-commutativity-audit/{\"n\":6,\"table\":[[0,1,2,3,4,5],[1,3,4,2,5,0],[2,4,0,5,1,3],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,0,3,1,2,4]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0008","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-commutativity-audit；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,3,5,2,0,4],[2,5,0,4,3,1],[3,2,4,5,1,0],[4,0,3,1,5,2],[5,4,1,0,2,3]]}。請依 [n,identity,noncommuting_pairs,abelian] 欄序回答。","answer":{"kind":"expression","value":"(6, 0, [], 是)","explanation_zh":"結構深度核對=(6, 0, [], 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-commutativity-audit","seed":202610261008,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,3,5,2,0,4],[2,5,0,4,3,1],[3,2,4,5,1,0],[4,0,3,1,5,2],[5,4,1,0,2,3]],"domain":"Complete_finite_abelian_group_table","operation":"abelian-commutativity-audit","semantic_template":"qf-mod26-abelian-commutativity-audit/{\"n\":6,\"table\":[[0,1,2,3,4,5],[1,3,5,2,0,4],[2,5,0,4,3,1],[3,2,4,5,1,0],[4,0,3,1,5,2],[5,4,1,0,2,3]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0009","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-commutativity-audit；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,3,5,4,2,0],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,2,1,5,0,3],[5,0,4,1,3,2]]}。請依 [n,identity,noncommuting_pairs,abelian] 欄序回答。","answer":{"kind":"expression","value":"(6, 0, [], 是)","explanation_zh":"結構深度核對=(6, 0, [], 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-commutativity-audit","seed":202610261009,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,3,5,4,2,0],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,2,1,5,0,3],[5,0,4,1,3,2]],"domain":"Complete_finite_abelian_group_table","operation":"abelian-commutativity-audit","semantic_template":"qf-mod26-abelian-commutativity-audit/{\"n\":6,\"table\":[[0,1,2,3,4,5],[1,3,5,4,2,0],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,2,1,5,0,3],[5,0,4,1,3,2]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0010","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-commutativity-audit；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,3,0,4,5,2],[2,0,5,1,3,4],[3,4,1,5,2,0],[4,5,3,2,0,1],[5,2,4,0,1,3]]}。請依 [n,identity,noncommuting_pairs,abelian] 欄序回答。","answer":{"kind":"expression","value":"(6, 0, [], 是)","explanation_zh":"結構深度核對=(6, 0, [], 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-commutativity-audit","seed":202610261010,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,3,0,4,5,2],[2,0,5,1,3,4],[3,4,1,5,2,0],[4,5,3,2,0,1],[5,2,4,0,1,3]],"domain":"Complete_finite_abelian_group_table","operation":"abelian-commutativity-audit","semantic_template":"qf-mod26-abelian-commutativity-audit/{\"n\":6,\"table\":[[0,1,2,3,4,5],[1,3,0,4,5,2],[2,0,5,1,3,4],[3,4,1,5,2,0],[4,5,3,2,0,1],[5,2,4,0,1,3]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0011","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-commutativity-audit；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,3,4,5,0,2],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,0,5,1,2,3],[5,2,1,4,3,0]]}。請依 [n,identity,noncommuting_pairs,abelian] 欄序回答。","answer":{"kind":"expression","value":"(6, 0, [], 是)","explanation_zh":"結構深度核對=(6, 0, [], 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-commutativity-audit","seed":202610261011,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,3,4,5,0,2],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,0,5,1,2,3],[5,2,1,4,3,0]],"domain":"Complete_finite_abelian_group_table","operation":"abelian-commutativity-audit","semantic_template":"qf-mod26-abelian-commutativity-audit/{\"n\":6,\"table\":[[0,1,2,3,4,5],[1,3,4,5,0,2],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,0,5,1,2,3],[5,2,1,4,3,0]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0012","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-commutativity-audit；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,3,0,5,2,4],[2,0,4,1,5,3],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,4,3,2,1,0]]}。請依 [n,identity,noncommuting_pairs,abelian] 欄序回答。","answer":{"kind":"expression","value":"(6, 0, [], 是)","explanation_zh":"結構深度核對=(6, 0, [], 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-commutativity-audit","seed":202610261012,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,3,0,5,2,4],[2,0,4,1,5,3],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,4,3,2,1,0]],"domain":"Complete_finite_abelian_group_table","operation":"abelian-commutativity-audit","semantic_template":"qf-mod26-abelian-commutativity-audit/{\"n\":6,\"table\":[[0,1,2,3,4,5],[1,3,0,5,2,4],[2,0,4,1,5,3],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,4,3,2,1,0]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0013","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-commutativity-audit；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,4,3,5,2,0],[2,3,0,1,5,4],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,0,4,2,1,3]]}。請依 [n,identity,noncommuting_pairs,abelian] 欄序回答。","answer":{"kind":"expression","value":"(6, 0, [], 是)","explanation_zh":"結構深度核對=(6, 0, [], 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-commutativity-audit","seed":202610261013,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,4,3,5,2,0],[2,3,0,1,5,4],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,0,4,2,1,3]],"domain":"Complete_finite_abelian_group_table","operation":"abelian-commutativity-audit","semantic_template":"qf-mod26-abelian-commutativity-audit/{\"n\":6,\"table\":[[0,1,2,3,4,5],[1,4,3,5,2,0],[2,3,0,1,5,4],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,0,4,2,1,3]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0014","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-commutativity-audit；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,4,5,0,2,3],[2,5,0,4,3,1],[3,0,4,5,1,2],[4,2,3,1,5,0],[5,3,1,2,0,4]]}。請依 [n,identity,noncommuting_pairs,abelian] 欄序回答。","answer":{"kind":"expression","value":"(6, 0, [], 是)","explanation_zh":"結構深度核對=(6, 0, [], 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-commutativity-audit","seed":202610261014,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,4,5,0,2,3],[2,5,0,4,3,1],[3,0,4,5,1,2],[4,2,3,1,5,0],[5,3,1,2,0,4]],"domain":"Complete_finite_abelian_group_table","operation":"abelian-commutativity-audit","semantic_template":"qf-mod26-abelian-commutativity-audit/{\"n\":6,\"table\":[[0,1,2,3,4,5],[1,4,5,0,2,3],[2,5,0,4,3,1],[3,0,4,5,1,2],[4,2,3,1,5,0],[5,3,1,2,0,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0015","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-commutativity-audit；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,4,5,2,3,0],[2,5,4,1,0,3],[3,2,1,0,5,4],[4,3,0,5,2,1],[5,0,3,4,1,2]]}。請依 [n,identity,noncommuting_pairs,abelian] 欄序回答。","answer":{"kind":"expression","value":"(6, 0, [], 是)","explanation_zh":"結構深度核對=(6, 0, [], 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-commutativity-audit","seed":202610261015,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,4,5,2,3,0],[2,5,4,1,0,3],[3,2,1,0,5,4],[4,3,0,5,2,1],[5,0,3,4,1,2]],"domain":"Complete_finite_abelian_group_table","operation":"abelian-commutativity-audit","semantic_template":"qf-mod26-abelian-commutativity-audit/{\"n\":6,\"table\":[[0,1,2,3,4,5],[1,4,5,2,3,0],[2,5,4,1,0,3],[3,2,1,0,5,4],[4,3,0,5,2,1],[5,0,3,4,1,2]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0016","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-commutativity-audit；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,4,0,5,3,2],[2,0,5,4,1,3],[3,5,4,0,2,1],[4,3,1,2,5,0],[5,2,3,1,0,4]]}。請依 [n,identity,noncommuting_pairs,abelian] 欄序回答。","answer":{"kind":"expression","value":"(6, 0, [], 是)","explanation_zh":"結構深度核對=(6, 0, [], 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-commutativity-audit","seed":202610261016,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,4,0,5,3,2],[2,0,5,4,1,3],[3,5,4,0,2,1],[4,3,1,2,5,0],[5,2,3,1,0,4]],"domain":"Complete_finite_abelian_group_table","operation":"abelian-commutativity-audit","semantic_template":"qf-mod26-abelian-commutativity-audit/{\"n\":6,\"table\":[[0,1,2,3,4,5],[1,4,0,5,3,2],[2,0,5,4,1,3],[3,5,4,0,2,1],[4,3,1,2,5,0],[5,2,3,1,0,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0017","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-commutativity-audit；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,4,3,0,5,2],[2,3,4,5,0,1],[3,0,5,2,1,4],[4,5,0,1,2,3],[5,2,1,4,3,0]]}。請依 [n,identity,noncommuting_pairs,abelian] 欄序回答。","answer":{"kind":"expression","value":"(6, 0, [], 是)","explanation_zh":"結構深度核對=(6, 0, [], 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-commutativity-audit","seed":202610261017,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,4,3,0,5,2],[2,3,4,5,0,1],[3,0,5,2,1,4],[4,5,0,1,2,3],[5,2,1,4,3,0]],"domain":"Complete_finite_abelian_group_table","operation":"abelian-commutativity-audit","semantic_template":"qf-mod26-abelian-commutativity-audit/{\"n\":6,\"table\":[[0,1,2,3,4,5],[1,4,3,0,5,2],[2,3,4,5,0,1],[3,0,5,2,1,4],[4,5,0,1,2,3],[5,2,1,4,3,0]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0018","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-commutativity-audit；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,4,0,2,5,3],[2,0,3,5,1,4],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,3,4,1,2,0]]}。請依 [n,identity,noncommuting_pairs,abelian] 欄序回答。","answer":{"kind":"expression","value":"(6, 0, [], 是)","explanation_zh":"結構深度核對=(6, 0, [], 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-commutativity-audit","seed":202610261018,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,4,0,2,5,3],[2,0,3,5,1,4],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,3,4,1,2,0]],"domain":"Complete_finite_abelian_group_table","operation":"abelian-commutativity-audit","semantic_template":"qf-mod26-abelian-commutativity-audit/{\"n\":6,\"table\":[[0,1,2,3,4,5],[1,4,0,2,5,3],[2,0,3,5,1,4],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,3,4,1,2,0]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0019","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-commutativity-audit；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,5,3,4,0,2],[2,3,0,1,5,4],[3,4,1,5,2,0],[4,0,5,2,3,1],[5,2,4,0,1,3]]}。請依 [n,identity,noncommuting_pairs,abelian] 欄序回答。","answer":{"kind":"expression","value":"(6, 0, [], 是)","explanation_zh":"結構深度核對=(6, 0, [], 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-commutativity-audit","seed":202610261019,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,5,3,4,0,2],[2,3,0,1,5,4],[3,4,1,5,2,0],[4,0,5,2,3,1],[5,2,4,0,1,3]],"domain":"Complete_finite_abelian_group_table","operation":"abelian-commutativity-audit","semantic_template":"qf-mod26-abelian-commutativity-audit/{\"n\":6,\"table\":[[0,1,2,3,4,5],[1,5,3,4,0,2],[2,3,0,1,5,4],[3,4,1,5,2,0],[4,0,5,2,3,1],[5,2,4,0,1,3]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0020","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-commutativity-audit；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,5,4,0,3,2],[2,4,0,5,1,3],[3,0,5,4,2,1],[4,3,1,2,5,0],[5,2,3,1,0,4]]}。