{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0001","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-qualification；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,0,3,2,5,4,7,6],[2,3,0,1,6,7,4,5],[3,2,1,0,7,6,5,4],[4,5,6,7,0,1,2,3],[5,4,7,6,1,0,3,2],[6,7,4,5,2,3,0,1],[7,6,5,4,3,2,1,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,0,1,0,1,0,1],[0,2,0,2,0,2,0,2],[0,3,0,3,0,3,0,3],[0,0,2,2,4,4,6,6],[0,1,2,3,4,5,6,7],[0,2,2,0,4,6,6,4],[0,3,2,1,4,7,6,5]]}。請依 [n,zero,one,noncommuting_pairs,commutative,classification] 欄序回答。","answer":{"kind":"expression","value":"(8, 0, 5, [[1, 2, 0, 2], [1, 3, 1, 3], [1, 6, 0, 2], [1, 7, 1, 3], [2, 3, 2, 0], [2, 4, 0, 2], [2, 6, 0, 2], [3, 4, 0, 2], [3, 7, 3, 1], [4, 6, 6, 4], [4, 7, 6, 4], [6, 7, 4, 6]], 否, unital_ring_not_commutative)","explanation_zh":"Wave4第二層核對=(8, 0, 5, [[1, 2, 0, 2], [1, 3, 1, 3], [1, 6, 0, 2], [1, 7, 1, 3], [2, 3, 2, 0], [2, 4, 0, 2], [2, 6, 0, 2], [3, 4, 0, 2], [3, 7, 3, 1], [4, 6, 6, 4], [4, 7, 6, 4], [6, 7, 4, 6]], 否, unital_ring_not_commutative)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-qualification","seed":202610281001,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,0,3,2,5,4,7,6],[2,3,0,1,6,7,4,5],[3,2,1,0,7,6,5,4],[4,5,6,7,0,1,2,3],[5,4,7,6,1,0,3,2],[6,7,4,5,2,3,0,1],[7,6,5,4,3,2,1,0]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,0,1,0,1,0,1],[0,2,0,2,0,2,0,2],[0,3,0,3,0,3,0,3],[0,0,2,2,4,4,6,6],[0,1,2,3,4,5,6,7],[0,2,2,0,4,6,6,4],[0,3,2,1,4,7,6,5]],"domain":"Complete_finite_commutative_ring_tables","operation":"commring-qualification","semantic_template":"qf-mod28-commring-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,0,3,2,5,4,7,6],[2,3,0,1,6,7,4,5],[3,2,1,0,7,6,5,4],[4,5,6,7,0,1,2,3],[5,4,7,6,1,0,3,2],[6,7,4,5,2,3,0,1],[7,6,5,4,3,2,1,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,0,1,0,1,0,1],[0,2,0,2,0,2,0,2],[0,3,0,3,0,3,0,3],[0,0,2,2,4,4,6,6],[0,1,2,3,4,5,6,7],[0,2,2,0,4,6,6,4],[0,3,2,1,4,7,6,5]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0002","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-qualification；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,7,0,6],[2,3,4,5,7,6,1,0],[3,4,5,7,6,0,2,1],[4,5,7,6,0,1,3,2],[5,7,6,0,1,2,4,3],[6,0,1,2,3,4,7,5],[7,6,0,1,2,3,5,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,7,0,2,7,4],[0,3,7,1,4,6,5,2],[0,4,0,4,0,4,4,0],[0,5,2,6,4,1,3,7],[0,6,7,5,4,3,1,2],[0,7,4,2,0,7,2,4]]}。請依 [n,zero,one,noncommuting_pairs,commutative,classification] 欄序回答。","answer":{"kind":"expression","value":"(8, 0, 1, [], 是, commutative_ring)","explanation_zh":"Wave4第二層核對=(8, 0, 1, [], 是, commutative_ring)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-qualification","seed":202610281002,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,7,0,6],[2,3,4,5,7,6,1,0],[3,4,5,7,6,0,2,1],[4,5,7,6,0,1,3,2],[5,7,6,0,1,2,4,3],[6,0,1,2,3,4,7,5],[7,6,0,1,2,3,5,4]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,7,0,2,7,4],[0,3,7,1,4,6,5,2],[0,4,0,4,0,4,4,0],[0,5,2,6,4,1,3,7],[0,6,7,5,4,3,1,2],[0,7,4,2,0,7,2,4]],"domain":"Complete_finite_commutative_ring_tables","operation":"commring-qualification","semantic_template":"qf-mod28-commring-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,7,0,6],[2,3,4,5,7,6,1,0],[3,4,5,7,6,0,2,1],[4,5,7,6,0,1,3,2],[5,7,6,0,1,2,4,3],[6,0,1,2,3,4,7,5],[7,6,0,1,2,3,5,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,7,0,2,7,4],[0,3,7,1,4,6,5,2],[0,4,0,4,0,4,4,0],[0,5,2,6,4,1,3,7],[0,6,7,5,4,3,1,2],[0,7,4,2,0,7,2,4]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0003","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-qualification；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,6,7,5,0],[2,3,4,6,5,0,7,1],[3,4,6,5,7,1,0,2],[4,6,5,7,0,2,1,3],[5,7,0,1,2,4,3,6],[6,5,7,0,1,3,2,4],[7,0,1,2,3,6,4,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,5,0,4,2,5],[0,3,5,1,4,2,7,6],[0,4,0,4,0,0,4,4],[0,5,4,2,0,4,5,2],[0,6,2,7,4,5,1,3],[0,7,5,6,4,2,3,1]]}。請依 [n,zero,one,noncommuting_pairs,commutative,classification] 欄序回答。","answer":{"kind":"expression","value":"(8, 0, 1, [], 是, commutative_ring)","explanation_zh":"Wave4第二層核對=(8, 0, 1, [], 是, commutative_ring)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-qualification","seed":202610281003,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,4,6,7,5,0],[2,3,4,6,5,0,7,1],[3,4,6,5,7,1,0,2],[4,6,5,7,0,2,1,3],[5,7,0,1,2,4,3,6],[6,5,7,0,1,3,2,4],[7,0,1,2,3,6,4,5]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,5,0,4,2,5],[0,3,5,1,4,2,7,6],[0,4,0,4,0,0,4,4],[0,5,4,2,0,4,5,2],[0,6,2,7,4,5,1,3],[0,7,5,6,4,2,3,1]],"domain":"Complete_finite_commutative_ring_tables","operation":"commring-qualification","semantic_template":"qf-mod28-commring-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,6,7,5,0],[2,3,4,6,5,0,7,1],[3,4,6,5,7,1,0,2],[4,6,5,7,0,2,1,3],[5,7,0,1,2,4,3,6],[6,5,7,0,1,3,2,4],[7,0,1,2,3,6,4,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,5,0,4,2,5],[0,3,5,1,4,2,7,6],[0,4,0,4,0,0,4,4],[0,5,4,2,0,4,5,2],[0,6,2,7,4,5,1,3],[0,7,5,6,4,2,3,1]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0004","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-qualification；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,6,0,7,5],[2,3,4,6,7,1,5,0],[3,4,6,7,5,2,0,1],[4,6,7,5,0,3,1,2],[5,0,1,2,3,7,4,6],[6,7,5,0,1,4,2,3],[7,5,0,1,2,6,3,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,7,0,7,2,4],[0,3,7,1,4,6,5,2],[0,4,0,4,0,4,4,0],[0,5,7,6,4,1,3,2],[0,6,2,5,4,3,1,7],[0,7,4,2,0,2,7,4]]}。請依 [n,zero,one,noncommuting_pairs,commutative,classification] 欄序回答。","answer":{"kind":"expression","value":"(8, 0, 1, [], 是, commutative_ring)","explanation_zh":"Wave4第二層核對=(8, 0, 1, [], 是, commutative_ring)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-qualification","seed":202610281004,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,4,6,0,7,5],[2,3,4,6,7,1,5,0],[3,4,6,7,5,2,0,1],[4,6,7,5,0,3,1,2],[5,0,1,2,3,7,4,6],[6,7,5,0,1,4,2,3],[7,5,0,1,2,6,3,4]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,7,0,7,2,4],[0,3,7,1,4,6,5,2],[0,4,0,4,0,4,4,0],[0,5,7,6,4,1,3,2],[0,6,2,5,4,3,1,7],[0,7,4,2,0,2,7,4]],"domain":"Complete_finite_commutative_ring_tables","operation":"commring-qualification","semantic_template":"qf-mod28-commring-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,6,0,7,5],[2,3,4,6,7,1,5,0],[3,4,6,7,5,2,0,1],[4,6,7,5,0,3,1,2],[5,0,1,2,3,7,4,6],[6,7,5,0,1,4,2,3],[7,5,0,1,2,6,3,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,7,0,7,2,4],[0,3,7,1,4,6,5,2],[0,4,0,4,0,4,4,0],[0,5,7,6,4,1,3,2],[0,6,2,5,4,3,1,7],[0,7,4,2,0,2,7,4]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0005","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-qualification；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,7,6,0,5],[2,3,4,7,5,0,1,6],[3,4,7,5,6,1,2,0],[4,7,5,6,0,2,3,1],[5,6,0,1,2,4,7,3],[6,0,1,2,3,7,5,4],[7,5,6,0,1,3,4,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,5,0,4,5,2],[0,3,5,1,4,2,7,6],[0,4,0,4,0,0,4,4],[0,5,4,2,0,4,2,5],[0,6,5,7,4,2,1,3],[0,7,2,6,4,5,3,1]]}。請依 [n,zero,one,noncommuting_pairs,commutative,classification] 欄序回答。","answer":{"kind":"expression","value":"(8, 0, 1, [], 是, commutative_ring)","explanation_zh":"Wave4第二層核對=(8, 0, 1, [], 是, commutative_ring)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-qualification","seed":202610281005,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,4,7,6,0,5],[2,3,4,7,5,0,1,6],[3,4,7,5,6,1,2,0],[4,7,5,6,0,2,3,1],[5,6,0,1,2,4,7,3],[6,0,1,2,3,7,5,4],[7,5,6,0,1,3,4,2]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,5,0,4,5,2],[0,3,5,1,4,2,7,6],[0,4,0,4,0,0,4,4],[0,5,4,2,0,4,2,5],[0,6,5,7,4,2,1,3],[0,7,2,6,4,5,3,1]],"domain":"Complete_finite_commutative_ring_tables","operation":"commring-qualification","semantic_template":"qf-mod28-commring-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,7,6,0,5],[2,3,4,7,5,0,1,6],[3,4,7,5,6,1,2,0],[4,7,5,6,0,2,3,1],[5,6,0,1,2,4,7,3],[6,0,1,2,3,7,5,4],[7,5,6,0,1,3,4,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,5,0,4,5,2],[0,3,5,1,4,2,7,6],[0,4,0,4,0,0,4,4],[0,5,4,2,0,4,2,5],[0,6,5,7,4,2,1,3],[0,7,2,6,4,5,3,1]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0006","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-qualification；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,0,3,2,7,6,5,4],[2,3,0,1,6,7,4,5],[3,2,1,0,5,4,7,6],[4,7,6,5,0,3,2,1],[5,6,7,4,3,0,1,2],[6,5,4,7,2,1,0,3],[7,4,5,6,1,2,3,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,0,1,0,1,0,1],[0,2,0,2,0,2,0,2],[0,3,0,3,0,3,0,3],[0,0,2,2,4,6,6,4],[0,3,2,1,4,7,6,5],[0,2,2,0,4,4,6,6],[0,1,2,3,4,5,6,7]]}。請依 [n,zero,one,noncommuting_pairs,commutative,classification] 欄序回答。","answer":{"kind":"expression","value":"(8, 0, 7, [[1, 2, 0, 2], [1, 3, 1, 3], [1, 5, 1, 3], [1, 6, 0, 2], [2, 3, 2, 0], [2, 4, 0, 2], [2, 6, 0, 2], [3, 4, 0, 2], [3, 5, 3, 1], [4, 5, 6, 4], [4, 6, 6, 4], [5, 6, 6, 4]], 否, unital_ring_not_commutative)","explanation_zh":"Wave4第二層核對=(8, 0, 7, [[1, 2, 0, 2], [1, 3, 1, 3], [1, 5, 1, 3], [1, 6, 0, 2], [2, 3, 2, 0], [2, 4, 0, 2], [2, 6, 0, 2], [3, 4, 0, 2], [3, 5, 3, 1], [4, 5, 6, 4], [4, 6, 6, 4], [5, 6, 6, 4]], 否, unital_ring_not_commutative)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-qualification","seed":202610281006,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,0,3,2,7,6,5,4],[2,3,0,1,6,7,4,5],[3,2,1,0,5,4,7,6],[4,7,6,5,0,3,2,1],[5,6,7,4,3,0,1,2],[6,5,4,7,2,1,0,3],[7,4,5,6,1,2,3,0]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,0,1,0,1,0,1],[0,2,0,2,0,2,0,2],[0,3,0,3,0,3,0,3],[0,0,2,2,4,6,6,4],[0,3,2,1,4,7,6,5],[0,2,2,0,4,4,6,6],[0,1,2,3,4,5,6,7]],"domain":"Complete_finite_commutative_ring_tables","operation":"commring-qualification","semantic_template":"qf-mod28-commring-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,0,3,2,7,6,5,4],[2,3,0,1,6,7,4,5],[3,2,1,0,5,4,7,6],[4,7,6,5,0,3,2,1],[5,6,7,4,3,0,1,2],[6,5,4,7,2,1,0,3],[7,4,5,6,1,2,3,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,0,1,0,1,0,1],[0,2,0,2,0,2,0,2],[0,3,0,3,0,3,0,3],[0,0,2,2,4,6,6,4],[0,3,2,1,4,7,6,5],[0,2,2,0,4,4,6,6],[0,1,2,3,4,5,6,7]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0007","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-qualification；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,6,4,7,0],[2,3,5,4,7,6,0,1],[3,5,4,6,0,7,1,2],[4,6,7,0,2,1,3,5],[5,4,6,7,1,0,2,3],[6,7,0,1,3,2,5,4],[7,0,1,2,5,3,4,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,6,2,0,5,6],[0,3,6,1,7,5,2,4],[0,4,2,7,1,5,6,3],[0,5,0,5,5,0,0,5],[0,6,5,2,6,0,5,2],[0,7,6,4,3,5,2,1]]}。請依 [n,zero,one,noncommuting_pairs,commutative,classification] 欄序回答。","answer":{"kind":"expression","value":"(8, 0, 1, [], 是, commutative_ring)","explanation_zh":"Wave4第二層核對=(8, 0, 1, [], 是, commutative_ring)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-qualification","seed":202610281007,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,5,6,4,7,0],[2,3,5,4,7,6,0,1],[3,5,4,6,0,7,1,2],[4,6,7,0,2,1,3,5],[5,4,6,7,1,0,2,3],[6,7,0,1,3,2,5,4],[7,0,1,2,5,3,4,6]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,6,2,0,5,6],[0,3,6,1,7,5,2,4],[0,4,2,7,1,5,6,3],[0,5,0,5,5,0,0,5],[0,6,5,2,6,0,5,2],[0,7,6,4,3,5,2,1]],"domain":"Complete_finite_commutative_ring_tables","operation":"commring-qualification","semantic_template":"qf-mod28-commring-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,6,4,7,0],[2,3,5,4,7,6,0,1],[3,5,4,6,0,7,1,2],[4,6,7,0,2,1,3,5],[5,4,6,7,1,0,2,3],[6,7,0,1,3,2,5,4],[7,0,1,2,5,3,4,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,6,2,0,5,6],[0,3,6,1,7,5,2,4],[0,4,2,7,1,5,6,3],[0,5,0,5,5,0,0,5],[0,6,5,2,6,0,5,2],[0,7,6,4,3,5,2,1]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0008","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-qualification；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,7,4,0,6],[2,3,5,4,6,7,1,0],[3,5,4,7,0,6,2,1],[4,7,6,0,2,1,5,3],[5,4,7,6,1,0,3,2],[6,0,1,2,5,3,7,4],[7,6,0,1,3,2,4,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,7,2,0,7,5],[0,3,7,1,6,5,4,2],[0,4,2,6,1,5,3,7],[0,5,0,5,5,0,5,0],[0,6,7,4,3,5,1,2],[0,7,5,2,7,0,2,5]]}。