請依 [n,identity,noncommuting_pairs,abelian] 欄序回答。","answer":{"kind":"expression","value":"(6, 0, [], 是)","explanation_zh":"結構深度核對=(6, 0, [], 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-commutativity-audit","seed":202610261020,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,5,4,0,3,2],[2,4,0,5,1,3],[3,0,5,4,2,1],[4,3,1,2,5,0],[5,2,3,1,0,4]],"domain":"Complete_finite_abelian_group_table","operation":"abelian-commutativity-audit","semantic_template":"qf-mod26-abelian-commutativity-audit/{\"n\":6,\"table\":[[0,1,2,3,4,5],[1,5,4,0,3,2],[2,4,0,5,1,3],[3,0,5,4,2,1],[4,3,1,2,5,0],[5,2,3,1,0,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0021","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-commutativity-audit；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,5,4,2,0,3],[2,4,5,1,3,0],[3,2,1,0,5,4],[4,0,3,5,2,1],[5,3,0,4,1,2]]}。請依 [n,identity,noncommuting_pairs,abelian] 欄序回答。","answer":{"kind":"expression","value":"(6, 0, [], 是)","explanation_zh":"結構深度核對=(6, 0, [], 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-commutativity-audit","seed":202610261021,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,5,4,2,0,3],[2,4,5,1,3,0],[3,2,1,0,5,4],[4,0,3,5,2,1],[5,3,0,4,1,2]],"domain":"Complete_finite_abelian_group_table","operation":"abelian-commutativity-audit","semantic_template":"qf-mod26-abelian-commutativity-audit/{\"n\":6,\"table\":[[0,1,2,3,4,5],[1,5,4,2,0,3],[2,4,5,1,3,0],[3,2,1,0,5,4],[4,0,3,5,2,1],[5,3,0,4,1,2]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0022","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-commutativity-audit；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,5,0,4,2,3],[2,0,4,5,3,1],[3,4,5,0,1,2],[4,2,3,1,5,0],[5,3,1,2,0,4]]}。請依 [n,identity,noncommuting_pairs,abelian] 欄序回答。","answer":{"kind":"expression","value":"(6, 0, [], 是)","explanation_zh":"結構深度核對=(6, 0, [], 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-commutativity-audit","seed":202610261022,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,5,0,4,2,3],[2,0,4,5,3,1],[3,4,5,0,1,2],[4,2,3,1,5,0],[5,3,1,2,0,4]],"domain":"Complete_finite_abelian_group_table","operation":"abelian-commutativity-audit","semantic_template":"qf-mod26-abelian-commutativity-audit/{\"n\":6,\"table\":[[0,1,2,3,4,5],[1,5,0,4,2,3],[2,0,4,5,3,1],[3,4,5,0,1,2],[4,2,3,1,5,0],[5,3,1,2,0,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0023","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-commutativity-audit；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,5,3,0,2,4],[2,3,5,4,1,0],[3,0,4,2,5,1],[4,2,1,5,0,3],[5,4,0,1,3,2]]}。請依 [n,identity,noncommuting_pairs,abelian] 欄序回答。","answer":{"kind":"expression","value":"(6, 0, [], 是)","explanation_zh":"結構深度核對=(6, 0, [], 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-commutativity-audit","seed":202610261023,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,5,3,0,2,4],[2,3,5,4,1,0],[3,0,4,2,5,1],[4,2,1,5,0,3],[5,4,0,1,3,2]],"domain":"Complete_finite_abelian_group_table","operation":"abelian-commutativity-audit","semantic_template":"qf-mod26-abelian-commutativity-audit/{\"n\":6,\"table\":[[0,1,2,3,4,5],[1,5,3,0,2,4],[2,3,5,4,1,0],[3,0,4,2,5,1],[4,2,1,5,0,3],[5,4,0,1,3,2]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0024","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-commutativity-audit；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,5,0,2,3,4],[2,0,3,4,5,1],[3,2,4,5,1,0],[4,3,5,1,0,2],[5,4,1,0,2,3]]}。請依 [n,identity,noncommuting_pairs,abelian] 欄序回答。","answer":{"kind":"expression","value":"(6, 0, [], 是)","explanation_zh":"結構深度核對=(6, 0, [], 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-commutativity-audit","seed":202610261024,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,5,0,2,3,4],[2,0,3,4,5,1],[3,2,4,5,1,0],[4,3,5,1,0,2],[5,4,1,0,2,3]],"domain":"Complete_finite_abelian_group_table","operation":"abelian-commutativity-audit","semantic_template":"qf-mod26-abelian-commutativity-audit/{\"n\":6,\"table\":[[0,1,2,3,4,5],[1,5,0,2,3,4],[2,0,3,4,5,1],[3,2,4,5,1,0],[4,3,5,1,0,2],[5,4,1,0,2,3]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0025","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-commutativity-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]]}。請依 [n,identity,noncommuting_pairs,abelian] 欄序回答。","answer":{"kind":"expression","value":"(6, 0, [], 是)","explanation_zh":"結構深度核對=(6, 0, [], 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-commutativity-audit","seed":202610261025,"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]],"domain":"Complete_finite_abelian_group_table","operation":"abelian-commutativity-audit","semantic_template":"qf-mod26-abelian-commutativity-audit/{\"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]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0026","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-element-order；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]],\"element\":1}。請依 [element,identity,order,returns_identity] 欄序回答。","answer":{"kind":"expression","value":"(1, 0, 6, 是)","explanation_zh":"結構深度核對=(1, 0, 6, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-element-order","seed":202610261026,"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]],"element":1,"domain":"Complete_finite_abelian_group_table","operation":"abelian-element-order","semantic_template":"qf-mod26-abelian-element-order/{\"element\":1,\"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]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0027","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-element-order；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,2,3,5,0,4],[2,3,5,4,1,0],[3,5,4,0,2,1],[4,0,1,2,5,3],[5,4,0,1,3,2]],\"element\":0}。請依 [element,identity,order,returns_identity] 欄序回答。","answer":{"kind":"expression","value":"(0, 0, 1, 是)","explanation_zh":"結構深度核對=(0, 0, 1, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-element-order","seed":202610261027,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,2,3,5,0,4],[2,3,5,4,1,0],[3,5,4,0,2,1],[4,0,1,2,5,3],[5,4,0,1,3,2]],"element":0,"domain":"Complete_finite_abelian_group_table","operation":"abelian-element-order","semantic_template":"qf-mod26-abelian-element-order/{\"element\":0,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,2,3,5,0,4],[2,3,5,4,1,0],[3,5,4,0,2,1],[4,0,1,2,5,3],[5,4,0,1,3,2]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0028","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-element-order；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,2,4,5,3,0],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,3,5,1,0,2],[5,0,1,4,2,3]],\"element\":5}。請依 [element,identity,order,returns_identity] 欄序回答。","answer":{"kind":"expression","value":"(5, 0, 6, 是)","explanation_zh":"結構深度核對=(5, 0, 6, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-element-order","seed":202610261028,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,2,4,5,3,0],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,3,5,1,0,2],[5,0,1,4,2,3]],"element":5,"domain":"Complete_finite_abelian_group_table","operation":"abelian-element-order","semantic_template":"qf-mod26-abelian-element-order/{\"element\":5,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,2,4,5,3,0],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,3,5,1,0,2],[5,0,1,4,2,3]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0029","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-element-order；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,2,4,0,5,3],[2,4,5,1,3,0],[3,0,1,5,2,4],[4,5,3,2,0,1],[5,3,0,4,1,2]],\"element\":4}。請依 [element,identity,order,returns_identity] 欄序回答。","answer":{"kind":"expression","value":"(4, 0, 2, 是)","explanation_zh":"結構深度核對=(4, 0, 2, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-element-order","seed":202610261029,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,2,4,0,5,3],[2,4,5,1,3,0],[3,0,1,5,2,4],[4,5,3,2,0,1],[5,3,0,4,1,2]],"element":4,"domain":"Complete_finite_abelian_group_table","operation":"abelian-element-order","semantic_template":"qf-mod26-abelian-element-order/{\"element\":4,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,2,4,0,5,3],[2,4,5,1,3,0],[3,0,1,5,2,4],[4,5,3,2,0,1],[5,3,0,4,1,2]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0030","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-element-order；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,2,5,4,0,3],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,0,1,5,3,2],[5,3,4,1,2,0]],\"element\":3}。請依 [element,identity,order,returns_identity] 欄序回答。","answer":{"kind":"expression","value":"(3, 0, 3, 是)","explanation_zh":"結構深度核對=(3, 0, 3, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-element-order","seed":202610261030,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,2,5,4,0,3],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,0,1,5,3,2],[5,3,4,1,2,0]],"element":3,"domain":"Complete_finite_abelian_group_table","operation":"abelian-element-order","semantic_template":"qf-mod26-abelian-element-order/{\"element\":3,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,2,5,4,0,3],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,0,1,5,3,2],[5,3,4,1,2,0]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0031","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-element-order；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,2,5,0,3,4],[2,5,4,1,0,3],[3,0,1,4,5,2],[4,3,0,5,2,1],[5,4,3,2,1,0]],\"element\":2}。請依 [element,identity,order,returns_identity] 欄序回答。","answer":{"kind":"expression","value":"(2, 0, 3, 是)","explanation_zh":"結構深度核對=(2, 0, 3, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-element-order","seed":202610261031,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,2,5,0,3,4],[2,5,4,1,0,3],[3,0,1,4,5,2],[4,3,0,5,2,1],[5,4,3,2,1,0]],"element":2,"domain":"Complete_finite_abelian_group_table","operation":"abelian-element-order","semantic_template":"qf-mod26-abelian-element-order/{\"element\":2,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,2,5,0,3,4],[2,5,4,1,0,3],[3,0,1,4,5,2],[4,3,0,5,2,1],[5,4,3,2,1,0]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0032","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-element-order；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,3,4,2,5,0],[2,4,0,5,1,3],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,0,3,1,2,4]],\"element\":1}。