請依 [n,zero,one,noncommuting_pairs,commutative,classification] 欄序回答。","answer":{"kind":"expression","value":"(8, 0, 1, [], 是, commutative_ring)","explanation_zh":"Wave4第二層核對=(8, 0, 1, [], 是, commutative_ring)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-qualification","seed":202610281008,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,5,7,4,0,6],[2,3,5,4,6,7,1,0],[3,5,4,7,0,6,2,1],[4,7,6,0,2,1,5,3],[5,4,7,6,1,0,3,2],[6,0,1,2,5,3,7,4],[7,6,0,1,3,2,4,5]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,7,2,0,7,5],[0,3,7,1,6,5,4,2],[0,4,2,6,1,5,3,7],[0,5,0,5,5,0,5,0],[0,6,7,4,3,5,1,2],[0,7,5,2,7,0,2,5]],"domain":"Complete_finite_commutative_ring_tables","operation":"commring-qualification","semantic_template":"qf-mod28-commring-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,7,4,0,6],[2,3,5,4,6,7,1,0],[3,5,4,7,0,6,2,1],[4,7,6,0,2,1,5,3],[5,4,7,6,1,0,3,2],[6,0,1,2,5,3,7,4],[7,6,0,1,3,2,4,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,7,2,0,7,5],[0,3,7,1,6,5,4,2],[0,4,2,6,1,5,3,7],[0,5,0,5,5,0,5,0],[0,6,7,4,3,5,1,2],[0,7,5,2,7,0,2,5]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0009","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-qualification；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,7,6,4,0],[2,3,5,6,0,4,7,1],[3,5,6,4,1,7,0,2],[4,7,0,1,5,2,3,6],[5,6,4,7,2,0,1,3],[6,4,7,0,3,1,2,5],[7,0,1,2,6,3,5,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,4,5,0,2,4],[0,3,4,1,2,5,7,6],[0,4,5,2,5,0,4,2],[0,5,0,5,0,0,5,5],[0,6,2,7,4,5,1,3],[0,7,4,6,2,5,3,1]]}。請依 [n,zero,one,noncommuting_pairs,commutative,classification] 欄序回答。","answer":{"kind":"expression","value":"(8, 0, 1, [], 是, commutative_ring)","explanation_zh":"Wave4第二層核對=(8, 0, 1, [], 是, commutative_ring)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-qualification","seed":202610281009,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,5,7,6,4,0],[2,3,5,6,0,4,7,1],[3,5,6,4,1,7,0,2],[4,7,0,1,5,2,3,6],[5,6,4,7,2,0,1,3],[6,4,7,0,3,1,2,5],[7,0,1,2,6,3,5,4]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,4,5,0,2,4],[0,3,4,1,2,5,7,6],[0,4,5,2,5,0,4,2],[0,5,0,5,0,0,5,5],[0,6,2,7,4,5,1,3],[0,7,4,6,2,5,3,1]],"domain":"Complete_finite_commutative_ring_tables","operation":"commring-qualification","semantic_template":"qf-mod28-commring-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,7,6,4,0],[2,3,5,6,0,4,7,1],[3,5,6,4,1,7,0,2],[4,7,0,1,5,2,3,6],[5,6,4,7,2,0,1,3],[6,4,7,0,3,1,2,5],[7,0,1,2,6,3,5,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,4,5,0,2,4],[0,3,4,1,2,5,7,6],[0,4,5,2,5,0,4,2],[0,5,0,5,0,0,5,5],[0,6,2,7,4,5,1,3],[0,7,4,6,2,5,3,1]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0010","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-qualification；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,0,6,7,4],[2,3,5,6,1,7,4,0],[3,5,6,7,2,4,0,1],[4,0,1,2,7,3,5,6],[5,6,7,4,3,0,1,2],[6,7,4,0,5,1,2,3],[7,4,0,1,6,2,3,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,7,7,0,2,5],[0,3,7,1,6,5,4,2],[0,4,7,6,1,5,3,2],[0,5,0,5,5,0,5,0],[0,6,2,4,3,5,1,7],[0,7,5,2,2,0,7,5]]}。請依 [n,zero,one,noncommuting_pairs,commutative,classification] 欄序回答。","answer":{"kind":"expression","value":"(8, 0, 1, [], 是, commutative_ring)","explanation_zh":"Wave4第二層核對=(8, 0, 1, [], 是, commutative_ring)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-qualification","seed":202610281010,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,5,0,6,7,4],[2,3,5,6,1,7,4,0],[3,5,6,7,2,4,0,1],[4,0,1,2,7,3,5,6],[5,6,7,4,3,0,1,2],[6,7,4,0,5,1,2,3],[7,4,0,1,6,2,3,5]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,7,7,0,2,5],[0,3,7,1,6,5,4,2],[0,4,7,6,1,5,3,2],[0,5,0,5,5,0,5,0],[0,6,2,4,3,5,1,7],[0,7,5,2,2,0,7,5]],"domain":"Complete_finite_commutative_ring_tables","operation":"commring-qualification","semantic_template":"qf-mod28-commring-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,0,6,7,4],[2,3,5,6,1,7,4,0],[3,5,6,7,2,4,0,1],[4,0,1,2,7,3,5,6],[5,6,7,4,3,0,1,2],[6,7,4,0,5,1,2,3],[7,4,0,1,6,2,3,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,7,7,0,2,5],[0,3,7,1,6,5,4,2],[0,4,7,6,1,5,3,2],[0,5,0,5,5,0,5,0],[0,6,2,4,3,5,1,7],[0,7,5,2,2,0,7,5]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0011","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-qualification；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,0,3,2,6,7,4,5],[2,3,0,1,5,4,7,6],[3,2,1,0,7,6,5,4],[4,6,5,7,0,2,1,3],[5,7,4,6,2,0,3,1],[6,4,7,5,1,3,0,2],[7,5,6,4,3,1,2,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,0,1,0,0,1,1],[0,2,0,2,0,0,2,2],[0,3,0,3,0,0,3,3],[0,2,2,0,4,5,5,4],[0,0,2,2,4,5,4,5],[0,3,2,1,4,5,7,6],[0,1,2,3,4,5,6,7]]}。請依 [n,zero,one,noncommuting_pairs,commutative,classification] 欄序回答。","answer":{"kind":"expression","value":"(8, 0, 7, [[1, 2, 0, 2], [1, 3, 1, 3], [1, 4, 0, 2], [1, 6, 1, 3], [2, 3, 2, 0], [2, 4, 0, 2], [2, 5, 0, 2], [3, 5, 0, 2], [3, 6, 3, 1], [4, 5, 5, 4], [4, 6, 5, 4], [5, 6, 4, 5]], 否, unital_ring_not_commutative)","explanation_zh":"Wave4第二層核對=(8, 0, 7, [[1, 2, 0, 2], [1, 3, 1, 3], [1, 4, 0, 2], [1, 6, 1, 3], [2, 3, 2, 0], [2, 4, 0, 2], [2, 5, 0, 2], [3, 5, 0, 2], [3, 6, 3, 1], [4, 5, 5, 4], [4, 6, 5, 4], [5, 6, 4, 5]], 否, unital_ring_not_commutative)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-qualification","seed":202610281011,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,0,3,2,6,7,4,5],[2,3,0,1,5,4,7,6],[3,2,1,0,7,6,5,4],[4,6,5,7,0,2,1,3],[5,7,4,6,2,0,3,1],[6,4,7,5,1,3,0,2],[7,5,6,4,3,1,2,0]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,0,1,0,0,1,1],[0,2,0,2,0,0,2,2],[0,3,0,3,0,0,3,3],[0,2,2,0,4,5,5,4],[0,0,2,2,4,5,4,5],[0,3,2,1,4,5,7,6],[0,1,2,3,4,5,6,7]],"domain":"Complete_finite_commutative_ring_tables","operation":"commring-qualification","semantic_template":"qf-mod28-commring-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,0,3,2,6,7,4,5],[2,3,0,1,5,4,7,6],[3,2,1,0,7,6,5,4],[4,6,5,7,0,2,1,3],[5,7,4,6,2,0,3,1],[6,4,7,5,1,3,0,2],[7,5,6,4,3,1,2,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,0,1,0,0,1,1],[0,2,0,2,0,0,2,2],[0,3,0,3,0,0,3,3],[0,2,2,0,4,5,5,4],[0,0,2,2,4,5,4,5],[0,3,2,1,4,5,7,6],[0,1,2,3,4,5,6,7]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0012","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-qualification；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,0,7,4,6],[2,3,5,7,1,6,0,4],[3,5,7,6,2,4,1,0],[4,0,1,2,6,3,7,5],[5,7,6,4,3,0,2,1],[6,4,0,1,7,2,5,3],[7,6,4,0,5,1,3,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,6,6,0,5,2],[0,3,6,1,7,5,2,4],[0,4,6,7,1,5,2,3],[0,5,0,5,5,0,0,5],[0,6,5,2,2,0,5,6],[0,7,2,4,3,5,6,1]]}。請依 [n,zero,one,noncommuting_pairs,commutative,classification] 欄序回答。","answer":{"kind":"expression","value":"(8, 0, 1, [], 是, commutative_ring)","explanation_zh":"Wave4第二層核對=(8, 0, 1, [], 是, commutative_ring)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-qualification","seed":202610281012,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,5,0,7,4,6],[2,3,5,7,1,6,0,4],[3,5,7,6,2,4,1,0],[4,0,1,2,6,3,7,5],[5,7,6,4,3,0,2,1],[6,4,0,1,7,2,5,3],[7,6,4,0,5,1,3,2]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,6,6,0,5,2],[0,3,6,1,7,5,2,4],[0,4,6,7,1,5,2,3],[0,5,0,5,5,0,0,5],[0,6,5,2,2,0,5,6],[0,7,2,4,3,5,6,1]],"domain":"Complete_finite_commutative_ring_tables","operation":"commring-qualification","semantic_template":"qf-mod28-commring-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,0,7,4,6],[2,3,5,7,1,6,0,4],[3,5,7,6,2,4,1,0],[4,0,1,2,6,3,7,5],[5,7,6,4,3,0,2,1],[6,4,0,1,7,2,5,3],[7,6,4,0,5,1,3,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,6,6,0,5,2],[0,3,6,1,7,5,2,4],[0,4,6,7,1,5,2,3],[0,5,0,5,5,0,0,5],[0,6,5,2,2,0,5,6],[0,7,2,4,3,5,6,1]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0013","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-qualification；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,5,7,4,0],[2,3,6,4,7,0,5,1],[3,6,4,5,0,1,7,2],[4,5,7,0,2,3,1,6],[5,7,0,1,3,6,2,4],[6,4,5,7,1,2,0,3],[7,0,1,2,6,4,3,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,5,2,6,0,5],[0,3,5,1,7,2,6,4],[0,4,2,7,1,5,6,3],[0,5,6,2,5,6,0,2],[0,6,0,6,6,0,0,6],[0,7,5,4,3,2,6,1]]}。請依 [n,zero,one,noncommuting_pairs,commutative,classification] 欄序回答。","answer":{"kind":"expression","value":"(8, 0, 1, [], 是, commutative_ring)","explanation_zh":"Wave4第二層核對=(8, 0, 1, [], 是, commutative_ring)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-qualification","seed":202610281013,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,6,5,7,4,0],[2,3,6,4,7,0,5,1],[3,6,4,5,0,1,7,2],[4,5,7,0,2,3,1,6],[5,7,0,1,3,6,2,4],[6,4,5,7,1,2,0,3],[7,0,1,2,6,4,3,5]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,5,2,6,0,5],[0,3,5,1,7,2,6,4],[0,4,2,7,1,5,6,3],[0,5,6,2,5,6,0,2],[0,6,0,6,6,0,0,6],[0,7,5,4,3,2,6,1]],"domain":"Complete_finite_commutative_ring_tables","operation":"commring-qualification","semantic_template":"qf-mod28-commring-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,5,7,4,0],[2,3,6,4,7,0,5,1],[3,6,4,5,0,1,7,2],[4,5,7,0,2,3,1,6],[5,7,0,1,3,6,2,4],[6,4,5,7,1,2,0,3],[7,0,1,2,6,4,3,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,5,2,6,0,5],[0,3,5,1,7,2,6,4],[0,4,2,7,1,5,6,3],[0,5,6,2,5,6,0,2],[0,6,0,6,6,0,0,6],[0,7,5,4,3,2,6,1]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0014","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-qualification；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,7,0,4,5],[2,3,6,4,5,1,7,0],[3,6,4,7,0,2,5,1],[4,7,5,0,2,6,1,3],[5,0,1,2,6,7,3,4],[6,4,7,5,1,3,0,2],[7,5,0,1,3,4,2,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,7,2,7,0,6],[0,3,7,1,5,4,6,2],[0,4,2,5,1,3,6,7],[0,5,7,4,3,1,6,2],[0,6,0,6,6,6,0,0],[0,7,6,2,7,2,0,6]]}。請依 [n,zero,one,noncommuting_pairs,commutative,classification] 欄序回答。","answer":{"kind":"expression","value":"(8, 0, 1, [], 是, commutative_ring)","explanation_zh":"Wave4第二層核對=(8, 0, 1, [], 是, commutative_ring)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-qualification","seed":202610281014,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,6,7,0,4,5],[2,3,6,4,5,1,7,0],[3,6,4,7,0,2,5,1],[4,7,5,0,2,6,1,3],[5,0,1,2,6,7,3,4],[6,4,7,5,1,3,0,2],[7,5,0,1,3,4,2,6]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,7,2,7,0,6],[0,3,7,1,5,4,6,2],[0,4,2,5,1,3,6,7],[0,5,7,4,3,1,6,2],[0,6,0,6,6,6,0,0],[0,7,6,2,7,2,0,6]],"domain":"Complete_finite_commutative_ring_tables","operation":"commring-qualification","semantic_template":"qf-mod28-commring-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,7,0,4,5],[2,3,6,4,5,1,7,0],[3,6,4,7,0,2,5,1],[4,7,5,0,2,6,1,3],[5,0,1,2,6,7,3,4],[6,4,7,5,1,3,0,2],[7,5,0,1,3,4,2,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,7,2,7,0,6],[0,3,7,1,5,4,6,2],[0,4,2,5,1,3,6,7],[0,5,7,4,3,1,6,2],[0,6,0,6,6,6,0,0],[0,7,6,2,7,2,0,6]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0015","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-qualification；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,7,4,5,0],[2,3,6,5,0,7,4,1],[3,6,5,4,1,0,7,2],[4,7,0,1,6,3,2,5],[5,4,7,0,3,2,1,6],[6,5,4,7,2,1,0,3],[7,0,1,2,5,6,3,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,4,6,2,0,4],[0,3,4,1,2,7,6,5],[0,4,6,2,6,4,0,2],[0,5,2,7,4,1,6,3],[0,6,0,6,0,6,0,6],[0,7,4,5,2,3,6,1]]}。請依 [n,zero,one,noncommuting_pairs,commutative,classification] 欄序回答。","answer":{"kind":"expression","value":"(8, 0, 1, [], 是, commutative_ring)","explanation_zh":"Wave4第二層核對=(8, 0, 1, [], 是, commutative_ring)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-qualification","seed":202610281015,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,6,7,4,5,0],[2,3,6,5,0,7,4,1],[3,6,5,4,1,0,7,2],[4,7,0,1,6,3,2,5],[5,4,7,0,3,2,1,6],[6,5,4,7,2,1,0,3],[7,0,1,2,5,6,3,4]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,4,6,2,0,4],[0,3,4,1,2,7,6,5],[0,4,6,2,6,4,0,2],[0,5,2,7,4,1,6,3],[0,6,0,6,0,6,0,6],[0,7,4,5,2,3,6,1]],"domain":"Complete_finite_commutative_ring_tables","operation":"commring-qualification","semantic_template":"qf-mod28-commring-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,7,4,5,0],[2,3,6,5,0,7,4,1],[3,6,5,4,1,0,7,2],[4,7,0,1,6,3,2,5],[5,4,7,0,3,2,1,6],[6,5,4,7,2,1,0,3],[7,0,1,2,5,6,3,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,4,6,2,0,4],[0,3,4,1,2,7,6,5],[0,4,6,2,6,4,0,2],[0,5,2,7,4,1,6,3],[0,6,0,6,0,6,0,6],[0,7,4,5,2,3,6,1]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0016","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-qualification；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,0,3,2,7,6,5,4],[2,3,0,1,5,4,7,6],[3,2,1,0,6,7,4,5],[4,7,5,6,0,2,3,1],[5,6,4,7,2,0,1,3],[6,5,7,4,3,1,0,2],[7,4,6,5,1,3,2,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,0,1,1,1,0,0],[0,2,0,2,2,2,0,0],[0,3,0,3,3,3,0,0],[0,3,2,1,5,4,6,7],[0,1,2,3,4,5,6,7],[0,0,2,2,7,6,6,7],[0,2,2,0,6,7,6,7]]}。