請依 [element,identity,order,returns_identity] 欄序回答。","answer":{"kind":"expression","value":"(1, 0, 6, 是)","explanation_zh":"結構深度核對=(1, 0, 6, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-element-order","seed":202610261032,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,3,4,2,5,0],[2,4,0,5,1,3],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,0,3,1,2,4]],"element":1,"domain":"Complete_finite_abelian_group_table","operation":"abelian-element-order","semantic_template":"qf-mod26-abelian-element-order/{\"element\":1,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,3,4,2,5,0],[2,4,0,5,1,3],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,0,3,1,2,4]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0033","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-element-order；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,3,5,2,0,4],[2,5,0,4,3,1],[3,2,4,5,1,0],[4,0,3,1,5,2],[5,4,1,0,2,3]],\"element\":0}。請依 [element,identity,order,returns_identity] 欄序回答。","answer":{"kind":"expression","value":"(0, 0, 1, 是)","explanation_zh":"結構深度核對=(0, 0, 1, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-element-order","seed":202610261033,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,3,5,2,0,4],[2,5,0,4,3,1],[3,2,4,5,1,0],[4,0,3,1,5,2],[5,4,1,0,2,3]],"element":0,"domain":"Complete_finite_abelian_group_table","operation":"abelian-element-order","semantic_template":"qf-mod26-abelian-element-order/{\"element\":0,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,3,5,2,0,4],[2,5,0,4,3,1],[3,2,4,5,1,0],[4,0,3,1,5,2],[5,4,1,0,2,3]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0034","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-element-order；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,3,5,4,2,0],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,2,1,5,0,3],[5,0,4,1,3,2]],\"element\":5}。請依 [element,identity,order,returns_identity] 欄序回答。","answer":{"kind":"expression","value":"(5, 0, 6, 是)","explanation_zh":"結構深度核對=(5, 0, 6, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-element-order","seed":202610261034,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,3,5,4,2,0],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,2,1,5,0,3],[5,0,4,1,3,2]],"element":5,"domain":"Complete_finite_abelian_group_table","operation":"abelian-element-order","semantic_template":"qf-mod26-abelian-element-order/{\"element\":5,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,3,5,4,2,0],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,2,1,5,0,3],[5,0,4,1,3,2]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0035","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-element-order；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,3,0,4,5,2],[2,0,5,1,3,4],[3,4,1,5,2,0],[4,5,3,2,0,1],[5,2,4,0,1,3]],\"element\":4}。請依 [element,identity,order,returns_identity] 欄序回答。","answer":{"kind":"expression","value":"(4, 0, 2, 是)","explanation_zh":"結構深度核對=(4, 0, 2, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-element-order","seed":202610261035,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,3,0,4,5,2],[2,0,5,1,3,4],[3,4,1,5,2,0],[4,5,3,2,0,1],[5,2,4,0,1,3]],"element":4,"domain":"Complete_finite_abelian_group_table","operation":"abelian-element-order","semantic_template":"qf-mod26-abelian-element-order/{\"element\":4,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,3,0,4,5,2],[2,0,5,1,3,4],[3,4,1,5,2,0],[4,5,3,2,0,1],[5,2,4,0,1,3]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0036","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-element-order；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,3,4,5,0,2],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,0,5,1,2,3],[5,2,1,4,3,0]],\"element\":3}。請依 [element,identity,order,returns_identity] 欄序回答。","answer":{"kind":"expression","value":"(3, 0, 3, 是)","explanation_zh":"結構深度核對=(3, 0, 3, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-element-order","seed":202610261036,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,3,4,5,0,2],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,0,5,1,2,3],[5,2,1,4,3,0]],"element":3,"domain":"Complete_finite_abelian_group_table","operation":"abelian-element-order","semantic_template":"qf-mod26-abelian-element-order/{\"element\":3,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,3,4,5,0,2],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,0,5,1,2,3],[5,2,1,4,3,0]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0037","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-element-order；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,3,0,5,2,4],[2,0,4,1,5,3],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,4,3,2,1,0]],\"element\":2}。請依 [element,identity,order,returns_identity] 欄序回答。","answer":{"kind":"expression","value":"(2, 0, 6, 是)","explanation_zh":"結構深度核對=(2, 0, 6, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-element-order","seed":202610261037,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,3,0,5,2,4],[2,0,4,1,5,3],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,4,3,2,1,0]],"element":2,"domain":"Complete_finite_abelian_group_table","operation":"abelian-element-order","semantic_template":"qf-mod26-abelian-element-order/{\"element\":2,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,3,0,5,2,4],[2,0,4,1,5,3],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,4,3,2,1,0]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0038","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-element-order；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,4,3,5,2,0],[2,3,0,1,5,4],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,0,4,2,1,3]],\"element\":1}。請依 [element,identity,order,returns_identity] 欄序回答。","answer":{"kind":"expression","value":"(1, 0, 6, 是)","explanation_zh":"結構深度核對=(1, 0, 6, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-element-order","seed":202610261038,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,4,3,5,2,0],[2,3,0,1,5,4],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,0,4,2,1,3]],"element":1,"domain":"Complete_finite_abelian_group_table","operation":"abelian-element-order","semantic_template":"qf-mod26-abelian-element-order/{\"element\":1,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,4,3,5,2,0],[2,3,0,1,5,4],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,0,4,2,1,3]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0039","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-element-order；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,4,5,0,2,3],[2,5,0,4,3,1],[3,0,4,5,1,2],[4,2,3,1,5,0],[5,3,1,2,0,4]],\"element\":0}。請依 [element,identity,order,returns_identity] 欄序回答。","answer":{"kind":"expression","value":"(0, 0, 1, 是)","explanation_zh":"結構深度核對=(0, 0, 1, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-element-order","seed":202610261039,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,4,5,0,2,3],[2,5,0,4,3,1],[3,0,4,5,1,2],[4,2,3,1,5,0],[5,3,1,2,0,4]],"element":0,"domain":"Complete_finite_abelian_group_table","operation":"abelian-element-order","semantic_template":"qf-mod26-abelian-element-order/{\"element\":0,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,4,5,0,2,3],[2,5,0,4,3,1],[3,0,4,5,1,2],[4,2,3,1,5,0],[5,3,1,2,0,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0040","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-element-order；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,4,5,2,3,0],[2,5,4,1,0,3],[3,2,1,0,5,4],[4,3,0,5,2,1],[5,0,3,4,1,2]],\"element\":5}。請依 [element,identity,order,returns_identity] 欄序回答。","answer":{"kind":"expression","value":"(5, 0, 6, 是)","explanation_zh":"結構深度核對=(5, 0, 6, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-element-order","seed":202610261040,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,4,5,2,3,0],[2,5,4,1,0,3],[3,2,1,0,5,4],[4,3,0,5,2,1],[5,0,3,4,1,2]],"element":5,"domain":"Complete_finite_abelian_group_table","operation":"abelian-element-order","semantic_template":"qf-mod26-abelian-element-order/{\"element\":5,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,4,5,2,3,0],[2,5,4,1,0,3],[3,2,1,0,5,4],[4,3,0,5,2,1],[5,0,3,4,1,2]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0041","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-element-order；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,4,0,5,3,2],[2,0,5,4,1,3],[3,5,4,0,2,1],[4,3,1,2,5,0],[5,2,3,1,0,4]],\"element\":4}。請依 [element,identity,order,returns_identity] 欄序回答。","answer":{"kind":"expression","value":"(4, 0, 3, 是)","explanation_zh":"結構深度核對=(4, 0, 3, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-element-order","seed":202610261041,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,4,0,5,3,2],[2,0,5,4,1,3],[3,5,4,0,2,1],[4,3,1,2,5,0],[5,2,3,1,0,4]],"element":4,"domain":"Complete_finite_abelian_group_table","operation":"abelian-element-order","semantic_template":"qf-mod26-abelian-element-order/{\"element\":4,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,4,0,5,3,2],[2,0,5,4,1,3],[3,5,4,0,2,1],[4,3,1,2,5,0],[5,2,3,1,0,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0042","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-element-order；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,4,3,0,5,2],[2,3,4,5,0,1],[3,0,5,2,1,4],[4,5,0,1,2,3],[5,2,1,4,3,0]],\"element\":3}。請依 [element,identity,order,returns_identity] 欄序回答。","answer":{"kind":"expression","value":"(3, 0, 6, 是)","explanation_zh":"結構深度核對=(3, 0, 6, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-element-order","seed":202610261042,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,4,3,0,5,2],[2,3,4,5,0,1],[3,0,5,2,1,4],[4,5,0,1,2,3],[5,2,1,4,3,0]],"element":3,"domain":"Complete_finite_abelian_group_table","operation":"abelian-element-order","semantic_template":"qf-mod26-abelian-element-order/{\"element\":3,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,4,3,0,5,2],[2,3,4,5,0,1],[3,0,5,2,1,4],[4,5,0,1,2,3],[5,2,1,4,3,0]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0043","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-element-order；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,4,0,2,5,3],[2,0,3,5,1,4],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,3,4,1,2,0]],\"element\":2}。請依 [element,identity,order,returns_identity] 欄序回答。","answer":{"kind":"expression","value":"(2, 0, 6, 是)","explanation_zh":"結構深度核對=(2, 0, 6, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-element-order","seed":202610261043,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,4,0,2,5,3],[2,0,3,5,1,4],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,3,4,1,2,0]],"element":2,"domain":"Complete_finite_abelian_group_table","operation":"abelian-element-order","semantic_template":"qf-mod26-abelian-element-order/{\"element\":2,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,4,0,2,5,3],[2,0,3,5,1,4],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,3,4,1,2,0]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0044","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-element-order；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,5,3,4,0,2],[2,3,0,1,5,4],[3,4,1,5,2,0],[4,0,5,2,3,1],[5,2,4,0,1,3]],\"element\":1}。