請依 [n,zero,one,noncommuting_pairs,commutative,classification] 欄序回答。","answer":{"kind":"expression","value":"(8, 0, 5, [[1, 2, 0, 2], [1, 3, 1, 3], [1, 4, 1, 3], [1, 7, 0, 2], [2, 3, 2, 0], [2, 6, 0, 2], [2, 7, 0, 2], [3, 4, 3, 1], [3, 6, 0, 2], [4, 6, 6, 7], [4, 7, 7, 6], [6, 7, 7, 6]], 否, unital_ring_not_commutative)","explanation_zh":"Wave4第二層核對=(8, 0, 5, [[1, 2, 0, 2], [1, 3, 1, 3], [1, 4, 1, 3], [1, 7, 0, 2], [2, 3, 2, 0], [2, 6, 0, 2], [2, 7, 0, 2], [3, 4, 3, 1], [3, 6, 0, 2], [4, 6, 6, 7], [4, 7, 7, 6], [6, 7, 7, 6]], 否, unital_ring_not_commutative)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-qualification","seed":202610281016,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,0,3,2,7,6,5,4],[2,3,0,1,5,4,7,6],[3,2,1,0,6,7,4,5],[4,7,5,6,0,2,3,1],[5,6,4,7,2,0,1,3],[6,5,7,4,3,1,0,2],[7,4,6,5,1,3,2,0]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,0,1,1,1,0,0],[0,2,0,2,2,2,0,0],[0,3,0,3,3,3,0,0],[0,3,2,1,5,4,6,7],[0,1,2,3,4,5,6,7],[0,0,2,2,7,6,6,7],[0,2,2,0,6,7,6,7]],"domain":"Complete_finite_commutative_ring_tables","operation":"commring-qualification","semantic_template":"qf-mod28-commring-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,0,3,2,7,6,5,4],[2,3,0,1,5,4,7,6],[3,2,1,0,6,7,4,5],[4,7,5,6,0,2,3,1],[5,6,4,7,2,0,1,3],[6,5,7,4,3,1,0,2],[7,4,6,5,1,3,2,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,0,1,1,1,0,0],[0,2,0,2,2,2,0,0],[0,3,0,3,3,3,0,0],[0,3,2,1,5,4,6,7],[0,1,2,3,4,5,6,7],[0,0,2,2,7,6,6,7],[0,2,2,0,6,7,6,7]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0017","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-qualification；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,5,0,7,4],[2,3,6,7,0,1,4,5],[3,6,7,4,1,2,5,0],[4,5,0,1,6,7,2,3],[5,0,1,2,7,4,3,6],[6,7,4,5,2,3,0,1],[7,4,5,0,3,6,1,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,4,6,4,0,2],[0,3,4,1,2,7,6,5],[0,4,6,2,6,2,0,4],[0,5,4,7,2,1,6,3],[0,6,0,6,0,6,0,6],[0,7,2,5,4,3,6,1]]}。請依 [n,zero,one,noncommuting_pairs,commutative,classification] 欄序回答。","answer":{"kind":"expression","value":"(8, 0, 1, [], 是, commutative_ring)","explanation_zh":"Wave4第二層核對=(8, 0, 1, [], 是, commutative_ring)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-qualification","seed":202610281017,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,6,5,0,7,4],[2,3,6,7,0,1,4,5],[3,6,7,4,1,2,5,0],[4,5,0,1,6,7,2,3],[5,0,1,2,7,4,3,6],[6,7,4,5,2,3,0,1],[7,4,5,0,3,6,1,2]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,4,6,4,0,2],[0,3,4,1,2,7,6,5],[0,4,6,2,6,2,0,4],[0,5,4,7,2,1,6,3],[0,6,0,6,0,6,0,6],[0,7,2,5,4,3,6,1]],"domain":"Complete_finite_commutative_ring_tables","operation":"commring-qualification","semantic_template":"qf-mod28-commring-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,5,0,7,4],[2,3,6,7,0,1,4,5],[3,6,7,4,1,2,5,0],[4,5,0,1,6,7,2,3],[5,0,1,2,7,4,3,6],[6,7,4,5,2,3,0,1],[7,4,5,0,3,6,1,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,4,6,4,0,2],[0,3,4,1,2,7,6,5],[0,4,6,2,6,2,0,4],[0,5,4,7,2,1,6,3],[0,6,0,6,0,6,0,6],[0,7,2,5,4,3,6,1]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0018","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-qualification；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,0,4,7,5],[2,3,6,7,1,0,5,4],[3,6,7,5,2,1,4,0],[4,0,1,2,5,7,3,6],[5,4,0,1,7,6,2,3],[6,7,5,4,3,2,0,1],[7,5,4,0,6,3,1,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,5,5,6,0,2],[0,3,5,1,7,2,6,4],[0,4,5,7,1,2,6,3],[0,5,6,2,2,6,0,5],[0,6,0,6,6,0,0,6],[0,7,2,4,3,5,6,1]]}。請依 [n,zero,one,noncommuting_pairs,commutative,classification] 欄序回答。","answer":{"kind":"expression","value":"(8, 0, 1, [], 是, commutative_ring)","explanation_zh":"Wave4第二層核對=(8, 0, 1, [], 是, commutative_ring)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-qualification","seed":202610281018,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,6,0,4,7,5],[2,3,6,7,1,0,5,4],[3,6,7,5,2,1,4,0],[4,0,1,2,5,7,3,6],[5,4,0,1,7,6,2,3],[6,7,5,4,3,2,0,1],[7,5,4,0,6,3,1,2]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,5,5,6,0,2],[0,3,5,1,7,2,6,4],[0,4,5,7,1,2,6,3],[0,5,6,2,2,6,0,5],[0,6,0,6,6,0,0,6],[0,7,2,4,3,5,6,1]],"domain":"Complete_finite_commutative_ring_tables","operation":"commring-qualification","semantic_template":"qf-mod28-commring-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,0,4,7,5],[2,3,6,7,1,0,5,4],[3,6,7,5,2,1,4,0],[4,0,1,2,5,7,3,6],[5,4,0,1,7,6,2,3],[6,7,5,4,3,2,0,1],[7,5,4,0,6,3,1,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,5,5,6,0,2],[0,3,5,1,7,2,6,4],[0,4,5,7,1,2,6,3],[0,5,6,2,2,6,0,5],[0,6,0,6,6,0,0,6],[0,7,2,4,3,5,6,1]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0019","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-qualification；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,5,6,0,4],[2,3,7,4,6,0,1,5],[3,7,4,5,0,1,2,6],[4,5,6,0,2,3,7,1],[5,6,0,1,3,7,4,2],[6,0,1,2,7,4,5,3],[7,4,5,6,1,2,3,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,5,2,7,5,0],[0,3,5,1,6,2,4,7],[0,4,2,6,1,5,3,7],[0,5,7,2,5,7,2,0],[0,6,5,4,3,2,1,7],[0,7,0,7,7,0,7,0]]}。請依 [n,zero,one,noncommuting_pairs,commutative,classification] 欄序回答。","answer":{"kind":"expression","value":"(8, 0, 1, [], 是, commutative_ring)","explanation_zh":"Wave4第二層核對=(8, 0, 1, [], 是, commutative_ring)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-qualification","seed":202610281019,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,7,5,6,0,4],[2,3,7,4,6,0,1,5],[3,7,4,5,0,1,2,6],[4,5,6,0,2,3,7,1],[5,6,0,1,3,7,4,2],[6,0,1,2,7,4,5,3],[7,4,5,6,1,2,3,0]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,5,2,7,5,0],[0,3,5,1,6,2,4,7],[0,4,2,6,1,5,3,7],[0,5,7,2,5,7,2,0],[0,6,5,4,3,2,1,7],[0,7,0,7,7,0,7,0]],"domain":"Complete_finite_commutative_ring_tables","operation":"commring-qualification","semantic_template":"qf-mod28-commring-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,5,6,0,4],[2,3,7,4,6,0,1,5],[3,7,4,5,0,1,2,6],[4,5,6,0,2,3,7,1],[5,6,0,1,3,7,4,2],[6,0,1,2,7,4,5,3],[7,4,5,6,1,2,3,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,5,2,7,5,0],[0,3,5,1,6,2,4,7],[0,4,2,6,1,5,3,7],[0,5,7,2,5,7,2,0],[0,6,5,4,3,2,1,7],[0,7,0,7,7,0,7,0]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0020","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-qualification；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,6,0,5,4],[2,3,7,4,5,1,0,6],[3,7,4,6,0,2,1,5],[4,6,5,0,2,7,3,1],[5,0,1,2,7,6,4,3],[6,5,0,1,3,4,7,2],[7,4,6,5,1,3,2,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,6,2,6,7,0],[0,3,6,1,5,4,2,7],[0,4,2,5,1,3,6,7],[0,5,6,4,3,1,2,7],[0,6,7,2,6,2,7,0],[0,7,0,7,7,7,0,0]]}。請依 [n,zero,one,noncommuting_pairs,commutative,classification] 欄序回答。","answer":{"kind":"expression","value":"(8, 0, 1, [], 是, commutative_ring)","explanation_zh":"Wave4第二層核對=(8, 0, 1, [], 是, commutative_ring)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-qualification","seed":202610281020,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,7,6,0,5,4],[2,3,7,4,5,1,0,6],[3,7,4,6,0,2,1,5],[4,6,5,0,2,7,3,1],[5,0,1,2,7,6,4,3],[6,5,0,1,3,4,7,2],[7,4,6,5,1,3,2,0]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,6,2,6,7,0],[0,3,6,1,5,4,2,7],[0,4,2,5,1,3,6,7],[0,5,6,4,3,1,2,7],[0,6,7,2,6,2,7,0],[0,7,0,7,7,7,0,0]],"domain":"Complete_finite_commutative_ring_tables","operation":"commring-qualification","semantic_template":"qf-mod28-commring-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,6,0,5,4],[2,3,7,4,5,1,0,6],[3,7,4,6,0,2,1,5],[4,6,5,0,2,7,3,1],[5,0,1,2,7,6,4,3],[6,5,0,1,3,4,7,2],[7,4,6,5,1,3,2,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,6,2,6,7,0],[0,3,6,1,5,4,2,7],[0,4,2,5,1,3,6,7],[0,5,6,4,3,1,2,7],[0,6,7,2,6,2,7,0],[0,7,0,7,7,7,0,0]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0021","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-qualification；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,0,3,2,6,7,4,5],[2,3,0,1,7,6,5,4],[3,2,1,0,5,4,7,6],[4,6,7,5,0,3,1,2],[5,7,6,4,3,0,2,1],[6,4,5,7,1,2,0,3],[7,5,4,6,2,1,3,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,0,1,0,1,1,0],[0,2,0,2,0,2,2,0],[0,3,0,3,0,3,3,0],[0,2,2,0,4,4,7,7],[0,1,2,3,4,5,6,7],[0,3,2,1,4,6,5,7],[0,0,2,2,4,7,4,7]]}。請依 [n,zero,one,noncommuting_pairs,commutative,classification] 欄序回答。","answer":{"kind":"expression","value":"(8, 0, 5, [[1, 2, 0, 2], [1, 3, 1, 3], [1, 4, 0, 2], [1, 6, 1, 3], [2, 3, 2, 0], [2, 4, 0, 2], [2, 7, 0, 2], [3, 6, 3, 1], [3, 7, 0, 2], [4, 6, 7, 4], [4, 7, 7, 4], [6, 7, 7, 4]], 否, unital_ring_not_commutative)","explanation_zh":"Wave4第二層核對=(8, 0, 5, [[1, 2, 0, 2], [1, 3, 1, 3], [1, 4, 0, 2], [1, 6, 1, 3], [2, 3, 2, 0], [2, 4, 0, 2], [2, 7, 0, 2], [3, 6, 3, 1], [3, 7, 0, 2], [4, 6, 7, 4], [4, 7, 7, 4], [6, 7, 7, 4]], 否, unital_ring_not_commutative)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-qualification","seed":202610281021,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,0,3,2,6,7,4,5],[2,3,0,1,7,6,5,4],[3,2,1,0,5,4,7,6],[4,6,7,5,0,3,1,2],[5,7,6,4,3,0,2,1],[6,4,5,7,1,2,0,3],[7,5,4,6,2,1,3,0]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,0,1,0,1,1,0],[0,2,0,2,0,2,2,0],[0,3,0,3,0,3,3,0],[0,2,2,0,4,4,7,7],[0,1,2,3,4,5,6,7],[0,3,2,1,4,6,5,7],[0,0,2,2,4,7,4,7]],"domain":"Complete_finite_commutative_ring_tables","operation":"commring-qualification","semantic_template":"qf-mod28-commring-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,0,3,2,6,7,4,5],[2,3,0,1,7,6,5,4],[3,2,1,0,5,4,7,6],[4,6,7,5,0,3,1,2],[5,7,6,4,3,0,2,1],[6,4,5,7,1,2,0,3],[7,5,4,6,2,1,3,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,0,1,0,1,1,0],[0,2,0,2,0,2,2,0],[0,3,0,3,0,3,3,0],[0,2,2,0,4,4,7,7],[0,1,2,3,4,5,6,7],[0,3,2,1,4,6,5,7],[0,0,2,2,4,7,4,7]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0022","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-qualification；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,0,6,4,5],[2,3,7,5,1,4,0,6],[3,7,5,6,2,0,1,4],[4,0,1,2,6,7,5,3],[5,6,4,0,7,2,3,1],[6,4,0,1,5,3,7,2],[7,5,6,4,3,1,2,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,6,6,2,7,0],[0,3,6,1,5,4,2,7],[0,4,6,5,1,3,2,7],[0,5,2,4,3,1,6,7],[0,6,7,2,2,6,7,0],[0,7,0,7,7,7,0,0]]}。請依 [n,zero,one,noncommuting_pairs,commutative,classification] 欄序回答。","answer":{"kind":"expression","value":"(8, 0, 1, [], 是, commutative_ring)","explanation_zh":"Wave4第二層核對=(8, 0, 1, [], 是, commutative_ring)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-qualification","seed":202610281022,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,7,0,6,4,5],[2,3,7,5,1,4,0,6],[3,7,5,6,2,0,1,4],[4,0,1,2,6,7,5,3],[5,6,4,0,7,2,3,1],[6,4,0,1,5,3,7,2],[7,5,6,4,3,1,2,0]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,6,6,2,7,0],[0,3,6,1,5,4,2,7],[0,4,6,5,1,3,2,7],[0,5,2,4,3,1,6,7],[0,6,7,2,2,6,7,0],[0,7,0,7,7,7,0,0]],"domain":"Complete_finite_commutative_ring_tables","operation":"commring-qualification","semantic_template":"qf-mod28-commring-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,0,6,4,5],[2,3,7,5,1,4,0,6],[3,7,5,6,2,0,1,4],[4,0,1,2,6,7,5,3],[5,6,4,0,7,2,3,1],[6,4,0,1,5,3,7,2],[7,5,6,4,3,1,2,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,6,6,2,7,0],[0,3,6,1,5,4,2,7],[0,4,6,5,1,3,2,7],[0,5,2,4,3,1,6,7],[0,6,7,2,2,6,7,0],[0,7,0,7,7,7,0,0]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0023","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-qualification；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,5,0,4,6],[2,3,7,6,0,1,5,4],[3,7,6,4,1,2,0,5],[4,5,0,1,7,6,3,2],[5,0,1,2,6,4,7,3],[6,4,5,0,3,7,2,1],[7,6,4,5,2,3,1,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,4,7,4,2,0],[0,3,4,1,2,6,5,7],[0,4,7,2,7,2,4,0],[0,5,4,6,2,1,3,7],[0,6,2,5,4,3,1,7],[0,7,0,7,0,7,7,0]]}。