請依 [element,identity,order,returns_identity] 欄序回答。","answer":{"kind":"expression","value":"(1, 0, 6, 是)","explanation_zh":"結構深度核對=(1, 0, 6, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-element-order","seed":202610261044,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,5,3,4,0,2],[2,3,0,1,5,4],[3,4,1,5,2,0],[4,0,5,2,3,1],[5,2,4,0,1,3]],"element":1,"domain":"Complete_finite_abelian_group_table","operation":"abelian-element-order","semantic_template":"qf-mod26-abelian-element-order/{\"element\":1,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,5,3,4,0,2],[2,3,0,1,5,4],[3,4,1,5,2,0],[4,0,5,2,3,1],[5,2,4,0,1,3]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0045","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-element-order；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,5,4,0,3,2],[2,4,0,5,1,3],[3,0,5,4,2,1],[4,3,1,2,5,0],[5,2,3,1,0,4]],\"element\":0}。請依 [element,identity,order,returns_identity] 欄序回答。","answer":{"kind":"expression","value":"(0, 0, 1, 是)","explanation_zh":"結構深度核對=(0, 0, 1, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-element-order","seed":202610261045,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,5,4,0,3,2],[2,4,0,5,1,3],[3,0,5,4,2,1],[4,3,1,2,5,0],[5,2,3,1,0,4]],"element":0,"domain":"Complete_finite_abelian_group_table","operation":"abelian-element-order","semantic_template":"qf-mod26-abelian-element-order/{\"element\":0,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,5,4,0,3,2],[2,4,0,5,1,3],[3,0,5,4,2,1],[4,3,1,2,5,0],[5,2,3,1,0,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0046","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-element-order；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,5,4,2,0,3],[2,4,5,1,3,0],[3,2,1,0,5,4],[4,0,3,5,2,1],[5,3,0,4,1,2]],\"element\":5}。請依 [element,identity,order,returns_identity] 欄序回答。","answer":{"kind":"expression","value":"(5, 0, 3, 是)","explanation_zh":"結構深度核對=(5, 0, 3, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-element-order","seed":202610261046,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,5,4,2,0,3],[2,4,5,1,3,0],[3,2,1,0,5,4],[4,0,3,5,2,1],[5,3,0,4,1,2]],"element":5,"domain":"Complete_finite_abelian_group_table","operation":"abelian-element-order","semantic_template":"qf-mod26-abelian-element-order/{\"element\":5,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,5,4,2,0,3],[2,4,5,1,3,0],[3,2,1,0,5,4],[4,0,3,5,2,1],[5,3,0,4,1,2]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0047","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-element-order；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,5,0,4,2,3],[2,0,4,5,3,1],[3,4,5,0,1,2],[4,2,3,1,5,0],[5,3,1,2,0,4]],\"element\":4}。請依 [element,identity,order,returns_identity] 欄序回答。","answer":{"kind":"expression","value":"(4, 0, 3, 是)","explanation_zh":"結構深度核對=(4, 0, 3, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-element-order","seed":202610261047,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,5,0,4,2,3],[2,0,4,5,3,1],[3,4,5,0,1,2],[4,2,3,1,5,0],[5,3,1,2,0,4]],"element":4,"domain":"Complete_finite_abelian_group_table","operation":"abelian-element-order","semantic_template":"qf-mod26-abelian-element-order/{\"element\":4,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,5,0,4,2,3],[2,0,4,5,3,1],[3,4,5,0,1,2],[4,2,3,1,5,0],[5,3,1,2,0,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0048","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-element-order；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,5,3,0,2,4],[2,3,5,4,1,0],[3,0,4,2,5,1],[4,2,1,5,0,3],[5,4,0,1,3,2]],\"element\":3}。請依 [element,identity,order,returns_identity] 欄序回答。","answer":{"kind":"expression","value":"(3, 0, 6, 是)","explanation_zh":"結構深度核對=(3, 0, 6, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-element-order","seed":202610261048,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,5,3,0,2,4],[2,3,5,4,1,0],[3,0,4,2,5,1],[4,2,1,5,0,3],[5,4,0,1,3,2]],"element":3,"domain":"Complete_finite_abelian_group_table","operation":"abelian-element-order","semantic_template":"qf-mod26-abelian-element-order/{\"element\":3,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,5,3,0,2,4],[2,3,5,4,1,0],[3,0,4,2,5,1],[4,2,1,5,0,3],[5,4,0,1,3,2]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0049","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-element-order；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,5,0,2,3,4],[2,0,3,4,5,1],[3,2,4,5,1,0],[4,3,5,1,0,2],[5,4,1,0,2,3]],\"element\":2}。請依 [element,identity,order,returns_identity] 欄序回答。","answer":{"kind":"expression","value":"(2, 0, 6, 是)","explanation_zh":"結構深度核對=(2, 0, 6, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-element-order","seed":202610261049,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,5,0,2,3,4],[2,0,3,4,5,1],[3,2,4,5,1,0],[4,3,5,1,0,2],[5,4,1,0,2,3]],"element":2,"domain":"Complete_finite_abelian_group_table","operation":"abelian-element-order","semantic_template":"qf-mod26-abelian-element-order/{\"element\":2,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,5,0,2,3,4],[2,0,3,4,5,1],[3,2,4,5,1,0],[4,3,5,1,0,2],[5,4,1,0,2,3]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0050","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-element-order；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]],\"element\":1}。請依 [element,identity,order,returns_identity] 欄序回答。","answer":{"kind":"expression","value":"(1, 0, 3, 是)","explanation_zh":"結構深度核對=(1, 0, 3, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-element-order","seed":202610261050,"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]],"element":1,"domain":"Complete_finite_abelian_group_table","operation":"abelian-element-order","semantic_template":"qf-mod26-abelian-element-order/{\"element\":1,\"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]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0051","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-integer-multiple；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]],\"element\":1,\"multiplier\":3}。請依 [element,multiplier,identity,result] 欄序回答。","answer":{"kind":"expression","value":"(1, 3, 0, 3)","explanation_zh":"結構深度核對=(1, 3, 0, 3)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-integer-multiple","seed":202610261051,"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]],"element":1,"multiplier":3,"domain":"Complete_finite_abelian_group_table","operation":"abelian-integer-multiple","semantic_template":"qf-mod26-abelian-integer-multiple/{\"element\":1,\"multiplier\":3,\"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]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0052","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-integer-multiple；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,2,3,5,0,4],[2,3,5,4,1,0],[3,5,4,0,2,1],[4,0,1,2,5,3],[5,4,0,1,3,2]],\"element\":0,\"multiplier\":4}。請依 [element,multiplier,identity,result] 欄序回答。","answer":{"kind":"expression","value":"(0, 4, 0, 0)","explanation_zh":"結構深度核對=(0, 4, 0, 0)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-integer-multiple","seed":202610261052,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,2,3,5,0,4],[2,3,5,4,1,0],[3,5,4,0,2,1],[4,0,1,2,5,3],[5,4,0,1,3,2]],"element":0,"multiplier":4,"domain":"Complete_finite_abelian_group_table","operation":"abelian-integer-multiple","semantic_template":"qf-mod26-abelian-integer-multiple/{\"element\":0,\"multiplier\":4,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,2,3,5,0,4],[2,3,5,4,1,0],[3,5,4,0,2,1],[4,0,1,2,5,3],[5,4,0,1,3,2]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0053","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-integer-multiple；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,2,4,5,3,0],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,3,5,1,0,2],[5,0,1,4,2,3]],\"element\":5,\"multiplier\":5}。請依 [element,multiplier,identity,result] 欄序回答。","answer":{"kind":"expression","value":"(5, 5, 0, 1)","explanation_zh":"結構深度核對=(5, 5, 0, 1)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-integer-multiple","seed":202610261053,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,2,4,5,3,0],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,3,5,1,0,2],[5,0,1,4,2,3]],"element":5,"multiplier":5,"domain":"Complete_finite_abelian_group_table","operation":"abelian-integer-multiple","semantic_template":"qf-mod26-abelian-integer-multiple/{\"element\":5,\"multiplier\":5,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,2,4,5,3,0],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,3,5,1,0,2],[5,0,1,4,2,3]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0054","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-integer-multiple；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,2,4,0,5,3],[2,4,5,1,3,0],[3,0,1,5,2,4],[4,5,3,2,0,1],[5,3,0,4,1,2]],\"element\":4,\"multiplier\":6}。請依 [element,multiplier,identity,result] 欄序回答。","answer":{"kind":"expression","value":"(4, 6, 0, 0)","explanation_zh":"結構深度核對=(4, 6, 0, 0)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-integer-multiple","seed":202610261054,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,2,4,0,5,3],[2,4,5,1,3,0],[3,0,1,5,2,4],[4,5,3,2,0,1],[5,3,0,4,1,2]],"element":4,"multiplier":6,"domain":"Complete_finite_abelian_group_table","operation":"abelian-integer-multiple","semantic_template":"qf-mod26-abelian-integer-multiple/{\"element\":4,\"multiplier\":6,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,2,4,0,5,3],[2,4,5,1,3,0],[3,0,1,5,2,4],[4,5,3,2,0,1],[5,3,0,4,1,2]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0055","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-integer-multiple；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,2,5,4,0,3],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,0,1,5,3,2],[5,3,4,1,2,0]],\"element\":3,\"multiplier\":7}。請依 [element,multiplier,identity,result] 欄序回答。","answer":{"kind":"expression","value":"(3, 7, 0, 3)","explanation_zh":"結構深度核對=(3, 7, 0, 3)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-integer-multiple","seed":202610261055,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,2,5,4,0,3],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,0,1,5,3,2],[5,3,4,1,2,0]],"element":3,"multiplier":7,"domain":"Complete_finite_abelian_group_table","operation":"abelian-integer-multiple","semantic_template":"qf-mod26-abelian-integer-multiple/{\"element\":3,\"multiplier\":7,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,2,5,4,0,3],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,0,1,5,3,2],[5,3,4,1,2,0]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0056","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-integer-multiple；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,2,5,0,3,4],[2,5,4,1,0,3],[3,0,1,4,5,2],[4,3,0,5,2,1],[5,4,3,2,1,0]],\"element\":2,\"multiplier\":8}。請依 [element,multiplier,identity,result] 欄序回答。","answer":{"kind":"expression","value":"(2, 8, 0, 4)","explanation_zh":"結構深度核對=(2, 8, 0, 4)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-integer-multiple","seed":202610261056,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,2,5,0,3,4],[2,5,4,1,0,3],[3,0,1,4,5,2],[4,3,0,5,2,1],[5,4,3,2,1,0]],"element":2,"multiplier":8,"domain":"Complete_finite_abelian_group_table","operation":"abelian-integer-multiple","semantic_template":"qf-mod26-abelian-integer-multiple/{\"element\":2,\"multiplier\":8,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,2,5,0,3,4],[2,5,4,1,0,3],[3,0,1,4,5,2],[4,3,0,5,2,1],[5,4,3,2,1,0]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0057","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-integer-multiple；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,3,4,2,5,0],[2,4,0,5,1,3],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,0,3,1,2,4]],\"element\":1,\"multiplier\":9}。請依 [element,multiplier,identity,result] 欄序回答。","answer":{"kind":"expression","value":"(1, 9, 0, 2)","explanation_zh":"結構深度核對=(1, 9, 0, 2)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-integer-multiple","seed":202610261057,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,3,4,2,5,0],[2,4,0,5,1,3],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,0,3,1,2,4]],"element":1,"multiplier":9,"domain":"Complete_finite_abelian_group_table","operation":"abelian-integer-multiple","semantic_template":"qf-mod26-abelian-integer-multiple/{\"element\":1,\"multiplier\":9,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,3,4,2,5,0],[2,4,0,5,1,3],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,0,3,1,2,4]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0058","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-integer-multiple；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,3,5,2,0,4],[2,5,0,4,3,1],[3,2,4,5,1,0],[4,0,3,1,5,2],[5,4,1,0,2,3]],\"element\":0,\"multiplier\":10}。請依 [element,multiplier,identity,result] 欄序回答。","answer":{"kind":"expression","value":"(0, 10, 0, 0)","explanation_zh":"結構深度核對=(0, 10, 0, 0)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-integer-multiple","seed":202610261058,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,3,5,2,0,4],[2,5,0,4,3,1],[3,2,4,5,1,0],[4,0,3,1,5,2],[5,4,1,0,2,3]],"element":0,"multiplier":10,"domain":"Complete_finite_abelian_group_table","operation":"abelian-integer-multiple","semantic_template":"qf-mod26-abelian-integer-multiple/{\"element\":0,\"multiplier\":10,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,3,5,2,0,4],[2,5,0,4,3,1],[3,2,4,5,1,0],[4,0,3,1,5,2],[5,4,1,0,2,3]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0059","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-integer-multiple；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,3,5,4,2,0],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,2,1,5,0,3],[5,0,4,1,3,2]],\"element\":5,\"multiplier\":11}。請依 [element,multiplier,identity,result] 欄序回答。","answer":{"kind":"expression","value":"(5, 11, 0, 1)","explanation_zh":"結構深度核對=(5, 11, 0, 1)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-integer-multiple","seed":202610261059,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,3,5,4,2,0],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,2,1,5,0,3],[5,0,4,1,3,2]],"element":5,"multiplier":11,"domain":"Complete_finite_abelian_group_table","operation":"abelian-integer-multiple","semantic_template":"qf-mod26-abelian-integer-multiple/{\"element\":5,\"multiplier\":11,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,3,5,4,2,0],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,2,1,5,0,3],[5,0,4,1,3,2]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0060","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-integer-multiple；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,3,0,4,5,2],[2,0,5,1,3,4],[3,4,1,5,2,0],[4,5,3,2,0,1],[5,2,4,0,1,3]],\"element\":4,\"multiplier\":12}。請依 [element,multiplier,identity,result] 欄序回答。","answer":{"kind":"expression","value":"(4, 12, 0, 0)","explanation_zh":"結構深度核對=(4, 12, 0, 0)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-integer-multiple","seed":202610261060,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,3,0,4,5,2],[2,0,5,1,3,4],[3,4,1,5,2,0],[4,5,3,2,0,1],[5,2,4,0,1,3]],"element":4,"multiplier":12,"domain":"Complete_finite_abelian_group_table","operation":"abelian-integer-multiple","semantic_template":"qf-mod26-abelian-integer-multiple/{\"element\":4,\"multiplier\":12,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,3,0,4,5,2],[2,0,5,1,3,4],[3,4,1,5,2,0],[4,5,3,2,0,1],[5,2,4,0,1,3]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0061","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-integer-multiple；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,3,4,5,0,2],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,0,5,1,2,3],[5,2,1,4,3,0]],\"element\":3,\"multiplier\":13}。請依 [element,multiplier,identity,result] 欄序回答。","answer":{"kind":"expression","value":"(3, 13, 0, 3)","explanation_zh":"結構深度核對=(3, 13, 0, 3)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-integer-multiple","seed":202610261061,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,3,4,5,0,2],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,0,5,1,2,3],[5,2,1,4,3,0]],"element":3,"multiplier":13,"domain":"Complete_finite_abelian_group_table","operation":"abelian-integer-multiple","semantic_template":"qf-mod26-abelian-integer-multiple/{\"element\":3,\"multiplier\":13,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,3,4,5,0,2],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,0,5,1,2,3],[5,2,1,4,3,0]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0062","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-integer-multiple；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,3,0,5,2,4],[2,0,4,1,5,3],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,4,3,2,1,0]],\"element\":2,\"multiplier\":14}。請依 [element,multiplier,identity,result] 欄序回答。","answer":{"kind":"expression","value":"(2, 14, 0, 4)","explanation_zh":"結構深度核對=(2, 14, 0, 4)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-integer-multiple","seed":202610261062,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,3,0,5,2,4],[2,0,4,1,5,3],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,4,3,2,1,0]],"element":2,"multiplier":14,"domain":"Complete_finite_abelian_group_table","operation":"abelian-integer-multiple","semantic_template":"qf-mod26-abelian-integer-multiple/{\"element\":2,\"multiplier\":14,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,3,0,5,2,4],[2,0,4,1,5,3],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,4,3,2,1,0]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0063","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-integer-multiple；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,4,3,5,2,0],[2,3,0,1,5,4],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,0,4,2,1,3]],\"element\":1,\"multiplier\":15}。請依 [element,multiplier,identity,result] 欄序回答。","answer":{"kind":"expression","value":"(1, 15, 0, 2)","explanation_zh":"結構深度核對=(1, 15, 0, 2)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-integer-multiple","seed":202610261063,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,4,3,5,2,0],[2,3,0,1,5,4],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,0,4,2,1,3]],"element":1,"multiplier":15,"domain":"Complete_finite_abelian_group_table","operation":"abelian-integer-multiple","semantic_template":"qf-mod26-abelian-integer-multiple/{\"element\":1,\"multiplier\":15,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,4,3,5,2,0],[2,3,0,1,5,4],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,0,4,2,1,3]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0064","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-integer-multiple；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,4,5,0,2,3],[2,5,0,4,3,1],[3,0,4,5,1,2],[4,2,3,1,5,0],[5,3,1,2,0,4]],\"element\":0,\"multiplier\":16}。請依 [element,multiplier,identity,result] 欄序回答。","answer":{"kind":"expression","value":"(0, 16, 0, 0)","explanation_zh":"結構深度核對=(0, 16, 0, 0)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-integer-multiple","seed":202610261064,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,4,5,0,2,3],[2,5,0,4,3,1],[3,0,4,5,1,2],[4,2,3,1,5,0],[5,3,1,2,0,4]],"element":0,"multiplier":16,"domain":"Complete_finite_abelian_group_table","operation":"abelian-integer-multiple","semantic_template":"qf-mod26-abelian-integer-multiple/{\"element\":0,\"multiplier\":16,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,4,5,0,2,3],[2,5,0,4,3,1],[3,0,4,5,1,2],[4,2,3,1,5,0],[5,3,1,2,0,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0065","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-integer-multiple；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,4,5,2,3,0],[2,5,4,1,0,3],[3,2,1,0,5,4],[4,3,0,5,2,1],[5,0,3,4,1,2]],\"element\":5,\"multiplier\":17}。請依 [element,multiplier,identity,result] 欄序回答。","answer":{"kind":"expression","value":"(5, 17, 0, 1)","explanation_zh":"結構深度核對=(5, 17, 0, 1)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-integer-multiple","seed":202610261065,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,4,5,2,3,0],[2,5,4,1,0,3],[3,2,1,0,5,4],[4,3,0,5,2,1],[5,0,3,4,1,2]],"element":5,"multiplier":17,"domain":"Complete_finite_abelian_group_table","operation":"abelian-integer-multiple","semantic_template":"qf-mod26-abelian-integer-multiple/{\"element\":5,\"multiplier\":17,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,4,5,2,3,0],[2,5,4,1,0,3],[3,2,1,0,5,4],[4,3,0,5,2,1],[5,0,3,4,1,2]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0066","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-integer-multiple；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,4,0,5,3,2],[2,0,5,4,1,3],[3,5,4,0,2,1],[4,3,1,2,5,0],[5,2,3,1,0,4]],\"element\":4,\"multiplier\":18}。請依 [element,multiplier,identity,result] 欄序回答。","answer":{"kind":"expression","value":"(4, 18, 0, 0)","explanation_zh":"結構深度核對=(4, 18, 0, 0)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-integer-multiple","seed":202610261066,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,4,0,5,3,2],[2,0,5,4,1,3],[3,5,4,0,2,1],[4,3,1,2,5,0],[5,2,3,1,0,4]],"element":4,"multiplier":18,"domain":"Complete_finite_abelian_group_table","operation":"abelian-integer-multiple","semantic_template":"qf-mod26-abelian-integer-multiple/{\"element\":4,\"multiplier\":18,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,4,0,5,3,2],[2,0,5,4,1,3],[3,5,4,0,2,1],[4,3,1,2,5,0],[5,2,3,1,0,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0067","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-integer-multiple；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,4,3,0,5,2],[2,3,4,5,0,1],[3,0,5,2,1,4],[4,5,0,1,2,3],[5,2,1,4,3,0]],\"element\":3,\"multiplier\":19}。