請依 [n,zero,one,noncommuting_pairs,commutative,classification] 欄序回答。","answer":{"kind":"expression","value":"(8, 0, 1, [], 是, commutative_ring)","explanation_zh":"Wave4第二層核對=(8, 0, 1, [], 是, commutative_ring)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-qualification","seed":202610281023,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,7,5,0,4,6],[2,3,7,6,0,1,5,4],[3,7,6,4,1,2,0,5],[4,5,0,1,7,6,3,2],[5,0,1,2,6,4,7,3],[6,4,5,0,3,7,2,1],[7,6,4,5,2,3,1,0]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,4,7,4,2,0],[0,3,4,1,2,6,5,7],[0,4,7,2,7,2,4,0],[0,5,4,6,2,1,3,7],[0,6,2,5,4,3,1,7],[0,7,0,7,0,7,7,0]],"domain":"Complete_finite_commutative_ring_tables","operation":"commring-qualification","semantic_template":"qf-mod28-commring-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,5,0,4,6],[2,3,7,6,0,1,5,4],[3,7,6,4,1,2,0,5],[4,5,0,1,7,6,3,2],[5,0,1,2,6,4,7,3],[6,4,5,0,3,7,2,1],[7,6,4,5,2,3,1,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,4,7,4,2,0],[0,3,4,1,2,6,5,7],[0,4,7,2,7,2,4,0],[0,5,4,6,2,1,3,7],[0,6,2,5,4,3,1,7],[0,7,0,7,0,7,7,0]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0024","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-qualification；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,0,4,5,6],[2,3,7,6,1,0,4,5],[3,7,6,5,2,1,0,4],[4,0,1,2,5,6,7,3],[5,4,0,1,6,7,3,2],[6,5,4,0,7,3,2,1],[7,6,5,4,3,2,1,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,5,5,7,2,0],[0,3,5,1,6,2,4,7],[0,4,5,6,1,2,3,7],[0,5,7,2,2,7,5,0],[0,6,2,4,3,5,1,7],[0,7,0,7,7,0,7,0]]}。請依 [n,zero,one,noncommuting_pairs,commutative,classification] 欄序回答。","answer":{"kind":"expression","value":"(8, 0, 1, [], 是, commutative_ring)","explanation_zh":"Wave4第二層核對=(8, 0, 1, [], 是, commutative_ring)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-qualification","seed":202610281024,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,7,0,4,5,6],[2,3,7,6,1,0,4,5],[3,7,6,5,2,1,0,4],[4,0,1,2,5,6,7,3],[5,4,0,1,6,7,3,2],[6,5,4,0,7,3,2,1],[7,6,5,4,3,2,1,0]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,5,5,7,2,0],[0,3,5,1,6,2,4,7],[0,4,5,6,1,2,3,7],[0,5,7,2,2,7,5,0],[0,6,2,4,3,5,1,7],[0,7,0,7,7,0,7,0]],"domain":"Complete_finite_commutative_ring_tables","operation":"commring-qualification","semantic_template":"qf-mod28-commring-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,0,4,5,6],[2,3,7,6,1,0,4,5],[3,7,6,5,2,1,0,4],[4,0,1,2,5,6,7,3],[5,4,0,1,6,7,3,2],[6,5,4,0,7,3,2,1],[7,6,5,4,3,2,1,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,5,5,7,2,0],[0,3,5,1,6,2,4,7],[0,4,5,6,1,2,3,7],[0,5,7,2,2,7,5,0],[0,6,2,4,3,5,1,7],[0,7,0,7,7,0,7,0]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0025","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-qualification；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,4,5,3,6,7,0],[2,4,3,6,5,7,0,1],[3,5,6,0,7,1,2,4],[4,3,5,7,6,0,1,2],[5,6,7,1,0,2,4,3],[6,7,0,2,1,4,3,5],[7,0,1,4,2,3,5,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,3,0,6,2,3,6],[0,3,0,0,3,3,0,3],[0,4,6,3,1,7,2,5],[0,5,2,3,7,1,6,4],[0,6,3,0,2,6,3,2],[0,7,6,3,5,4,2,1]]}。請依 [n,zero,one,noncommuting_pairs,commutative,classification] 欄序回答。","answer":{"kind":"expression","value":"(8, 0, 1, [], 是, commutative_ring)","explanation_zh":"Wave4第二層核對=(8, 0, 1, [], 是, commutative_ring)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-qualification","seed":202610281025,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,4,5,3,6,7,0],[2,4,3,6,5,7,0,1],[3,5,6,0,7,1,2,4],[4,3,5,7,6,0,1,2],[5,6,7,1,0,2,4,3],[6,7,0,2,1,4,3,5],[7,0,1,4,2,3,5,6]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,3,0,6,2,3,6],[0,3,0,0,3,3,0,3],[0,4,6,3,1,7,2,5],[0,5,2,3,7,1,6,4],[0,6,3,0,2,6,3,2],[0,7,6,3,5,4,2,1]],"domain":"Complete_finite_commutative_ring_tables","operation":"commring-qualification","semantic_template":"qf-mod28-commring-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,4,5,3,6,7,0],[2,4,3,6,5,7,0,1],[3,5,6,0,7,1,2,4],[4,3,5,7,6,0,1,2],[5,6,7,1,0,2,4,3],[6,7,0,2,1,4,3,5],[7,0,1,4,2,3,5,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,3,0,6,2,3,6],[0,3,0,0,3,3,0,3],[0,4,6,3,1,7,2,5],[0,5,2,3,7,1,6,4],[0,6,3,0,2,6,3,2],[0,7,6,3,5,4,2,1]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0026","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-product-swap；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,6,0,2,4,6],[0,3,6,1,4,7,2,5],[0,4,0,4,0,4,0,4],[0,5,2,7,4,1,6,3],[0,6,4,2,0,6,4,2],[0,7,6,5,4,3,2,1]],\"a\":1,\"b\":2}。請依 [a,b,ab,ba,equal] 欄序回答。","answer":{"kind":"expression","value":"(1, 2, 2, 2, 是)","explanation_zh":"Wave4第二層核對=(1, 2, 2, 2, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-product-swap","seed":202610281026,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,6,0,2,4,6],[0,3,6,1,4,7,2,5],[0,4,0,4,0,4,0,4],[0,5,2,7,4,1,6,3],[0,6,4,2,0,6,4,2],[0,7,6,5,4,3,2,1]],"a":1,"b":2,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-product-swap","semantic_template":"qf-mod28-commring-product-swap/{\"a\":1,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],\"b\":2,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,6,0,2,4,6],[0,3,6,1,4,7,2,5],[0,4,0,4,0,4,0,4],[0,5,2,7,4,1,6,3],[0,6,4,2,0,6,4,2],[0,7,6,5,4,3,2,1]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0027","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-product-swap；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,7,0,6],[2,3,4,5,7,6,1,0],[3,4,5,7,6,0,2,1],[4,5,7,6,0,1,3,2],[5,7,6,0,1,2,4,3],[6,0,1,2,3,4,7,5],[7,6,0,1,2,3,5,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,7,0,2,7,4],[0,3,7,1,4,6,5,2],[0,4,0,4,0,4,4,0],[0,5,2,6,4,1,3,7],[0,6,7,5,4,3,1,2],[0,7,4,2,0,7,2,4]],\"a\":2,\"b\":5}。請依 [a,b,ab,ba,equal] 欄序回答。","answer":{"kind":"expression","value":"(2, 5, 2, 2, 是)","explanation_zh":"Wave4第二層核對=(2, 5, 2, 2, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-product-swap","seed":202610281027,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,7,0,6],[2,3,4,5,7,6,1,0],[3,4,5,7,6,0,2,1],[4,5,7,6,0,1,3,2],[5,7,6,0,1,2,4,3],[6,0,1,2,3,4,7,5],[7,6,0,1,2,3,5,4]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,7,0,2,7,4],[0,3,7,1,4,6,5,2],[0,4,0,4,0,4,4,0],[0,5,2,6,4,1,3,7],[0,6,7,5,4,3,1,2],[0,7,4,2,0,7,2,4]],"a":2,"b":5,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-product-swap","semantic_template":"qf-mod28-commring-product-swap/{\"a\":2,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,7,0,6],[2,3,4,5,7,6,1,0],[3,4,5,7,6,0,2,1],[4,5,7,6,0,1,3,2],[5,7,6,0,1,2,4,3],[6,0,1,2,3,4,7,5],[7,6,0,1,2,3,5,4]],\"b\":5,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,7,0,2,7,4],[0,3,7,1,4,6,5,2],[0,4,0,4,0,4,4,0],[0,5,2,6,4,1,3,7],[0,6,7,5,4,3,1,2],[0,7,4,2,0,7,2,4]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0028","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-product-swap；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,6,7,5,0],[2,3,4,6,5,0,7,1],[3,4,6,5,7,1,0,2],[4,6,5,7,0,2,1,3],[5,7,0,1,2,4,3,6],[6,5,7,0,1,3,2,4],[7,0,1,2,3,6,4,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,5,0,4,2,5],[0,3,5,1,4,2,7,6],[0,4,0,4,0,0,4,4],[0,5,4,2,0,4,5,2],[0,6,2,7,4,5,1,3],[0,7,5,6,4,2,3,1]],\"a\":3,\"b\":0}。請依 [a,b,ab,ba,equal] 欄序回答。","answer":{"kind":"expression","value":"(3, 0, 0, 0, 是)","explanation_zh":"Wave4第二層核對=(3, 0, 0, 0, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-product-swap","seed":202610281028,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,4,6,7,5,0],[2,3,4,6,5,0,7,1],[3,4,6,5,7,1,0,2],[4,6,5,7,0,2,1,3],[5,7,0,1,2,4,3,6],[6,5,7,0,1,3,2,4],[7,0,1,2,3,6,4,5]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,5,0,4,2,5],[0,3,5,1,4,2,7,6],[0,4,0,4,0,0,4,4],[0,5,4,2,0,4,5,2],[0,6,2,7,4,5,1,3],[0,7,5,6,4,2,3,1]],"a":3,"b":0,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-product-swap","semantic_template":"qf-mod28-commring-product-swap/{\"a\":3,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,6,7,5,0],[2,3,4,6,5,0,7,1],[3,4,6,5,7,1,0,2],[4,6,5,7,0,2,1,3],[5,7,0,1,2,4,3,6],[6,5,7,0,1,3,2,4],[7,0,1,2,3,6,4,5]],\"b\":0,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,5,0,4,2,5],[0,3,5,1,4,2,7,6],[0,4,0,4,0,0,4,4],[0,5,4,2,0,4,5,2],[0,6,2,7,4,5,1,3],[0,7,5,6,4,2,3,1]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0029","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-product-swap；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,6,0,7,5],[2,3,4,6,7,1,5,0],[3,4,6,7,5,2,0,1],[4,6,7,5,0,3,1,2],[5,0,1,2,3,7,4,6],[6,7,5,0,1,4,2,3],[7,5,0,1,2,6,3,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,7,0,7,2,4],[0,3,7,1,4,6,5,2],[0,4,0,4,0,4,4,0],[0,5,7,6,4,1,3,2],[0,6,2,5,4,3,1,7],[0,7,4,2,0,2,7,4]],\"a\":4,\"b\":3}。請依 [a,b,ab,ba,equal] 欄序回答。","answer":{"kind":"expression","value":"(4, 3, 4, 4, 是)","explanation_zh":"Wave4第二層核對=(4, 3, 4, 4, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-product-swap","seed":202610281029,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,4,6,0,7,5],[2,3,4,6,7,1,5,0],[3,4,6,7,5,2,0,1],[4,6,7,5,0,3,1,2],[5,0,1,2,3,7,4,6],[6,7,5,0,1,4,2,3],[7,5,0,1,2,6,3,4]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,7,0,7,2,4],[0,3,7,1,4,6,5,2],[0,4,0,4,0,4,4,0],[0,5,7,6,4,1,3,2],[0,6,2,5,4,3,1,7],[0,7,4,2,0,2,7,4]],"a":4,"b":3,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-product-swap","semantic_template":"qf-mod28-commring-product-swap/{\"a\":4,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,6,0,7,5],[2,3,4,6,7,1,5,0],[3,4,6,7,5,2,0,1],[4,6,7,5,0,3,1,2],[5,0,1,2,3,7,4,6],[6,7,5,0,1,4,2,3],[7,5,0,1,2,6,3,4]],\"b\":3,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,7,0,7,2,4],[0,3,7,1,4,6,5,2],[0,4,0,4,0,4,4,0],[0,5,7,6,4,1,3,2],[0,6,2,5,4,3,1,7],[0,7,4,2,0,2,7,4]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0030","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-product-swap；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,7,6,0,5],[2,3,4,7,5,0,1,6],[3,4,7,5,6,1,2,0],[4,7,5,6,0,2,3,1],[5,6,0,1,2,4,7,3],[6,0,1,2,3,7,5,4],[7,5,6,0,1,3,4,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,5,0,4,5,2],[0,3,5,1,4,2,7,6],[0,4,0,4,0,0,4,4],[0,5,4,2,0,4,2,5],[0,6,5,7,4,2,1,3],[0,7,2,6,4,5,3,1]],\"a\":7,\"b\":5}。請依 [a,b,ab,ba,equal] 欄序回答。","answer":{"kind":"expression","value":"(7, 5, 5, 5, 是)","explanation_zh":"Wave4第二層核對=(7, 5, 5, 5, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-product-swap","seed":202610281030,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,4,7,6,0,5],[2,3,4,7,5,0,1,6],[3,4,7,5,6,1,2,0],[4,7,5,6,0,2,3,1],[5,6,0,1,2,4,7,3],[6,0,1,2,3,7,5,4],[7,5,6,0,1,3,4,2]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,5,0,4,5,2],[0,3,5,1,4,2,7,6],[0,4,0,4,0,0,4,4],[0,5,4,2,0,4,2,5],[0,6,5,7,4,2,1,3],[0,7,2,6,4,5,3,1]],"a":7,"b":5,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-product-swap","semantic_template":"qf-mod28-commring-product-swap/{\"a\":7,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,7,6,0,5],[2,3,4,7,5,0,1,6],[3,4,7,5,6,1,2,0],[4,7,5,6,0,2,3,1],[5,6,0,1,2,4,7,3],[6,0,1,2,3,7,5,4],[7,5,6,0,1,3,4,2]],\"b\":5,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,5,0,4,5,2],[0,3,5,1,4,2,7,6],[0,4,0,4,0,0,4,4],[0,5,4,2,0,4,2,5],[0,6,5,7,4,2,1,3],[0,7,2,6,4,5,3,1]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0031","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-product-swap；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,7,0,5,6],[2,3,4,7,6,1,0,5],[3,4,7,6,5,2,1,0],[4,7,6,5,0,3,2,1],[5,0,1,2,3,6,7,4],[6,5,0,1,2,7,4,3],[7,6,5,0,1,4,3,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,6,0,6,4,2],[0,3,6,1,4,7,2,5],[0,4,0,4,0,4,0,4],[0,5,6,7,4,1,2,3],[0,6,4,2,0,2,4,6],[0,7,2,5,4,3,6,1]],\"a\":6,\"b\":1}。請依 [a,b,ab,ba,equal] 欄序回答。","answer":{"kind":"expression","value":"(6, 1, 6, 6, 是)","explanation_zh":"Wave4第二層核對=(6, 1, 6, 6, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-product-swap","seed":202610281031,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,4,7,0,5,6],[2,3,4,7,6,1,0,5],[3,4,7,6,5,2,1,0],[4,7,6,5,0,3,2,1],[5,0,1,2,3,6,7,4],[6,5,0,1,2,7,4,3],[7,6,5,0,1,4,3,2]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,6,0,6,4,2],[0,3,6,1,4,7,2,5],[0,4,0,4,0,4,0,4],[0,5,6,7,4,1,2,3],[0,6,4,2,0,2,4,6],[0,7,2,5,4,3,6,1]],"a":6,"b":1,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-product-swap","semantic_template":"qf-mod28-commring-product-swap/{\"a\":6,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,7,0,5,6],[2,3,4,7,6,1,0,5],[3,4,7,6,5,2,1,0],[4,7,6,5,0,3,2,1],[5,0,1,2,3,6,7,4],[6,5,0,1,2,7,4,3],[7,6,5,0,1,4,3,2]],\"b\":1,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,6,0,6,4,2],[0,3,6,1,4,7,2,5],[0,4,0,4,0,4,0,4],[0,5,6,7,4,1,2,3],[0,6,4,2,0,2,4,6],[0,7,2,5,4,3,6,1]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0032","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-product-swap；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,6,4,7,0],[2,3,5,4,7,6,0,1],[3,5,4,6,0,7,1,2],[4,6,7,0,2,1,3,5],[5,4,6,7,1,0,2,3],[6,7,0,1,3,2,5,4],[7,0,1,2,5,3,4,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,6,2,0,5,6],[0,3,6,1,7,5,2,4],[0,4,2,7,1,5,6,3],[0,5,0,5,5,0,0,5],[0,6,5,2,6,0,5,2],[0,7,6,4,3,5,2,1]],\"a\":7,\"b\":5}。請依 [a,b,ab,ba,equal] 欄序回答。","answer":{"kind":"expression","value":"(7, 5, 5, 5, 是)","explanation_zh":"Wave4第二層核對=(7, 5, 5, 5, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-product-swap","seed":202610281032,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,5,6,4,7,0],[2,3,5,4,7,6,0,1],[3,5,4,6,0,7,1,2],[4,6,7,0,2,1,3,5],[5,4,6,7,1,0,2,3],[6,7,0,1,3,2,5,4],[7,0,1,2,5,3,4,6]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,6,2,0,5,6],[0,3,6,1,7,5,2,4],[0,4,2,7,1,5,6,3],[0,5,0,5,5,0,0,5],[0,6,5,2,6,0,5,2],[0,7,6,4,3,5,2,1]],"a":7,"b":5,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-product-swap","semantic_template":"qf-mod28-commring-product-swap/{\"a\":7,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,6,4,7,0],[2,3,5,4,7,6,0,1],[3,5,4,6,0,7,1,2],[4,6,7,0,2,1,3,5],[5,4,6,7,1,0,2,3],[6,7,0,1,3,2,5,4],[7,0,1,2,5,3,4,6]],\"b\":5,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,6,2,0,5,6],[0,3,6,1,7,5,2,4],[0,4,2,7,1,5,6,3],[0,5,0,5,5,0,0,5],[0,6,5,2,6,0,5,2],[0,7,6,4,3,5,2,1]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0033","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-product-swap；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,7,4,0,6],[2,3,5,4,6,7,1,0],[3,5,4,7,0,6,2,1],[4,7,6,0,2,1,5,3],[5,4,7,6,1,0,3,2],[6,0,1,2,5,3,7,4],[7,6,0,1,3,2,4,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,7,2,0,7,5],[0,3,7,1,6,5,4,2],[0,4,2,6,1,5,3,7],[0,5,0,5,5,0,5,0],[0,6,7,4,3,5,1,2],[0,7,5,2,7,0,2,5]],\"a\":0,\"b\":6}。請依 [a,b,ab,ba,equal] 欄序回答。","answer":{"kind":"expression","value":"(0, 6, 0, 0, 是)","explanation_zh":"Wave4第二層核對=(0, 6, 0, 0, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-product-swap","seed":202610281033,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,5,7,4,0,6],[2,3,5,4,6,7,1,0],[3,5,4,7,0,6,2,1],[4,7,6,0,2,1,5,3],[5,4,7,6,1,0,3,2],[6,0,1,2,5,3,7,4],[7,6,0,1,3,2,4,5]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,7,2,0,7,5],[0,3,7,1,6,5,4,2],[0,4,2,6,1,5,3,7],[0,5,0,5,5,0,5,0],[0,6,7,4,3,5,1,2],[0,7,5,2,7,0,2,5]],"a":0,"b":6,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-product-swap","semantic_template":"qf-mod28-commring-product-swap/{\"a\":0,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,7,4,0,6],[2,3,5,4,6,7,1,0],[3,5,4,7,0,6,2,1],[4,7,6,0,2,1,5,3],[5,4,7,6,1,0,3,2],[6,0,1,2,5,3,7,4],[7,6,0,1,3,2,4,5]],\"b\":6,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,7,2,0,7,5],[0,3,7,1,6,5,4,2],[0,4,2,6,1,5,3,7],[0,5,0,5,5,0,5,0],[0,6,7,4,3,5,1,2],[0,7,5,2,7,0,2,5]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0034","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-product-swap；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,7,6,4,0],[2,3,5,6,0,4,7,1],[3,5,6,4,1,7,0,2],[4,7,0,1,5,2,3,6],[5,6,4,7,2,0,1,3],[6,4,7,0,3,1,2,5],[7,0,1,2,6,3,5,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,4,5,0,2,4],[0,3,4,1,2,5,7,6],[0,4,5,2,5,0,4,2],[0,5,0,5,0,0,5,5],[0,6,2,7,4,5,1,3],[0,7,4,6,2,5,3,1]],\"a\":1,\"b\":2}。請依 [a,b,ab,ba,equal] 欄序回答。","answer":{"kind":"expression","value":"(1, 2, 2, 2, 是)","explanation_zh":"Wave4第二層核對=(1, 2, 2, 2, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-product-swap","seed":202610281034,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,5,7,6,4,0],[2,3,5,6,0,4,7,1],[3,5,6,4,1,7,0,2],[4,7,0,1,5,2,3,6],[5,6,4,7,2,0,1,3],[6,4,7,0,3,1,2,5],[7,0,1,2,6,3,5,4]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,4,5,0,2,4],[0,3,4,1,2,5,7,6],[0,4,5,2,5,0,4,2],[0,5,0,5,0,0,5,5],[0,6,2,7,4,5,1,3],[0,7,4,6,2,5,3,1]],"a":1,"b":2,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-product-swap","semantic_template":"qf-mod28-commring-product-swap/{\"a\":1,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,7,6,4,0],[2,3,5,6,0,4,7,1],[3,5,6,4,1,7,0,2],[4,7,0,1,5,2,3,6],[5,6,4,7,2,0,1,3],[6,4,7,0,3,1,2,5],[7,0,1,2,6,3,5,4]],\"b\":2,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,4,5,0,2,4],[0,3,4,1,2,5,7,6],[0,4,5,2,5,0,4,2],[0,5,0,5,0,0,5,5],[0,6,2,7,4,5,1,3],[0,7,4,6,2,5,3,1]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0035","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-product-swap；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,0,6,7,4],[2,3,5,6,1,7,4,0],[3,5,6,7,2,4,0,1],[4,0,1,2,7,3,5,6],[5,6,7,4,3,0,1,2],[6,7,4,0,5,1,2,3],[7,4,0,1,6,2,3,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,7,7,0,2,5],[0,3,7,1,6,5,4,2],[0,4,7,6,1,5,3,2],[0,5,0,5,5,0,5,0],[0,6,2,4,3,5,1,7],[0,7,5,2,2,0,7,5]],\"a\":2,\"b\":6}。請依 [a,b,ab,ba,equal] 欄序回答。","answer":{"kind":"expression","value":"(2, 6, 2, 2, 是)","explanation_zh":"Wave4第二層核對=(2, 6, 2, 2, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-product-swap","seed":202610281035,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,5,0,6,7,4],[2,3,5,6,1,7,4,0],[3,5,6,7,2,4,0,1],[4,0,1,2,7,3,5,6],[5,6,7,4,3,0,1,2],[6,7,4,0,5,1,2,3],[7,4,0,1,6,2,3,5]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,7,7,0,2,5],[0,3,7,1,6,5,4,2],[0,4,7,6,1,5,3,2],[0,5,0,5,5,0,5,0],[0,6,2,4,3,5,1,7],[0,7,5,2,2,0,7,5]],"a":2,"b":6,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-product-swap","semantic_template":"qf-mod28-commring-product-swap/{\"a\":2,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,0,6,7,4],[2,3,5,6,1,7,4,0],[3,5,6,7,2,4,0,1],[4,0,1,2,7,3,5,6],[5,6,7,4,3,0,1,2],[6,7,4,0,5,1,2,3],[7,4,0,1,6,2,3,5]],\"b\":6,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,7,7,0,2,5],[0,3,7,1,6,5,4,2],[0,4,7,6,1,5,3,2],[0,5,0,5,5,0,5,0],[0,6,2,4,3,5,1,7],[0,7,5,2,2,0,7,5]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0036","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-product-swap；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,6,7,0,4],[2,3,5,7,0,4,1,6],[3,5,7,4,1,6,2,0],[4,6,0,1,5,2,7,3],[5,7,4,6,2,0,3,1],[6,0,1,2,7,3,4,5],[7,4,6,0,3,1,5,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,4,5,0,4,2],[0,3,4,1,2,5,7,6],[0,4,5,2,5,0,2,4],[0,5,0,5,0,0,5,5],[0,6,4,7,2,5,1,3],[0,7,2,6,4,5,3,1]],\"a\":3,\"b\":0}。請依 [a,b,ab,ba,equal] 欄序回答。","answer":{"kind":"expression","value":"(3, 0, 0, 0, 是)","explanation_zh":"Wave4第二層核對=(3, 0, 0, 0, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-product-swap","seed":202610281036,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,5,6,7,0,4],[2,3,5,7,0,4,1,6],[3,5,7,4,1,6,2,0],[4,6,0,1,5,2,7,3],[5,7,4,6,2,0,3,1],[6,0,1,2,7,3,4,5],[7,4,6,0,3,1,5,2]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,4,5,0,4,2],[0,3,4,1,2,5,7,6],[0,4,5,2,5,0,2,4],[0,5,0,5,0,0,5,5],[0,6,4,7,2,5,1,3],[0,7,2,6,4,5,3,1]],"a":3,"b":0,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-product-swap","semantic_template":"qf-mod28-commring-product-swap/{\"a\":3,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,6,7,0,4],[2,3,5,7,0,4,1,6],[3,5,7,4,1,6,2,0],[4,6,0,1,5,2,7,3],[5,7,4,6,2,0,3,1],[6,0,1,2,7,3,4,5],[7,4,6,0,3,1,5,2]],\"b\":0,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,4,5,0,4,2],[0,3,4,1,2,5,7,6],[0,4,5,2,5,0,2,4],[0,5,0,5,0,0,5,5],[0,6,4,7,2,5,1,3],[0,7,2,6,4,5,3,1]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0037","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-product-swap；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,0,7,4,6],[2,3,5,7,1,6,0,4],[3,5,7,6,2,4,1,0],[4,0,1,2,6,3,7,5],[5,7,6,4,3,0,2,1],[6,4,0,1,7,2,5,3],[7,6,4,0,5,1,3,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,6,6,0,5,2],[0,3,6,1,7,5,2,4],[0,4,6,7,1,5,2,3],[0,5,0,5,5,0,0,5],[0,6,5,2,2,0,5,6],[0,7,2,4,3,5,6,1]],\"a\":5,\"b\":3}。請依 [a,b,ab,ba,equal] 欄序回答。","answer":{"kind":"expression","value":"(5, 3, 5, 5, 是)","explanation_zh":"Wave4第二層核對=(5, 3, 5, 5, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-product-swap","seed":202610281037,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,5,0,7,4,6],[2,3,5,7,1,6,0,4],[3,5,7,6,2,4,1,0],[4,0,1,2,6,3,7,5],[5,7,6,4,3,0,2,1],[6,4,0,1,7,2,5,3],[7,6,4,0,5,1,3,2]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,6,6,0,5,2],[0,3,6,1,7,5,2,4],[0,4,6,7,1,5,2,3],[0,5,0,5,5,0,0,5],[0,6,5,2,2,0,5,6],[0,7,2,4,3,5,6,1]],"a":5,"b":3,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-product-swap","semantic_template":"qf-mod28-commring-product-swap/{\"a\":5,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,0,7,4,6],[2,3,5,7,1,6,0,4],[3,5,7,6,2,4,1,0],[4,0,1,2,6,3,7,5],[5,7,6,4,3,0,2,1],[6,4,0,1,7,2,5,3],[7,6,4,0,5,1,3,2]],\"b\":3,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,6,6,0,5,2],[0,3,6,1,7,5,2,4],[0,4,6,7,1,5,2,3],[0,5,0,5,5,0,0,5],[0,6,5,2,2,0,5,6],[0,7,2,4,3,5,6,1]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0038","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-product-swap；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,5,7,4,0],[2,3,6,4,7,0,5,1],[3,6,4,5,0,1,7,2],[4,5,7,0,2,3,1,6],[5,7,0,1,3,6,2,4],[6,4,5,7,1,2,0,3],[7,0,1,2,6,4,3,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,5,2,6,0,5],[0,3,5,1,7,2,6,4],[0,4,2,7,1,5,6,3],[0,5,6,2,5,6,0,2],[0,6,0,6,6,0,0,6],[0,7,5,4,3,2,6,1]],\"a\":4,\"b\":5}。請依 [a,b,ab,ba,equal] 欄序回答。","answer":{"kind":"expression","value":"(4, 5, 5, 5, 是)","explanation_zh":"Wave4第二層核對=(4, 5, 5, 5, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-product-swap","seed":202610281038,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,6,5,7,4,0],[2,3,6,4,7,0,5,1],[3,6,4,5,0,1,7,2],[4,5,7,0,2,3,1,6],[5,7,0,1,3,6,2,4],[6,4,5,7,1,2,0,3],[7,0,1,2,6,4,3,5]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,5,2,6,0,5],[0,3,5,1,7,2,6,4],[0,4,2,7,1,5,6,3],[0,5,6,2,5,6,0,2],[0,6,0,6,6,0,0,6],[0,7,5,4,3,2,6,1]],"a":4,"b":5,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-product-swap","semantic_template":"qf-mod28-commring-product-swap/{\"a\":4,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,5,7,4,0],[2,3,6,4,7,0,5,1],[3,6,4,5,0,1,7,2],[4,5,7,0,2,3,1,6],[5,7,0,1,3,6,2,4],[6,4,5,7,1,2,0,3],[7,0,1,2,6,4,3,5]],\"b\":5,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,5,2,6,0,5],[0,3,5,1,7,2,6,4],[0,4,2,7,1,5,6,3],[0,5,6,2,5,6,0,2],[0,6,0,6,6,0,0,6],[0,7,5,4,3,2,6,1]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0039","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-product-swap；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,7,0,4,5],[2,3,6,4,5,1,7,0],[3,6,4,7,0,2,5,1],[4,7,5,0,2,6,1,3],[5,0,1,2,6,7,3,4],[6,4,7,5,1,3,0,2],[7,5,0,1,3,4,2,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,7,2,7,0,6],[0,3,7,1,5,4,6,2],[0,4,2,5,1,3,6,7],[0,5,7,4,3,1,6,2],[0,6,0,6,6,6,0,0],[0,7,6,2,7,2,0,6]],\"a\":7,\"b\":1}。請依 [a,b,ab,ba,equal] 欄序回答。","answer":{"kind":"expression","value":"(7, 1, 7, 7, 是)","explanation_zh":"Wave4第二層核對=(7, 1, 7, 7, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-product-swap","seed":202610281039,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,6,7,0,4,5],[2,3,6,4,5,1,7,0],[3,6,4,7,0,2,5,1],[4,7,5,0,2,6,1,3],[5,0,1,2,6,7,3,4],[6,4,7,5,1,3,0,2],[7,5,0,1,3,4,2,6]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,7,2,7,0,6],[0,3,7,1,5,4,6,2],[0,4,2,5,1,3,6,7],[0,5,7,4,3,1,6,2],[0,6,0,6,6,6,0,0],[0,7,6,2,7,2,0,6]],"a":7,"b":1,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-product-swap","semantic_template":"qf-mod28-commring-product-swap/{\"a\":7,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,7,0,4,5],[2,3,6,4,5,1,7,0],[3,6,4,7,0,2,5,1],[4,7,5,0,2,6,1,3],[5,0,1,2,6,7,3,4],[6,4,7,5,1,3,0,2],[7,5,0,1,3,4,2,6]],\"b\":1,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,7,2,7,0,6],[0,3,7,1,5,4,6,2],[0,4,2,5,1,3,6,7],[0,5,7,4,3,1,6,2],[0,6,0,6,6,6,0,0],[0,7,6,2,7,2,0,6]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0040","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-product-swap；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,7,4,5,0],[2,3,6,5,0,7,4,1],[3,6,5,4,1,0,7,2],[4,7,0,1,6,3,2,5],[5,4,7,0,3,2,1,6],[6,5,4,7,2,1,0,3],[7,0,1,2,5,6,3,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,4,6,2,0,4],[0,3,4,1,2,7,6,5],[0,4,6,2,6,4,0,2],[0,5,2,7,4,1,6,3],[0,6,0,6,0,6,0,6],[0,7,4,5,2,3,6,1]],\"a\":7,\"b\":6}。