請依 [element,multiplier,identity,result] 欄序回答。","answer":{"kind":"expression","value":"(3, 19, 0, 3)","explanation_zh":"結構深度核對=(3, 19, 0, 3)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-integer-multiple","seed":202610261067,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,4,3,0,5,2],[2,3,4,5,0,1],[3,0,5,2,1,4],[4,5,0,1,2,3],[5,2,1,4,3,0]],"element":3,"multiplier":19,"domain":"Complete_finite_abelian_group_table","operation":"abelian-integer-multiple","semantic_template":"qf-mod26-abelian-integer-multiple/{\"element\":3,\"multiplier\":19,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,4,3,0,5,2],[2,3,4,5,0,1],[3,0,5,2,1,4],[4,5,0,1,2,3],[5,2,1,4,3,0]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0068","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-integer-multiple；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,4,0,2,5,3],[2,0,3,5,1,4],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,3,4,1,2,0]],\"element\":2,\"multiplier\":20}。請依 [element,multiplier,identity,result] 欄序回答。","answer":{"kind":"expression","value":"(2, 20, 0, 3)","explanation_zh":"結構深度核對=(2, 20, 0, 3)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-integer-multiple","seed":202610261068,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,4,0,2,5,3],[2,0,3,5,1,4],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,3,4,1,2,0]],"element":2,"multiplier":20,"domain":"Complete_finite_abelian_group_table","operation":"abelian-integer-multiple","semantic_template":"qf-mod26-abelian-integer-multiple/{\"element\":2,\"multiplier\":20,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,4,0,2,5,3],[2,0,3,5,1,4],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,3,4,1,2,0]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0069","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-integer-multiple；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,5,3,4,0,2],[2,3,0,1,5,4],[3,4,1,5,2,0],[4,0,5,2,3,1],[5,2,4,0,1,3]],\"element\":1,\"multiplier\":21}。請依 [element,multiplier,identity,result] 欄序回答。","answer":{"kind":"expression","value":"(1, 21, 0, 2)","explanation_zh":"結構深度核對=(1, 21, 0, 2)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-integer-multiple","seed":202610261069,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,5,3,4,0,2],[2,3,0,1,5,4],[3,4,1,5,2,0],[4,0,5,2,3,1],[5,2,4,0,1,3]],"element":1,"multiplier":21,"domain":"Complete_finite_abelian_group_table","operation":"abelian-integer-multiple","semantic_template":"qf-mod26-abelian-integer-multiple/{\"element\":1,\"multiplier\":21,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,5,3,4,0,2],[2,3,0,1,5,4],[3,4,1,5,2,0],[4,0,5,2,3,1],[5,2,4,0,1,3]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0070","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-integer-multiple；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,5,4,0,3,2],[2,4,0,5,1,3],[3,0,5,4,2,1],[4,3,1,2,5,0],[5,2,3,1,0,4]],\"element\":0,\"multiplier\":22}。請依 [element,multiplier,identity,result] 欄序回答。","answer":{"kind":"expression","value":"(0, 22, 0, 0)","explanation_zh":"結構深度核對=(0, 22, 0, 0)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-integer-multiple","seed":202610261070,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,5,4,0,3,2],[2,4,0,5,1,3],[3,0,5,4,2,1],[4,3,1,2,5,0],[5,2,3,1,0,4]],"element":0,"multiplier":22,"domain":"Complete_finite_abelian_group_table","operation":"abelian-integer-multiple","semantic_template":"qf-mod26-abelian-integer-multiple/{\"element\":0,\"multiplier\":22,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,5,4,0,3,2],[2,4,0,5,1,3],[3,0,5,4,2,1],[4,3,1,2,5,0],[5,2,3,1,0,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0071","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-integer-multiple；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,5,4,2,0,3],[2,4,5,1,3,0],[3,2,1,0,5,4],[4,0,3,5,2,1],[5,3,0,4,1,2]],\"element\":5,\"multiplier\":23}。請依 [element,multiplier,identity,result] 欄序回答。","answer":{"kind":"expression","value":"(5, 23, 0, 2)","explanation_zh":"結構深度核對=(5, 23, 0, 2)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-integer-multiple","seed":202610261071,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,5,4,2,0,3],[2,4,5,1,3,0],[3,2,1,0,5,4],[4,0,3,5,2,1],[5,3,0,4,1,2]],"element":5,"multiplier":23,"domain":"Complete_finite_abelian_group_table","operation":"abelian-integer-multiple","semantic_template":"qf-mod26-abelian-integer-multiple/{\"element\":5,\"multiplier\":23,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,5,4,2,0,3],[2,4,5,1,3,0],[3,2,1,0,5,4],[4,0,3,5,2,1],[5,3,0,4,1,2]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0072","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-integer-multiple；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,5,0,4,2,3],[2,0,4,5,3,1],[3,4,5,0,1,2],[4,2,3,1,5,0],[5,3,1,2,0,4]],\"element\":4,\"multiplier\":24}。請依 [element,multiplier,identity,result] 欄序回答。","answer":{"kind":"expression","value":"(4, 24, 0, 0)","explanation_zh":"結構深度核對=(4, 24, 0, 0)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-integer-multiple","seed":202610261072,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,5,0,4,2,3],[2,0,4,5,3,1],[3,4,5,0,1,2],[4,2,3,1,5,0],[5,3,1,2,0,4]],"element":4,"multiplier":24,"domain":"Complete_finite_abelian_group_table","operation":"abelian-integer-multiple","semantic_template":"qf-mod26-abelian-integer-multiple/{\"element\":4,\"multiplier\":24,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,5,0,4,2,3],[2,0,4,5,3,1],[3,4,5,0,1,2],[4,2,3,1,5,0],[5,3,1,2,0,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0073","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-integer-multiple；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,5,3,0,2,4],[2,3,5,4,1,0],[3,0,4,2,5,1],[4,2,1,5,0,3],[5,4,0,1,3,2]],\"element\":3,\"multiplier\":25}。請依 [element,multiplier,identity,result] 欄序回答。","answer":{"kind":"expression","value":"(3, 25, 0, 3)","explanation_zh":"結構深度核對=(3, 25, 0, 3)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-integer-multiple","seed":202610261073,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,5,3,0,2,4],[2,3,5,4,1,0],[3,0,4,2,5,1],[4,2,1,5,0,3],[5,4,0,1,3,2]],"element":3,"multiplier":25,"domain":"Complete_finite_abelian_group_table","operation":"abelian-integer-multiple","semantic_template":"qf-mod26-abelian-integer-multiple/{\"element\":3,\"multiplier\":25,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,5,3,0,2,4],[2,3,5,4,1,0],[3,0,4,2,5,1],[4,2,1,5,0,3],[5,4,0,1,3,2]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0074","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-integer-multiple；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,5,0,2,3,4],[2,0,3,4,5,1],[3,2,4,5,1,0],[4,3,5,1,0,2],[5,4,1,0,2,3]],\"element\":2,\"multiplier\":26}。請依 [element,multiplier,identity,result] 欄序回答。","answer":{"kind":"expression","value":"(2, 26, 0, 3)","explanation_zh":"結構深度核對=(2, 26, 0, 3)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-integer-multiple","seed":202610261074,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,5,0,2,3,4],[2,0,3,4,5,1],[3,2,4,5,1,0],[4,3,5,1,0,2],[5,4,1,0,2,3]],"element":2,"multiplier":26,"domain":"Complete_finite_abelian_group_table","operation":"abelian-integer-multiple","semantic_template":"qf-mod26-abelian-integer-multiple/{\"element\":2,\"multiplier\":26,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,5,0,2,3,4],[2,0,3,4,5,1],[3,2,4,5,1,0],[4,3,5,1,0,2],[5,4,1,0,2,3]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0075","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-integer-multiple；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]],\"element\":1,\"multiplier\":27}。請依 [element,multiplier,identity,result] 欄序回答。","answer":{"kind":"expression","value":"(1, 27, 0, 0)","explanation_zh":"結構深度核對=(1, 27, 0, 0)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-integer-multiple","seed":202610261075,"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]],"element":1,"multiplier":27,"domain":"Complete_finite_abelian_group_table","operation":"abelian-integer-multiple","semantic_template":"qf-mod26-abelian-integer-multiple/{\"element\":1,\"multiplier\":27,\"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]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0076","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-cyclic-subgroup；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]],\"element\":1}。請依 [element,identity,generated_elements,next_repeat,order] 欄序回答。","answer":{"kind":"expression","value":"(1, 0, [0, 1, 2, 3, 4, 5], 0, 6)","explanation_zh":"結構深度核對=(1, 0, [0, 1, 2, 3, 4, 5], 0, 6)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-cyclic-subgroup","seed":202610261076,"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]],"element":1,"domain":"Complete_finite_abelian_group_table","operation":"abelian-cyclic-subgroup","semantic_template":"qf-mod26-abelian-cyclic-subgroup/{\"element\":1,\"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]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0077","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-cyclic-subgroup；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,2,3,5,0,4],[2,3,5,4,1,0],[3,5,4,0,2,1],[4,0,1,2,5,3],[5,4,0,1,3,2]],\"element\":0}。請依 [element,identity,generated_elements,next_repeat,order] 欄序回答。","answer":{"kind":"expression","value":"(0, 0, [0], 0, 1)","explanation_zh":"結構深度核對=(0, 0, [0], 0, 1)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-cyclic-subgroup","seed":202610261077,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,2,3,5,0,4],[2,3,5,4,1,0],[3,5,4,0,2,1],[4,0,1,2,5,3],[5,4,0,1,3,2]],"element":0,"domain":"Complete_finite_abelian_group_table","operation":"abelian-cyclic-subgroup","semantic_template":"qf-mod26-abelian-cyclic-subgroup/{\"element\":0,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,2,3,5,0,4],[2,3,5,4,1,0],[3,5,4,0,2,1],[4,0,1,2,5,3],[5,4,0,1,3,2]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0078","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-cyclic-subgroup；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,2,4,5,3,0],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,3,5,1,0,2],[5,0,1,4,2,3]],\"element\":5}。請依 [element,identity,generated_elements,next_repeat,order] 欄序回答。","answer":{"kind":"expression","value":"(5, 0, [0, 5, 3, 4, 2, 1], 0, 6)","explanation_zh":"結構深度核對=(5, 0, [0, 5, 3, 4, 2, 1], 0, 6)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-cyclic-subgroup","seed":202610261078,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,2,4,5,3,0],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,3,5,1,0,2],[5,0,1,4,2,3]],"element":5,"domain":"Complete_finite_abelian_group_table","operation":"abelian-cyclic-subgroup","semantic_template":"qf-mod26-abelian-cyclic-subgroup/{\"element\":5,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,2,4,5,3,0],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,3,5,1,0,2],[5,0,1,4,2,3]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0079","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-cyclic-subgroup；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,2,4,0,5,3],[2,4,5,1,3,0],[3,0,1,5,2,4],[4,5,3,2,0,1],[5,3,0,4,1,2]],\"element\":4}。請依 [element,identity,generated_elements,next_repeat,order] 欄序回答。","answer":{"kind":"expression","value":"(4, 0, [0, 4], 0, 2)","explanation_zh":"結構深度核對=(4, 0, [0, 4], 0, 2)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-cyclic-subgroup","seed":202610261079,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,2,4,0,5,3],[2,4,5,1,3,0],[3,0,1,5,2,4],[4,5,3,2,0,1],[5,3,0,4,1,2]],"element":4,"domain":"Complete_finite_abelian_group_table","operation":"abelian-cyclic-subgroup","semantic_template":"qf-mod26-abelian-cyclic-subgroup/{\"element\":4,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,2,4,0,5,3],[2,4,5,1,3,0],[3,0,1,5,2,4],[4,5,3,2,0,1],[5,3,0,4,1,2]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0080","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-cyclic-subgroup；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,2,5,4,0,3],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,0,1,5,3,2],[5,3,4,1,2,0]],\"element\":3}。請依 [element,identity,generated_elements,next_repeat,order] 欄序回答。","answer":{"kind":"expression","value":"(3, 0, [0, 3, 2], 0, 3)","explanation_zh":"結構深度核對=(3, 0, [0, 3, 2], 0, 3)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-cyclic-subgroup","seed":202610261080,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,2,5,4,0,3],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,0,1,5,3,2],[5,3,4,1,2,0]],"element":3,"domain":"Complete_finite_abelian_group_table","operation":"abelian-cyclic-subgroup","semantic_template":"qf-mod26-abelian-cyclic-subgroup/{\"element\":3,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,2,5,4,0,3],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,0,1,5,3,2],[5,3,4,1,2,0]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0081","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-cyclic-subgroup；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,2,5,0,3,4],[2,5,4,1,0,3],[3,0,1,4,5,2],[4,3,0,5,2,1],[5,4,3,2,1,0]],\"element\":2}。請依 [element,identity,generated_elements,next_repeat,order] 欄序回答。","answer":{"kind":"expression","value":"(2, 0, [0, 2, 4], 0, 3)","explanation_zh":"結構深度核對=(2, 0, [0, 2, 4], 0, 3)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-cyclic-subgroup","seed":202610261081,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,2,5,0,3,4],[2,5,4,1,0,3],[3,0,1,4,5,2],[4,3,0,5,2,1],[5,4,3,2,1,0]],"element":2,"domain":"Complete_finite_abelian_group_table","operation":"abelian-cyclic-subgroup","semantic_template":"qf-mod26-abelian-cyclic-subgroup/{\"element\":2,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,2,5,0,3,4],[2,5,4,1,0,3],[3,0,1,4,5,2],[4,3,0,5,2,1],[5,4,3,2,1,0]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0082","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-cyclic-subgroup；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,3,4,2,5,0],[2,4,0,5,1,3],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,0,3,1,2,4]],\"element\":1}。請依 [element,identity,generated_elements,next_repeat,order] 欄序回答。","answer":{"kind":"expression","value":"(1, 0, [0, 1, 3, 2, 4, 5], 0, 6)","explanation_zh":"結構深度核對=(1, 0, [0, 1, 3, 2, 4, 5], 0, 6)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-cyclic-subgroup","seed":202610261082,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,3,4,2,5,0],[2,4,0,5,1,3],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,0,3,1,2,4]],"element":1,"domain":"Complete_finite_abelian_group_table","operation":"abelian-cyclic-subgroup","semantic_template":"qf-mod26-abelian-cyclic-subgroup/{\"element\":1,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,3,4,2,5,0],[2,4,0,5,1,3],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,0,3,1,2,4]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0083","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-cyclic-subgroup；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,3,5,2,0,4],[2,5,0,4,3,1],[3,2,4,5,1,0],[4,0,3,1,5,2],[5,4,1,0,2,3]],\"element\":0}。請依 [element,identity,generated_elements,next_repeat,order] 欄序回答。","answer":{"kind":"expression","value":"(0, 0, [0], 0, 1)","explanation_zh":"結構深度核對=(0, 0, [0], 0, 1)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-cyclic-subgroup","seed":202610261083,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,3,5,2,0,4],[2,5,0,4,3,1],[3,2,4,5,1,0],[4,0,3,1,5,2],[5,4,1,0,2,3]],"element":0,"domain":"Complete_finite_abelian_group_table","operation":"abelian-cyclic-subgroup","semantic_template":"qf-mod26-abelian-cyclic-subgroup/{\"element\":0,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,3,5,2,0,4],[2,5,0,4,3,1],[3,2,4,5,1,0],[4,0,3,1,5,2],[5,4,1,0,2,3]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0084","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-cyclic-subgroup；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,3,5,4,2,0],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,2,1,5,0,3],[5,0,4,1,3,2]],\"element\":5}。請依 [element,identity,generated_elements,next_repeat,order] 欄序回答。","answer":{"kind":"expression","value":"(5, 0, [0, 5, 2, 4, 3, 1], 0, 6)","explanation_zh":"結構深度核對=(5, 0, [0, 5, 2, 4, 3, 1], 0, 6)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-cyclic-subgroup","seed":202610261084,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,3,5,4,2,0],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,2,1,5,0,3],[5,0,4,1,3,2]],"element":5,"domain":"Complete_finite_abelian_group_table","operation":"abelian-cyclic-subgroup","semantic_template":"qf-mod26-abelian-cyclic-subgroup/{\"element\":5,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,3,5,4,2,0],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,2,1,5,0,3],[5,0,4,1,3,2]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0085","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-cyclic-subgroup；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,3,0,4,5,2],[2,0,5,1,3,4],[3,4,1,5,2,0],[4,5,3,2,0,1],[5,2,4,0,1,3]],\"element\":4}。請依 [element,identity,generated_elements,next_repeat,order] 欄序回答。","answer":{"kind":"expression","value":"(4, 0, [0, 4], 0, 2)","explanation_zh":"結構深度核對=(4, 0, [0, 4], 0, 2)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-cyclic-subgroup","seed":202610261085,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,3,0,4,5,2],[2,0,5,1,3,4],[3,4,1,5,2,0],[4,5,3,2,0,1],[5,2,4,0,1,3]],"element":4,"domain":"Complete_finite_abelian_group_table","operation":"abelian-cyclic-subgroup","semantic_template":"qf-mod26-abelian-cyclic-subgroup/{\"element\":4,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,3,0,4,5,2],[2,0,5,1,3,4],[3,4,1,5,2,0],[4,5,3,2,0,1],[5,2,4,0,1,3]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0086","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-cyclic-subgroup；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,3,4,5,0,2],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,0,5,1,2,3],[5,2,1,4,3,0]],\"element\":3}。請依 [element,identity,generated_elements,next_repeat,order] 欄序回答。","answer":{"kind":"expression","value":"(3, 0, [0, 3, 2], 0, 3)","explanation_zh":"結構深度核對=(3, 0, [0, 3, 2], 0, 3)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-cyclic-subgroup","seed":202610261086,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,3,4,5,0,2],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,0,5,1,2,3],[5,2,1,4,3,0]],"element":3,"domain":"Complete_finite_abelian_group_table","operation":"abelian-cyclic-subgroup","semantic_template":"qf-mod26-abelian-cyclic-subgroup/{\"element\":3,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,3,4,5,0,2],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,0,5,1,2,3],[5,2,1,4,3,0]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0087","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-cyclic-subgroup；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,3,0,5,2,4],[2,0,4,1,5,3],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,4,3,2,1,0]],\"element\":2}。請依 [element,identity,generated_elements,next_repeat,order] 欄序回答。","answer":{"kind":"expression","value":"(2, 0, [0, 2, 4, 5, 3, 1], 0, 6)","explanation_zh":"結構深度核對=(2, 0, [0, 2, 4, 5, 3, 1], 0, 6)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-cyclic-subgroup","seed":202610261087,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,3,0,5,2,4],[2,0,4,1,5,3],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,4,3,2,1,0]],"element":2,"domain":"Complete_finite_abelian_group_table","operation":"abelian-cyclic-subgroup","semantic_template":"qf-mod26-abelian-cyclic-subgroup/{\"element\":2,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,3,0,5,2,4],[2,0,4,1,5,3],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,4,3,2,1,0]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0088","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-cyclic-subgroup；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,4,3,5,2,0],[2,3,0,1,5,4],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,0,4,2,1,3]],\"element\":1}。請依 [element,identity,generated_elements,next_repeat,order] 欄序回答。","answer":{"kind":"expression","value":"(1, 0, [0, 1, 4, 2, 3, 5], 0, 6)","explanation_zh":"結構深度核對=(1, 0, [0, 1, 4, 2, 3, 5], 0, 6)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-cyclic-subgroup","seed":202610261088,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,4,3,5,2,0],[2,3,0,1,5,4],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,0,4,2,1,3]],"element":1,"domain":"Complete_finite_abelian_group_table","operation":"abelian-cyclic-subgroup","semantic_template":"qf-mod26-abelian-cyclic-subgroup/{\"element\":1,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,4,3,5,2,0],[2,3,0,1,5,4],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,0,4,2,1,3]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0089","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-cyclic-subgroup；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,4,5,0,2,3],[2,5,0,4,3,1],[3,0,4,5,1,2],[4,2,3,1,5,0],[5,3,1,2,0,4]],\"element\":0}。請依 [element,identity,generated_elements,next_repeat,order] 欄序回答。","answer":{"kind":"expression","value":"(0, 0, [0], 0, 1)","explanation_zh":"結構深度核對=(0, 0, [0], 0, 1)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-cyclic-subgroup","seed":202610261089,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,4,5,0,2,3],[2,5,0,4,3,1],[3,0,4,5,1,2],[4,2,3,1,5,0],[5,3,1,2,0,4]],"element":0,"domain":"Complete_finite_abelian_group_table","operation":"abelian-cyclic-subgroup","semantic_template":"qf-mod26-abelian-cyclic-subgroup/{\"element\":0,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,4,5,0,2,3],[2,5,0,4,3,1],[3,0,4,5,1,2],[4,2,3,1,5,0],[5,3,1,2,0,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0090","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-cyclic-subgroup；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,4,5,2,3,0],[2,5,4,1,0,3],[3,2,1,0,5,4],[4,3,0,5,2,1],[5,0,3,4,1,2]],\"element\":5}。