請依 [a,b,ab,ba,equal] 欄序回答。","answer":{"kind":"expression","value":"(7, 6, 6, 6, 是)","explanation_zh":"Wave4第二層核對=(7, 6, 6, 6, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-product-swap","seed":202610281040,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,6,7,4,5,0],[2,3,6,5,0,7,4,1],[3,6,5,4,1,0,7,2],[4,7,0,1,6,3,2,5],[5,4,7,0,3,2,1,6],[6,5,4,7,2,1,0,3],[7,0,1,2,5,6,3,4]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,4,6,2,0,4],[0,3,4,1,2,7,6,5],[0,4,6,2,6,4,0,2],[0,5,2,7,4,1,6,3],[0,6,0,6,0,6,0,6],[0,7,4,5,2,3,6,1]],"a":7,"b":6,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-product-swap","semantic_template":"qf-mod28-commring-product-swap/{\"a\":7,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,7,4,5,0],[2,3,6,5,0,7,4,1],[3,6,5,4,1,0,7,2],[4,7,0,1,6,3,2,5],[5,4,7,0,3,2,1,6],[6,5,4,7,2,1,0,3],[7,0,1,2,5,6,3,4]],\"b\":6,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,4,6,2,0,4],[0,3,4,1,2,7,6,5],[0,4,6,2,6,4,0,2],[0,5,2,7,4,1,6,3],[0,6,0,6,0,6,0,6],[0,7,4,5,2,3,6,1]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0041","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-product-swap；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,0,7,5,4],[2,3,6,5,1,4,7,0],[3,6,5,7,2,0,4,1],[4,0,1,2,7,6,3,5],[5,7,4,0,6,2,1,3],[6,5,7,4,3,1,0,2],[7,4,0,1,5,3,2,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,7,7,2,0,6],[0,3,7,1,5,4,6,2],[0,4,7,5,1,3,6,2],[0,5,2,4,3,1,6,7],[0,6,0,6,6,6,0,0],[0,7,6,2,2,7,0,6]],\"a\":0,\"b\":4}。請依 [a,b,ab,ba,equal] 欄序回答。","answer":{"kind":"expression","value":"(0, 4, 0, 0, 是)","explanation_zh":"Wave4第二層核對=(0, 4, 0, 0, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-product-swap","seed":202610281041,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,6,0,7,5,4],[2,3,6,5,1,4,7,0],[3,6,5,7,2,0,4,1],[4,0,1,2,7,6,3,5],[5,7,4,0,6,2,1,3],[6,5,7,4,3,1,0,2],[7,4,0,1,5,3,2,6]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,7,7,2,0,6],[0,3,7,1,5,4,6,2],[0,4,7,5,1,3,6,2],[0,5,2,4,3,1,6,7],[0,6,0,6,6,6,0,0],[0,7,6,2,2,7,0,6]],"a":0,"b":4,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-product-swap","semantic_template":"qf-mod28-commring-product-swap/{\"a\":0,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,0,7,5,4],[2,3,6,5,1,4,7,0],[3,6,5,7,2,0,4,1],[4,0,1,2,7,6,3,5],[5,7,4,0,6,2,1,3],[6,5,7,4,3,1,0,2],[7,4,0,1,5,3,2,6]],\"b\":4,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,7,7,2,0,6],[0,3,7,1,5,4,6,2],[0,4,7,5,1,3,6,2],[0,5,2,4,3,1,6,7],[0,6,0,6,6,6,0,0],[0,7,6,2,2,7,0,6]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0042","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-product-swap；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,5,0,7,4],[2,3,6,7,0,1,4,5],[3,6,7,4,1,2,5,0],[4,5,0,1,6,7,2,3],[5,0,1,2,7,4,3,6],[6,7,4,5,2,3,0,1],[7,4,5,0,3,6,1,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,4,6,4,0,2],[0,3,4,1,2,7,6,5],[0,4,6,2,6,2,0,4],[0,5,4,7,2,1,6,3],[0,6,0,6,0,6,0,6],[0,7,2,5,4,3,6,1]],\"a\":1,\"b\":2}。請依 [a,b,ab,ba,equal] 欄序回答。","answer":{"kind":"expression","value":"(1, 2, 2, 2, 是)","explanation_zh":"Wave4第二層核對=(1, 2, 2, 2, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-product-swap","seed":202610281042,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,6,5,0,7,4],[2,3,6,7,0,1,4,5],[3,6,7,4,1,2,5,0],[4,5,0,1,6,7,2,3],[5,0,1,2,7,4,3,6],[6,7,4,5,2,3,0,1],[7,4,5,0,3,6,1,2]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,4,6,4,0,2],[0,3,4,1,2,7,6,5],[0,4,6,2,6,2,0,4],[0,5,4,7,2,1,6,3],[0,6,0,6,0,6,0,6],[0,7,2,5,4,3,6,1]],"a":1,"b":2,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-product-swap","semantic_template":"qf-mod28-commring-product-swap/{\"a\":1,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,5,0,7,4],[2,3,6,7,0,1,4,5],[3,6,7,4,1,2,5,0],[4,5,0,1,6,7,2,3],[5,0,1,2,7,4,3,6],[6,7,4,5,2,3,0,1],[7,4,5,0,3,6,1,2]],\"b\":2,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,4,6,4,0,2],[0,3,4,1,2,7,6,5],[0,4,6,2,6,2,0,4],[0,5,4,7,2,1,6,3],[0,6,0,6,0,6,0,6],[0,7,2,5,4,3,6,1]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0043","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-product-swap；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,0,4,7,5],[2,3,6,7,1,0,5,4],[3,6,7,5,2,1,4,0],[4,0,1,2,5,7,3,6],[5,4,0,1,7,6,2,3],[6,7,5,4,3,2,0,1],[7,5,4,0,6,3,1,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,5,5,6,0,2],[0,3,5,1,7,2,6,4],[0,4,5,7,1,2,6,3],[0,5,6,2,2,6,0,5],[0,6,0,6,6,0,0,6],[0,7,2,4,3,5,6,1]],\"a\":2,\"b\":7}。請依 [a,b,ab,ba,equal] 欄序回答。","answer":{"kind":"expression","value":"(2, 7, 2, 2, 是)","explanation_zh":"Wave4第二層核對=(2, 7, 2, 2, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-product-swap","seed":202610281043,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,6,0,4,7,5],[2,3,6,7,1,0,5,4],[3,6,7,5,2,1,4,0],[4,0,1,2,5,7,3,6],[5,4,0,1,7,6,2,3],[6,7,5,4,3,2,0,1],[7,5,4,0,6,3,1,2]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,5,5,6,0,2],[0,3,5,1,7,2,6,4],[0,4,5,7,1,2,6,3],[0,5,6,2,2,6,0,5],[0,6,0,6,6,0,0,6],[0,7,2,4,3,5,6,1]],"a":2,"b":7,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-product-swap","semantic_template":"qf-mod28-commring-product-swap/{\"a\":2,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,0,4,7,5],[2,3,6,7,1,0,5,4],[3,6,7,5,2,1,4,0],[4,0,1,2,5,7,3,6],[5,4,0,1,7,6,2,3],[6,7,5,4,3,2,0,1],[7,5,4,0,6,3,1,2]],\"b\":7,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,5,5,6,0,2],[0,3,5,1,7,2,6,4],[0,4,5,7,1,2,6,3],[0,5,6,2,2,6,0,5],[0,6,0,6,6,0,0,6],[0,7,2,4,3,5,6,1]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0044","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-product-swap；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,5,6,0,4],[2,3,7,4,6,0,1,5],[3,7,4,5,0,1,2,6],[4,5,6,0,2,3,7,1],[5,6,0,1,3,7,4,2],[6,0,1,2,7,4,5,3],[7,4,5,6,1,2,3,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,5,2,7,5,0],[0,3,5,1,6,2,4,7],[0,4,2,6,1,5,3,7],[0,5,7,2,5,7,2,0],[0,6,5,4,3,2,1,7],[0,7,0,7,7,0,7,0]],\"a\":3,\"b\":0}。請依 [a,b,ab,ba,equal] 欄序回答。","answer":{"kind":"expression","value":"(3, 0, 0, 0, 是)","explanation_zh":"Wave4第二層核對=(3, 0, 0, 0, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-product-swap","seed":202610281044,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,7,5,6,0,4],[2,3,7,4,6,0,1,5],[3,7,4,5,0,1,2,6],[4,5,6,0,2,3,7,1],[5,6,0,1,3,7,4,2],[6,0,1,2,7,4,5,3],[7,4,5,6,1,2,3,0]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,5,2,7,5,0],[0,3,5,1,6,2,4,7],[0,4,2,6,1,5,3,7],[0,5,7,2,5,7,2,0],[0,6,5,4,3,2,1,7],[0,7,0,7,7,0,7,0]],"a":3,"b":0,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-product-swap","semantic_template":"qf-mod28-commring-product-swap/{\"a\":3,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,5,6,0,4],[2,3,7,4,6,0,1,5],[3,7,4,5,0,1,2,6],[4,5,6,0,2,3,7,1],[5,6,0,1,3,7,4,2],[6,0,1,2,7,4,5,3],[7,4,5,6,1,2,3,0]],\"b\":0,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,5,2,7,5,0],[0,3,5,1,6,2,4,7],[0,4,2,6,1,5,3,7],[0,5,7,2,5,7,2,0],[0,6,5,4,3,2,1,7],[0,7,0,7,7,0,7,0]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0045","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-product-swap；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,6,0,5,4],[2,3,7,4,5,1,0,6],[3,7,4,6,0,2,1,5],[4,6,5,0,2,7,3,1],[5,0,1,2,7,6,4,3],[6,5,0,1,3,4,7,2],[7,4,6,5,1,3,2,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,6,2,6,7,0],[0,3,6,1,5,4,2,7],[0,4,2,5,1,3,6,7],[0,5,6,4,3,1,2,7],[0,6,7,2,6,2,7,0],[0,7,0,7,7,7,0,0]],\"a\":7,\"b\":3}。請依 [a,b,ab,ba,equal] 欄序回答。","answer":{"kind":"expression","value":"(7, 3, 7, 7, 是)","explanation_zh":"Wave4第二層核對=(7, 3, 7, 7, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-product-swap","seed":202610281045,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,7,6,0,5,4],[2,3,7,4,5,1,0,6],[3,7,4,6,0,2,1,5],[4,6,5,0,2,7,3,1],[5,0,1,2,7,6,4,3],[6,5,0,1,3,4,7,2],[7,4,6,5,1,3,2,0]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,6,2,6,7,0],[0,3,6,1,5,4,2,7],[0,4,2,5,1,3,6,7],[0,5,6,4,3,1,2,7],[0,6,7,2,6,2,7,0],[0,7,0,7,7,7,0,0]],"a":7,"b":3,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-product-swap","semantic_template":"qf-mod28-commring-product-swap/{\"a\":7,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,6,0,5,4],[2,3,7,4,5,1,0,6],[3,7,4,6,0,2,1,5],[4,6,5,0,2,7,3,1],[5,0,1,2,7,6,4,3],[6,5,0,1,3,4,7,2],[7,4,6,5,1,3,2,0]],\"b\":3,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,6,2,6,7,0],[0,3,6,1,5,4,2,7],[0,4,2,5,1,3,6,7],[0,5,6,4,3,1,2,7],[0,6,7,2,6,2,7,0],[0,7,0,7,7,7,0,0]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0046","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-product-swap；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,6,4,0,5],[2,3,7,5,0,6,1,4],[3,7,5,4,1,0,2,6],[4,6,0,1,7,3,5,2],[5,4,6,0,3,2,7,1],[6,0,1,2,5,7,4,3],[7,5,4,6,2,1,3,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,4,7,2,4,0],[0,3,4,1,2,6,5,7],[0,4,7,2,7,4,2,0],[0,5,2,6,4,1,3,7],[0,6,4,5,2,3,1,7],[0,7,0,7,0,7,7,0]],\"a\":5,\"b\":4}。請依 [a,b,ab,ba,equal] 欄序回答。","answer":{"kind":"expression","value":"(5, 4, 4, 4, 是)","explanation_zh":"Wave4第二層核對=(5, 4, 4, 4, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-product-swap","seed":202610281046,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,7,6,4,0,5],[2,3,7,5,0,6,1,4],[3,7,5,4,1,0,2,6],[4,6,0,1,7,3,5,2],[5,4,6,0,3,2,7,1],[6,0,1,2,5,7,4,3],[7,5,4,6,2,1,3,0]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,4,7,2,4,0],[0,3,4,1,2,6,5,7],[0,4,7,2,7,4,2,0],[0,5,2,6,4,1,3,7],[0,6,4,5,2,3,1,7],[0,7,0,7,0,7,7,0]],"a":5,"b":4,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-product-swap","semantic_template":"qf-mod28-commring-product-swap/{\"a\":5,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,6,4,0,5],[2,3,7,5,0,6,1,4],[3,7,5,4,1,0,2,6],[4,6,0,1,7,3,5,2],[5,4,6,0,3,2,7,1],[6,0,1,2,5,7,4,3],[7,5,4,6,2,1,3,0]],\"b\":4,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,4,7,2,4,0],[0,3,4,1,2,6,5,7],[0,4,7,2,7,4,2,0],[0,5,2,6,4,1,3,7],[0,6,4,5,2,3,1,7],[0,7,0,7,0,7,7,0]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0047","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-product-swap；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,0,6,4,5],[2,3,7,5,1,4,0,6],[3,7,5,6,2,0,1,4],[4,0,1,2,6,7,5,3],[5,6,4,0,7,2,3,1],[6,4,0,1,5,3,7,2],[7,5,6,4,3,1,2,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,6,6,2,7,0],[0,3,6,1,5,4,2,7],[0,4,6,5,1,3,2,7],[0,5,2,4,3,1,6,7],[0,6,7,2,2,6,7,0],[0,7,0,7,7,7,0,0]],\"a\":6,\"b\":1}。請依 [a,b,ab,ba,equal] 欄序回答。","answer":{"kind":"expression","value":"(6, 1, 6, 6, 是)","explanation_zh":"Wave4第二層核對=(6, 1, 6, 6, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-product-swap","seed":202610281047,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,7,0,6,4,5],[2,3,7,5,1,4,0,6],[3,7,5,6,2,0,1,4],[4,0,1,2,6,7,5,3],[5,6,4,0,7,2,3,1],[6,4,0,1,5,3,7,2],[7,5,6,4,3,1,2,0]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,6,6,2,7,0],[0,3,6,1,5,4,2,7],[0,4,6,5,1,3,2,7],[0,5,2,4,3,1,6,7],[0,6,7,2,2,6,7,0],[0,7,0,7,7,7,0,0]],"a":6,"b":1,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-product-swap","semantic_template":"qf-mod28-commring-product-swap/{\"a\":6,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,0,6,4,5],[2,3,7,5,1,4,0,6],[3,7,5,6,2,0,1,4],[4,0,1,2,6,7,5,3],[5,6,4,0,7,2,3,1],[6,4,0,1,5,3,7,2],[7,5,6,4,3,1,2,0]],\"b\":1,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,6,6,2,7,0],[0,3,6,1,5,4,2,7],[0,4,6,5,1,3,2,7],[0,5,2,4,3,1,6,7],[0,6,7,2,2,6,7,0],[0,7,0,7,7,7,0,0]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0048","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-product-swap；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,5,0,4,6],[2,3,7,6,0,1,5,4],[3,7,6,4,1,2,0,5],[4,5,0,1,7,6,3,2],[5,0,1,2,6,4,7,3],[6,4,5,0,3,7,2,1],[7,6,4,5,2,3,1,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,4,7,4,2,0],[0,3,4,1,2,6,5,7],[0,4,7,2,7,2,4,0],[0,5,4,6,2,1,3,7],[0,6,2,5,4,3,1,7],[0,7,0,7,0,7,7,0]],\"a\":5,\"b\":7}。