請依 [element,identity,generated_elements,next_repeat,order] 欄序回答。","answer":{"kind":"expression","value":"(5, 0, [0, 5, 2, 3, 4, 1], 0, 6)","explanation_zh":"結構深度核對=(5, 0, [0, 5, 2, 3, 4, 1], 0, 6)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-cyclic-subgroup","seed":202610261090,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,4,5,2,3,0],[2,5,4,1,0,3],[3,2,1,0,5,4],[4,3,0,5,2,1],[5,0,3,4,1,2]],"element":5,"domain":"Complete_finite_abelian_group_table","operation":"abelian-cyclic-subgroup","semantic_template":"qf-mod26-abelian-cyclic-subgroup/{\"element\":5,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,4,5,2,3,0],[2,5,4,1,0,3],[3,2,1,0,5,4],[4,3,0,5,2,1],[5,0,3,4,1,2]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0091","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-cyclic-subgroup；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,4,0,5,3,2],[2,0,5,4,1,3],[3,5,4,0,2,1],[4,3,1,2,5,0],[5,2,3,1,0,4]],\"element\":4}。請依 [element,identity,generated_elements,next_repeat,order] 欄序回答。","answer":{"kind":"expression","value":"(4, 0, [0, 4, 5], 0, 3)","explanation_zh":"結構深度核對=(4, 0, [0, 4, 5], 0, 3)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-cyclic-subgroup","seed":202610261091,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,4,0,5,3,2],[2,0,5,4,1,3],[3,5,4,0,2,1],[4,3,1,2,5,0],[5,2,3,1,0,4]],"element":4,"domain":"Complete_finite_abelian_group_table","operation":"abelian-cyclic-subgroup","semantic_template":"qf-mod26-abelian-cyclic-subgroup/{\"element\":4,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,4,0,5,3,2],[2,0,5,4,1,3],[3,5,4,0,2,1],[4,3,1,2,5,0],[5,2,3,1,0,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0092","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-cyclic-subgroup；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,4,3,0,5,2],[2,3,4,5,0,1],[3,0,5,2,1,4],[4,5,0,1,2,3],[5,2,1,4,3,0]],\"element\":3}。請依 [element,identity,generated_elements,next_repeat,order] 欄序回答。","answer":{"kind":"expression","value":"(3, 0, [0, 3, 2, 5, 4, 1], 0, 6)","explanation_zh":"結構深度核對=(3, 0, [0, 3, 2, 5, 4, 1], 0, 6)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-cyclic-subgroup","seed":202610261092,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,4,3,0,5,2],[2,3,4,5,0,1],[3,0,5,2,1,4],[4,5,0,1,2,3],[5,2,1,4,3,0]],"element":3,"domain":"Complete_finite_abelian_group_table","operation":"abelian-cyclic-subgroup","semantic_template":"qf-mod26-abelian-cyclic-subgroup/{\"element\":3,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,4,3,0,5,2],[2,3,4,5,0,1],[3,0,5,2,1,4],[4,5,0,1,2,3],[5,2,1,4,3,0]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0093","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-cyclic-subgroup；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,4,0,2,5,3],[2,0,3,5,1,4],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,3,4,1,2,0]],\"element\":2}。請依 [element,identity,generated_elements,next_repeat,order] 欄序回答。","answer":{"kind":"expression","value":"(2, 0, [0, 2, 3, 5, 4, 1], 0, 6)","explanation_zh":"結構深度核對=(2, 0, [0, 2, 3, 5, 4, 1], 0, 6)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-cyclic-subgroup","seed":202610261093,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,4,0,2,5,3],[2,0,3,5,1,4],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,3,4,1,2,0]],"element":2,"domain":"Complete_finite_abelian_group_table","operation":"abelian-cyclic-subgroup","semantic_template":"qf-mod26-abelian-cyclic-subgroup/{\"element\":2,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,4,0,2,5,3],[2,0,3,5,1,4],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,3,4,1,2,0]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0094","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-cyclic-subgroup；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,5,3,4,0,2],[2,3,0,1,5,4],[3,4,1,5,2,0],[4,0,5,2,3,1],[5,2,4,0,1,3]],\"element\":1}。請依 [element,identity,generated_elements,next_repeat,order] 欄序回答。","answer":{"kind":"expression","value":"(1, 0, [0, 1, 5, 2, 3, 4], 0, 6)","explanation_zh":"結構深度核對=(1, 0, [0, 1, 5, 2, 3, 4], 0, 6)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-cyclic-subgroup","seed":202610261094,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,5,3,4,0,2],[2,3,0,1,5,4],[3,4,1,5,2,0],[4,0,5,2,3,1],[5,2,4,0,1,3]],"element":1,"domain":"Complete_finite_abelian_group_table","operation":"abelian-cyclic-subgroup","semantic_template":"qf-mod26-abelian-cyclic-subgroup/{\"element\":1,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,5,3,4,0,2],[2,3,0,1,5,4],[3,4,1,5,2,0],[4,0,5,2,3,1],[5,2,4,0,1,3]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0095","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-cyclic-subgroup；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,5,4,0,3,2],[2,4,0,5,1,3],[3,0,5,4,2,1],[4,3,1,2,5,0],[5,2,3,1,0,4]],\"element\":0}。請依 [element,identity,generated_elements,next_repeat,order] 欄序回答。","answer":{"kind":"expression","value":"(0, 0, [0], 0, 1)","explanation_zh":"結構深度核對=(0, 0, [0], 0, 1)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-cyclic-subgroup","seed":202610261095,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,5,4,0,3,2],[2,4,0,5,1,3],[3,0,5,4,2,1],[4,3,1,2,5,0],[5,2,3,1,0,4]],"element":0,"domain":"Complete_finite_abelian_group_table","operation":"abelian-cyclic-subgroup","semantic_template":"qf-mod26-abelian-cyclic-subgroup/{\"element\":0,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,5,4,0,3,2],[2,4,0,5,1,3],[3,0,5,4,2,1],[4,3,1,2,5,0],[5,2,3,1,0,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0096","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-cyclic-subgroup；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,5,4,2,0,3],[2,4,5,1,3,0],[3,2,1,0,5,4],[4,0,3,5,2,1],[5,3,0,4,1,2]],\"element\":5}。請依 [element,identity,generated_elements,next_repeat,order] 欄序回答。","answer":{"kind":"expression","value":"(5, 0, [0, 5, 2], 0, 3)","explanation_zh":"結構深度核對=(5, 0, [0, 5, 2], 0, 3)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-cyclic-subgroup","seed":202610261096,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,5,4,2,0,3],[2,4,5,1,3,0],[3,2,1,0,5,4],[4,0,3,5,2,1],[5,3,0,4,1,2]],"element":5,"domain":"Complete_finite_abelian_group_table","operation":"abelian-cyclic-subgroup","semantic_template":"qf-mod26-abelian-cyclic-subgroup/{\"element\":5,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,5,4,2,0,3],[2,4,5,1,3,0],[3,2,1,0,5,4],[4,0,3,5,2,1],[5,3,0,4,1,2]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0097","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-cyclic-subgroup；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,5,0,4,2,3],[2,0,4,5,3,1],[3,4,5,0,1,2],[4,2,3,1,5,0],[5,3,1,2,0,4]],\"element\":4}。請依 [element,identity,generated_elements,next_repeat,order] 欄序回答。","answer":{"kind":"expression","value":"(4, 0, [0, 4, 5], 0, 3)","explanation_zh":"結構深度核對=(4, 0, [0, 4, 5], 0, 3)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-cyclic-subgroup","seed":202610261097,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,5,0,4,2,3],[2,0,4,5,3,1],[3,4,5,0,1,2],[4,2,3,1,5,0],[5,3,1,2,0,4]],"element":4,"domain":"Complete_finite_abelian_group_table","operation":"abelian-cyclic-subgroup","semantic_template":"qf-mod26-abelian-cyclic-subgroup/{\"element\":4,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,5,0,4,2,3],[2,0,4,5,3,1],[3,4,5,0,1,2],[4,2,3,1,5,0],[5,3,1,2,0,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0098","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-cyclic-subgroup；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,5,3,0,2,4],[2,3,5,4,1,0],[3,0,4,2,5,1],[4,2,1,5,0,3],[5,4,0,1,3,2]],\"element\":3}。請依 [element,identity,generated_elements,next_repeat,order] 欄序回答。","answer":{"kind":"expression","value":"(3, 0, [0, 3, 2, 4, 5, 1], 0, 6)","explanation_zh":"結構深度核對=(3, 0, [0, 3, 2, 4, 5, 1], 0, 6)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-cyclic-subgroup","seed":202610261098,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,5,3,0,2,4],[2,3,5,4,1,0],[3,0,4,2,5,1],[4,2,1,5,0,3],[5,4,0,1,3,2]],"element":3,"domain":"Complete_finite_abelian_group_table","operation":"abelian-cyclic-subgroup","semantic_template":"qf-mod26-abelian-cyclic-subgroup/{\"element\":3,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,5,3,0,2,4],[2,3,5,4,1,0],[3,0,4,2,5,1],[4,2,1,5,0,3],[5,4,0,1,3,2]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0099","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-cyclic-subgroup；資料={\"n\":6,\"table\":[[0,1,2,3,4,5],[1,5,0,2,3,4],[2,0,3,4,5,1],[3,2,4,5,1,0],[4,3,5,1,0,2],[5,4,1,0,2,3]],\"element\":2}。請依 [element,identity,generated_elements,next_repeat,order] 欄序回答。","answer":{"kind":"expression","value":"(2, 0, [0, 2, 3, 4, 5, 1], 0, 6)","explanation_zh":"結構深度核對=(2, 0, [0, 2, 3, 4, 5, 1], 0, 6)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-cyclic-subgroup","seed":202610261099,"parameters":{"n":6,"table":[[0,1,2,3,4,5],[1,5,0,2,3,4],[2,0,3,4,5,1],[3,2,4,5,1,0],[4,3,5,1,0,2],[5,4,1,0,2,3]],"element":2,"domain":"Complete_finite_abelian_group_table","operation":"abelian-cyclic-subgroup","semantic_template":"qf-mod26-abelian-cyclic-subgroup/{\"element\":2,\"n\":6,\"table\":[[0,1,2,3,4,5],[1,5,0,2,3,4],[2,0,3,4,5,1],[3,2,4,5,1,0],[4,3,5,1,0,2],[5,4,1,0,2,3]]}"}},"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、獨立性、一般圖同構、不限維分析、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-20261026-r01-finite-abelian-group-0100","batch_id":"batch-20261013-advance-20261026-r01-finite-abelian-group-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-abelian-group"],"title_zh":"有限阿貝爾群","stem_zh":"完整有限Cayley表先通過全部群公理。操作=abelian-cyclic-subgroup；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]],\"element\":1}。請依 [element,identity,generated_elements,next_repeat,order] 欄序回答。","answer":{"kind":"expression","value":"(1, 0, [0, 1, 4], 0, 3)","explanation_zh":"結構深度核對=(1, 0, [0, 1, 4], 0, 3)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；交換律逐元素對核對；只分類指定有限群。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限阿貝爾群"]},"variant":{"family_id":"qf-mod26-abelian-cyclic-subgroup","seed":202610261100,"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]],"element":1,"domain":"Complete_finite_abelian_group_table","operation":"abelian-cyclic-subgroup","semantic_template":"qf-mod26-abelian-cyclic-subgroup/{\"element\":1,\"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]]}"}},"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、獨立性、一般圖同構、不限維分析、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"}}