請依 [a,b,ab,ba,equal] 欄序回答。","answer":{"kind":"expression","value":"(5, 7, 7, 7, 是)","explanation_zh":"Wave4第二層核對=(5, 7, 7, 7, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-product-swap","seed":202610281048,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,7,5,0,4,6],[2,3,7,6,0,1,5,4],[3,7,6,4,1,2,0,5],[4,5,0,1,7,6,3,2],[5,0,1,2,6,4,7,3],[6,4,5,0,3,7,2,1],[7,6,4,5,2,3,1,0]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,4,7,4,2,0],[0,3,4,1,2,6,5,7],[0,4,7,2,7,2,4,0],[0,5,4,6,2,1,3,7],[0,6,2,5,4,3,1,7],[0,7,0,7,0,7,7,0]],"a":5,"b":7,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-product-swap","semantic_template":"qf-mod28-commring-product-swap/{\"a\":5,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,5,0,4,6],[2,3,7,6,0,1,5,4],[3,7,6,4,1,2,0,5],[4,5,0,1,7,6,3,2],[5,0,1,2,6,4,7,3],[6,4,5,0,3,7,2,1],[7,6,4,5,2,3,1,0]],\"b\":7,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,4,7,4,2,0],[0,3,4,1,2,6,5,7],[0,4,7,2,7,2,4,0],[0,5,4,6,2,1,3,7],[0,6,2,5,4,3,1,7],[0,7,0,7,0,7,7,0]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0049","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-product-swap；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,0,4,5,6],[2,3,7,6,1,0,4,5],[3,7,6,5,2,1,0,4],[4,0,1,2,5,6,7,3],[5,4,0,1,6,7,3,2],[6,5,4,0,7,3,2,1],[7,6,5,4,3,2,1,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,5,5,7,2,0],[0,3,5,1,6,2,4,7],[0,4,5,6,1,2,3,7],[0,5,7,2,2,7,5,0],[0,6,2,4,3,5,1,7],[0,7,0,7,7,0,7,0]],\"a\":0,\"b\":4}。請依 [a,b,ab,ba,equal] 欄序回答。","answer":{"kind":"expression","value":"(0, 4, 0, 0, 是)","explanation_zh":"Wave4第二層核對=(0, 4, 0, 0, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-product-swap","seed":202610281049,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,7,0,4,5,6],[2,3,7,6,1,0,4,5],[3,7,6,5,2,1,0,4],[4,0,1,2,5,6,7,3],[5,4,0,1,6,7,3,2],[6,5,4,0,7,3,2,1],[7,6,5,4,3,2,1,0]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,5,5,7,2,0],[0,3,5,1,6,2,4,7],[0,4,5,6,1,2,3,7],[0,5,7,2,2,7,5,0],[0,6,2,4,3,5,1,7],[0,7,0,7,7,0,7,0]],"a":0,"b":4,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-product-swap","semantic_template":"qf-mod28-commring-product-swap/{\"a\":0,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,0,4,5,6],[2,3,7,6,1,0,4,5],[3,7,6,5,2,1,0,4],[4,0,1,2,5,6,7,3],[5,4,0,1,6,7,3,2],[6,5,4,0,7,3,2,1],[7,6,5,4,3,2,1,0]],\"b\":4,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,5,5,7,2,0],[0,3,5,1,6,2,4,7],[0,4,5,6,1,2,3,7],[0,5,7,2,2,7,5,0],[0,6,2,4,3,5,1,7],[0,7,0,7,7,0,7,0]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0050","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-product-swap；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,4,5,3,6,7,0],[2,4,3,6,5,7,0,1],[3,5,6,0,7,1,2,4],[4,3,5,7,6,0,1,2],[5,6,7,1,0,2,4,3],[6,7,0,2,1,4,3,5],[7,0,1,4,2,3,5,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,3,0,6,2,3,6],[0,3,0,0,3,3,0,3],[0,4,6,3,1,7,2,5],[0,5,2,3,7,1,6,4],[0,6,3,0,2,6,3,2],[0,7,6,3,5,4,2,1]],\"a\":1,\"b\":2}。請依 [a,b,ab,ba,equal] 欄序回答。","answer":{"kind":"expression","value":"(1, 2, 2, 2, 是)","explanation_zh":"Wave4第二層核對=(1, 2, 2, 2, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-product-swap","seed":202610281050,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,4,5,3,6,7,0],[2,4,3,6,5,7,0,1],[3,5,6,0,7,1,2,4],[4,3,5,7,6,0,1,2],[5,6,7,1,0,2,4,3],[6,7,0,2,1,4,3,5],[7,0,1,4,2,3,5,6]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,3,0,6,2,3,6],[0,3,0,0,3,3,0,3],[0,4,6,3,1,7,2,5],[0,5,2,3,7,1,6,4],[0,6,3,0,2,6,3,2],[0,7,6,3,5,4,2,1]],"a":1,"b":2,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-product-swap","semantic_template":"qf-mod28-commring-product-swap/{\"a\":1,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,4,5,3,6,7,0],[2,4,3,6,5,7,0,1],[3,5,6,0,7,1,2,4],[4,3,5,7,6,0,1,2],[5,6,7,1,0,2,4,3],[6,7,0,2,1,4,3,5],[7,0,1,4,2,3,5,6]],\"b\":2,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,3,0,6,2,3,6],[0,3,0,0,3,3,0,3],[0,4,6,3,1,7,2,5],[0,5,2,3,7,1,6,4],[0,6,3,0,2,6,3,2],[0,7,6,3,5,4,2,1]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0051","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-power-product；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,6,0,2,4,6],[0,3,6,1,4,7,2,5],[0,4,0,4,0,4,0,4],[0,5,2,7,4,1,6,3],[0,6,4,2,0,6,4,2],[0,7,6,5,4,3,2,1]],\"a\":1,\"b\":2,\"exponent\":1}。請依 [a,b,exponent,ab,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(1, 2, 1, 2, 2, 2, 是)","explanation_zh":"Wave4第二層核對=(1, 2, 1, 2, 2, 2, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-power-product","seed":202610281051,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,6,0,2,4,6],[0,3,6,1,4,7,2,5],[0,4,0,4,0,4,0,4],[0,5,2,7,4,1,6,3],[0,6,4,2,0,6,4,2],[0,7,6,5,4,3,2,1]],"a":1,"b":2,"exponent":1,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-power-product","semantic_template":"qf-mod28-commring-power-product/{\"a\":1,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],\"b\":2,\"exponent\":1,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,6,0,2,4,6],[0,3,6,1,4,7,2,5],[0,4,0,4,0,4,0,4],[0,5,2,7,4,1,6,3],[0,6,4,2,0,6,4,2],[0,7,6,5,4,3,2,1]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0052","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-power-product；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,7,0,6],[2,3,4,5,7,6,1,0],[3,4,5,7,6,0,2,1],[4,5,7,6,0,1,3,2],[5,7,6,0,1,2,4,3],[6,0,1,2,3,4,7,5],[7,6,0,1,2,3,5,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,7,0,2,7,4],[0,3,7,1,4,6,5,2],[0,4,0,4,0,4,4,0],[0,5,2,6,4,1,3,7],[0,6,7,5,4,3,1,2],[0,7,4,2,0,7,2,4]],\"a\":2,\"b\":5,\"exponent\":2}。請依 [a,b,exponent,ab,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(2, 5, 2, 2, 4, 4, 是)","explanation_zh":"Wave4第二層核對=(2, 5, 2, 2, 4, 4, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-power-product","seed":202610281052,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,7,0,6],[2,3,4,5,7,6,1,0],[3,4,5,7,6,0,2,1],[4,5,7,6,0,1,3,2],[5,7,6,0,1,2,4,3],[6,0,1,2,3,4,7,5],[7,6,0,1,2,3,5,4]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,7,0,2,7,4],[0,3,7,1,4,6,5,2],[0,4,0,4,0,4,4,0],[0,5,2,6,4,1,3,7],[0,6,7,5,4,3,1,2],[0,7,4,2,0,7,2,4]],"a":2,"b":5,"exponent":2,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-power-product","semantic_template":"qf-mod28-commring-power-product/{\"a\":2,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,7,0,6],[2,3,4,5,7,6,1,0],[3,4,5,7,6,0,2,1],[4,5,7,6,0,1,3,2],[5,7,6,0,1,2,4,3],[6,0,1,2,3,4,7,5],[7,6,0,1,2,3,5,4]],\"b\":5,\"exponent\":2,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,7,0,2,7,4],[0,3,7,1,4,6,5,2],[0,4,0,4,0,4,4,0],[0,5,2,6,4,1,3,7],[0,6,7,5,4,3,1,2],[0,7,4,2,0,7,2,4]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0053","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-power-product；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,6,7,5,0],[2,3,4,6,5,0,7,1],[3,4,6,5,7,1,0,2],[4,6,5,7,0,2,1,3],[5,7,0,1,2,4,3,6],[6,5,7,0,1,3,2,4],[7,0,1,2,3,6,4,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,5,0,4,2,5],[0,3,5,1,4,2,7,6],[0,4,0,4,0,0,4,4],[0,5,4,2,0,4,5,2],[0,6,2,7,4,5,1,3],[0,7,5,6,4,2,3,1]],\"a\":3,\"b\":0,\"exponent\":3}。請依 [a,b,exponent,ab,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(3, 0, 3, 0, 0, 0, 是)","explanation_zh":"Wave4第二層核對=(3, 0, 3, 0, 0, 0, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-power-product","seed":202610281053,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,4,6,7,5,0],[2,3,4,6,5,0,7,1],[3,4,6,5,7,1,0,2],[4,6,5,7,0,2,1,3],[5,7,0,1,2,4,3,6],[6,5,7,0,1,3,2,4],[7,0,1,2,3,6,4,5]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,5,0,4,2,5],[0,3,5,1,4,2,7,6],[0,4,0,4,0,0,4,4],[0,5,4,2,0,4,5,2],[0,6,2,7,4,5,1,3],[0,7,5,6,4,2,3,1]],"a":3,"b":0,"exponent":3,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-power-product","semantic_template":"qf-mod28-commring-power-product/{\"a\":3,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,6,7,5,0],[2,3,4,6,5,0,7,1],[3,4,6,5,7,1,0,2],[4,6,5,7,0,2,1,3],[5,7,0,1,2,4,3,6],[6,5,7,0,1,3,2,4],[7,0,1,2,3,6,4,5]],\"b\":0,\"exponent\":3,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,5,0,4,2,5],[0,3,5,1,4,2,7,6],[0,4,0,4,0,0,4,4],[0,5,4,2,0,4,5,2],[0,6,2,7,4,5,1,3],[0,7,5,6,4,2,3,1]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0054","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-power-product；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,6,0,7,5],[2,3,4,6,7,1,5,0],[3,4,6,7,5,2,0,1],[4,6,7,5,0,3,1,2],[5,0,1,2,3,7,4,6],[6,7,5,0,1,4,2,3],[7,5,0,1,2,6,3,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,7,0,7,2,4],[0,3,7,1,4,6,5,2],[0,4,0,4,0,4,4,0],[0,5,7,6,4,1,3,2],[0,6,2,5,4,3,1,7],[0,7,4,2,0,2,7,4]],\"a\":4,\"b\":3,\"exponent\":4}。請依 [a,b,exponent,ab,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(4, 3, 4, 4, 0, 0, 是)","explanation_zh":"Wave4第二層核對=(4, 3, 4, 4, 0, 0, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-power-product","seed":202610281054,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,4,6,0,7,5],[2,3,4,6,7,1,5,0],[3,4,6,7,5,2,0,1],[4,6,7,5,0,3,1,2],[5,0,1,2,3,7,4,6],[6,7,5,0,1,4,2,3],[7,5,0,1,2,6,3,4]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,7,0,7,2,4],[0,3,7,1,4,6,5,2],[0,4,0,4,0,4,4,0],[0,5,7,6,4,1,3,2],[0,6,2,5,4,3,1,7],[0,7,4,2,0,2,7,4]],"a":4,"b":3,"exponent":4,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-power-product","semantic_template":"qf-mod28-commring-power-product/{\"a\":4,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,6,0,7,5],[2,3,4,6,7,1,5,0],[3,4,6,7,5,2,0,1],[4,6,7,5,0,3,1,2],[5,0,1,2,3,7,4,6],[6,7,5,0,1,4,2,3],[7,5,0,1,2,6,3,4]],\"b\":3,\"exponent\":4,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,7,0,7,2,4],[0,3,7,1,4,6,5,2],[0,4,0,4,0,4,4,0],[0,5,7,6,4,1,3,2],[0,6,2,5,4,3,1,7],[0,7,4,2,0,2,7,4]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0055","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-power-product；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,7,6,0,5],[2,3,4,7,5,0,1,6],[3,4,7,5,6,1,2,0],[4,7,5,6,0,2,3,1],[5,6,0,1,2,4,7,3],[6,0,1,2,3,7,5,4],[7,5,6,0,1,3,4,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,5,0,4,5,2],[0,3,5,1,4,2,7,6],[0,4,0,4,0,0,4,4],[0,5,4,2,0,4,2,5],[0,6,5,7,4,2,1,3],[0,7,2,6,4,5,3,1]],\"a\":7,\"b\":5,\"exponent\":5}。請依 [a,b,exponent,ab,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(7, 5, 5, 5, 0, 0, 是)","explanation_zh":"Wave4第二層核對=(7, 5, 5, 5, 0, 0, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-power-product","seed":202610281055,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,4,7,6,0,5],[2,3,4,7,5,0,1,6],[3,4,7,5,6,1,2,0],[4,7,5,6,0,2,3,1],[5,6,0,1,2,4,7,3],[6,0,1,2,3,7,5,4],[7,5,6,0,1,3,4,2]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,5,0,4,5,2],[0,3,5,1,4,2,7,6],[0,4,0,4,0,0,4,4],[0,5,4,2,0,4,2,5],[0,6,5,7,4,2,1,3],[0,7,2,6,4,5,3,1]],"a":7,"b":5,"exponent":5,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-power-product","semantic_template":"qf-mod28-commring-power-product/{\"a\":7,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,7,6,0,5],[2,3,4,7,5,0,1,6],[3,4,7,5,6,1,2,0],[4,7,5,6,0,2,3,1],[5,6,0,1,2,4,7,3],[6,0,1,2,3,7,5,4],[7,5,6,0,1,3,4,2]],\"b\":5,\"exponent\":5,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,5,0,4,5,2],[0,3,5,1,4,2,7,6],[0,4,0,4,0,0,4,4],[0,5,4,2,0,4,2,5],[0,6,5,7,4,2,1,3],[0,7,2,6,4,5,3,1]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0056","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-power-product；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,7,0,5,6],[2,3,4,7,6,1,0,5],[3,4,7,6,5,2,1,0],[4,7,6,5,0,3,2,1],[5,0,1,2,3,6,7,4],[6,5,0,1,2,7,4,3],[7,6,5,0,1,4,3,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,6,0,6,4,2],[0,3,6,1,4,7,2,5],[0,4,0,4,0,4,0,4],[0,5,6,7,4,1,2,3],[0,6,4,2,0,2,4,6],[0,7,2,5,4,3,6,1]],\"a\":6,\"b\":1,\"exponent\":6}。請依 [a,b,exponent,ab,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(6, 1, 6, 6, 0, 0, 是)","explanation_zh":"Wave4第二層核對=(6, 1, 6, 6, 0, 0, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-power-product","seed":202610281056,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,4,7,0,5,6],[2,3,4,7,6,1,0,5],[3,4,7,6,5,2,1,0],[4,7,6,5,0,3,2,1],[5,0,1,2,3,6,7,4],[6,5,0,1,2,7,4,3],[7,6,5,0,1,4,3,2]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,6,0,6,4,2],[0,3,6,1,4,7,2,5],[0,4,0,4,0,4,0,4],[0,5,6,7,4,1,2,3],[0,6,4,2,0,2,4,6],[0,7,2,5,4,3,6,1]],"a":6,"b":1,"exponent":6,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-power-product","semantic_template":"qf-mod28-commring-power-product/{\"a\":6,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,7,0,5,6],[2,3,4,7,6,1,0,5],[3,4,7,6,5,2,1,0],[4,7,6,5,0,3,2,1],[5,0,1,2,3,6,7,4],[6,5,0,1,2,7,4,3],[7,6,5,0,1,4,3,2]],\"b\":1,\"exponent\":6,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,6,0,6,4,2],[0,3,6,1,4,7,2,5],[0,4,0,4,0,4,0,4],[0,5,6,7,4,1,2,3],[0,6,4,2,0,2,4,6],[0,7,2,5,4,3,6,1]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0057","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-power-product；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,6,4,7,0],[2,3,5,4,7,6,0,1],[3,5,4,6,0,7,1,2],[4,6,7,0,2,1,3,5],[5,4,6,7,1,0,2,3],[6,7,0,1,3,2,5,4],[7,0,1,2,5,3,4,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,6,2,0,5,6],[0,3,6,1,7,5,2,4],[0,4,2,7,1,5,6,3],[0,5,0,5,5,0,0,5],[0,6,5,2,6,0,5,2],[0,7,6,4,3,5,2,1]],\"a\":7,\"b\":5,\"exponent\":7}。請依 [a,b,exponent,ab,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(7, 5, 7, 5, 0, 0, 是)","explanation_zh":"Wave4第二層核對=(7, 5, 7, 5, 0, 0, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-power-product","seed":202610281057,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,5,6,4,7,0],[2,3,5,4,7,6,0,1],[3,5,4,6,0,7,1,2],[4,6,7,0,2,1,3,5],[5,4,6,7,1,0,2,3],[6,7,0,1,3,2,5,4],[7,0,1,2,5,3,4,6]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,6,2,0,5,6],[0,3,6,1,7,5,2,4],[0,4,2,7,1,5,6,3],[0,5,0,5,5,0,0,5],[0,6,5,2,6,0,5,2],[0,7,6,4,3,5,2,1]],"a":7,"b":5,"exponent":7,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-power-product","semantic_template":"qf-mod28-commring-power-product/{\"a\":7,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,6,4,7,0],[2,3,5,4,7,6,0,1],[3,5,4,6,0,7,1,2],[4,6,7,0,2,1,3,5],[5,4,6,7,1,0,2,3],[6,7,0,1,3,2,5,4],[7,0,1,2,5,3,4,6]],\"b\":5,\"exponent\":7,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,6,2,0,5,6],[0,3,6,1,7,5,2,4],[0,4,2,7,1,5,6,3],[0,5,0,5,5,0,0,5],[0,6,5,2,6,0,5,2],[0,7,6,4,3,5,2,1]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0058","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-power-product；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,7,4,0,6],[2,3,5,4,6,7,1,0],[3,5,4,7,0,6,2,1],[4,7,6,0,2,1,5,3],[5,4,7,6,1,0,3,2],[6,0,1,2,5,3,7,4],[7,6,0,1,3,2,4,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,7,2,0,7,5],[0,3,7,1,6,5,4,2],[0,4,2,6,1,5,3,7],[0,5,0,5,5,0,5,0],[0,6,7,4,3,5,1,2],[0,7,5,2,7,0,2,5]],\"a\":0,\"b\":6,\"exponent\":1}。請依 [a,b,exponent,ab,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(0, 6, 1, 0, 0, 0, 是)","explanation_zh":"Wave4第二層核對=(0, 6, 1, 0, 0, 0, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-power-product","seed":202610281058,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,5,7,4,0,6],[2,3,5,4,6,7,1,0],[3,5,4,7,0,6,2,1],[4,7,6,0,2,1,5,3],[5,4,7,6,1,0,3,2],[6,0,1,2,5,3,7,4],[7,6,0,1,3,2,4,5]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,7,2,0,7,5],[0,3,7,1,6,5,4,2],[0,4,2,6,1,5,3,7],[0,5,0,5,5,0,5,0],[0,6,7,4,3,5,1,2],[0,7,5,2,7,0,2,5]],"a":0,"b":6,"exponent":1,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-power-product","semantic_template":"qf-mod28-commring-power-product/{\"a\":0,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,7,4,0,6],[2,3,5,4,6,7,1,0],[3,5,4,7,0,6,2,1],[4,7,6,0,2,1,5,3],[5,4,7,6,1,0,3,2],[6,0,1,2,5,3,7,4],[7,6,0,1,3,2,4,5]],\"b\":6,\"exponent\":1,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,7,2,0,7,5],[0,3,7,1,6,5,4,2],[0,4,2,6,1,5,3,7],[0,5,0,5,5,0,5,0],[0,6,7,4,3,5,1,2],[0,7,5,2,7,0,2,5]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0059","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-power-product；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,7,6,4,0],[2,3,5,6,0,4,7,1],[3,5,6,4,1,7,0,2],[4,7,0,1,5,2,3,6],[5,6,4,7,2,0,1,3],[6,4,7,0,3,1,2,5],[7,0,1,2,6,3,5,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,4,5,0,2,4],[0,3,4,1,2,5,7,6],[0,4,5,2,5,0,4,2],[0,5,0,5,0,0,5,5],[0,6,2,7,4,5,1,3],[0,7,4,6,2,5,3,1]],\"a\":1,\"b\":2,\"exponent\":2}。請依 [a,b,exponent,ab,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(1, 2, 2, 2, 5, 5, 是)","explanation_zh":"Wave4第二層核對=(1, 2, 2, 2, 5, 5, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-power-product","seed":202610281059,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,5,7,6,4,0],[2,3,5,6,0,4,7,1],[3,5,6,4,1,7,0,2],[4,7,0,1,5,2,3,6],[5,6,4,7,2,0,1,3],[6,4,7,0,3,1,2,5],[7,0,1,2,6,3,5,4]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,4,5,0,2,4],[0,3,4,1,2,5,7,6],[0,4,5,2,5,0,4,2],[0,5,0,5,0,0,5,5],[0,6,2,7,4,5,1,3],[0,7,4,6,2,5,3,1]],"a":1,"b":2,"exponent":2,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-power-product","semantic_template":"qf-mod28-commring-power-product/{\"a\":1,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,7,6,4,0],[2,3,5,6,0,4,7,1],[3,5,6,4,1,7,0,2],[4,7,0,1,5,2,3,6],[5,6,4,7,2,0,1,3],[6,4,7,0,3,1,2,5],[7,0,1,2,6,3,5,4]],\"b\":2,\"exponent\":2,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,4,5,0,2,4],[0,3,4,1,2,5,7,6],[0,4,5,2,5,0,4,2],[0,5,0,5,0,0,5,5],[0,6,2,7,4,5,1,3],[0,7,4,6,2,5,3,1]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0060","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-power-product；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,0,6,7,4],[2,3,5,6,1,7,4,0],[3,5,6,7,2,4,0,1],[4,0,1,2,7,3,5,6],[5,6,7,4,3,0,1,2],[6,7,4,0,5,1,2,3],[7,4,0,1,6,2,3,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,7,7,0,2,5],[0,3,7,1,6,5,4,2],[0,4,7,6,1,5,3,2],[0,5,0,5,5,0,5,0],[0,6,2,4,3,5,1,7],[0,7,5,2,2,0,7,5]],\"a\":2,\"b\":6,\"exponent\":3}。請依 [a,b,exponent,ab,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(2, 6, 3, 2, 0, 0, 是)","explanation_zh":"Wave4第二層核對=(2, 6, 3, 2, 0, 0, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-power-product","seed":202610281060,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,5,0,6,7,4],[2,3,5,6,1,7,4,0],[3,5,6,7,2,4,0,1],[4,0,1,2,7,3,5,6],[5,6,7,4,3,0,1,2],[6,7,4,0,5,1,2,3],[7,4,0,1,6,2,3,5]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,7,7,0,2,5],[0,3,7,1,6,5,4,2],[0,4,7,6,1,5,3,2],[0,5,0,5,5,0,5,0],[0,6,2,4,3,5,1,7],[0,7,5,2,2,0,7,5]],"a":2,"b":6,"exponent":3,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-power-product","semantic_template":"qf-mod28-commring-power-product/{\"a\":2,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,0,6,7,4],[2,3,5,6,1,7,4,0],[3,5,6,7,2,4,0,1],[4,0,1,2,7,3,5,6],[5,6,7,4,3,0,1,2],[6,7,4,0,5,1,2,3],[7,4,0,1,6,2,3,5]],\"b\":6,\"exponent\":3,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,7,7,0,2,5],[0,3,7,1,6,5,4,2],[0,4,7,6,1,5,3,2],[0,5,0,5,5,0,5,0],[0,6,2,4,3,5,1,7],[0,7,5,2,2,0,7,5]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0061","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-power-product；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,6,7,0,4],[2,3,5,7,0,4,1,6],[3,5,7,4,1,6,2,0],[4,6,0,1,5,2,7,3],[5,7,4,6,2,0,3,1],[6,0,1,2,7,3,4,5],[7,4,6,0,3,1,5,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,4,5,0,4,2],[0,3,4,1,2,5,7,6],[0,4,5,2,5,0,2,4],[0,5,0,5,0,0,5,5],[0,6,4,7,2,5,1,3],[0,7,2,6,4,5,3,1]],\"a\":3,\"b\":0,\"exponent\":4}。請依 [a,b,exponent,ab,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(3, 0, 4, 0, 0, 0, 是)","explanation_zh":"Wave4第二層核對=(3, 0, 4, 0, 0, 0, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-power-product","seed":202610281061,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,5,6,7,0,4],[2,3,5,7,0,4,1,6],[3,5,7,4,1,6,2,0],[4,6,0,1,5,2,7,3],[5,7,4,6,2,0,3,1],[6,0,1,2,7,3,4,5],[7,4,6,0,3,1,5,2]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,4,5,0,4,2],[0,3,4,1,2,5,7,6],[0,4,5,2,5,0,2,4],[0,5,0,5,0,0,5,5],[0,6,4,7,2,5,1,3],[0,7,2,6,4,5,3,1]],"a":3,"b":0,"exponent":4,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-power-product","semantic_template":"qf-mod28-commring-power-product/{\"a\":3,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,6,7,0,4],[2,3,5,7,0,4,1,6],[3,5,7,4,1,6,2,0],[4,6,0,1,5,2,7,3],[5,7,4,6,2,0,3,1],[6,0,1,2,7,3,4,5],[7,4,6,0,3,1,5,2]],\"b\":0,\"exponent\":4,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,4,5,0,4,2],[0,3,4,1,2,5,7,6],[0,4,5,2,5,0,2,4],[0,5,0,5,0,0,5,5],[0,6,4,7,2,5,1,3],[0,7,2,6,4,5,3,1]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0062","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-power-product；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,0,7,4,6],[2,3,5,7,1,6,0,4],[3,5,7,6,2,4,1,0],[4,0,1,2,6,3,7,5],[5,7,6,4,3,0,2,1],[6,4,0,1,7,2,5,3],[7,6,4,0,5,1,3,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,6,6,0,5,2],[0,3,6,1,7,5,2,4],[0,4,6,7,1,5,2,3],[0,5,0,5,5,0,0,5],[0,6,5,2,2,0,5,6],[0,7,2,4,3,5,6,1]],\"a\":5,\"b\":3,\"exponent\":5}。請依 [a,b,exponent,ab,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(5, 3, 5, 5, 0, 0, 是)","explanation_zh":"Wave4第二層核對=(5, 3, 5, 5, 0, 0, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-power-product","seed":202610281062,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,5,0,7,4,6],[2,3,5,7,1,6,0,4],[3,5,7,6,2,4,1,0],[4,0,1,2,6,3,7,5],[5,7,6,4,3,0,2,1],[6,4,0,1,7,2,5,3],[7,6,4,0,5,1,3,2]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,6,6,0,5,2],[0,3,6,1,7,5,2,4],[0,4,6,7,1,5,2,3],[0,5,0,5,5,0,0,5],[0,6,5,2,2,0,5,6],[0,7,2,4,3,5,6,1]],"a":5,"b":3,"exponent":5,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-power-product","semantic_template":"qf-mod28-commring-power-product/{\"a\":5,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,0,7,4,6],[2,3,5,7,1,6,0,4],[3,5,7,6,2,4,1,0],[4,0,1,2,6,3,7,5],[5,7,6,4,3,0,2,1],[6,4,0,1,7,2,5,3],[7,6,4,0,5,1,3,2]],\"b\":3,\"exponent\":5,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,6,6,0,5,2],[0,3,6,1,7,5,2,4],[0,4,6,7,1,5,2,3],[0,5,0,5,5,0,0,5],[0,6,5,2,2,0,5,6],[0,7,2,4,3,5,6,1]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0063","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-power-product；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,5,7,4,0],[2,3,6,4,7,0,5,1],[3,6,4,5,0,1,7,2],[4,5,7,0,2,3,1,6],[5,7,0,1,3,6,2,4],[6,4,5,7,1,2,0,3],[7,0,1,2,6,4,3,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,5,2,6,0,5],[0,3,5,1,7,2,6,4],[0,4,2,7,1,5,6,3],[0,5,6,2,5,6,0,2],[0,6,0,6,6,0,0,6],[0,7,5,4,3,2,6,1]],\"a\":4,\"b\":5,\"exponent\":6}。請依 [a,b,exponent,ab,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(4, 5, 6, 5, 0, 0, 是)","explanation_zh":"Wave4第二層核對=(4, 5, 6, 5, 0, 0, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-power-product","seed":202610281063,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,6,5,7,4,0],[2,3,6,4,7,0,5,1],[3,6,4,5,0,1,7,2],[4,5,7,0,2,3,1,6],[5,7,0,1,3,6,2,4],[6,4,5,7,1,2,0,3],[7,0,1,2,6,4,3,5]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,5,2,6,0,5],[0,3,5,1,7,2,6,4],[0,4,2,7,1,5,6,3],[0,5,6,2,5,6,0,2],[0,6,0,6,6,0,0,6],[0,7,5,4,3,2,6,1]],"a":4,"b":5,"exponent":6,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-power-product","semantic_template":"qf-mod28-commring-power-product/{\"a\":4,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,5,7,4,0],[2,3,6,4,7,0,5,1],[3,6,4,5,0,1,7,2],[4,5,7,0,2,3,1,6],[5,7,0,1,3,6,2,4],[6,4,5,7,1,2,0,3],[7,0,1,2,6,4,3,5]],\"b\":5,\"exponent\":6,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,5,2,6,0,5],[0,3,5,1,7,2,6,4],[0,4,2,7,1,5,6,3],[0,5,6,2,5,6,0,2],[0,6,0,6,6,0,0,6],[0,7,5,4,3,2,6,1]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0064","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-power-product；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,7,0,4,5],[2,3,6,4,5,1,7,0],[3,6,4,7,0,2,5,1],[4,7,5,0,2,6,1,3],[5,0,1,2,6,7,3,4],[6,4,7,5,1,3,0,2],[7,5,0,1,3,4,2,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,7,2,7,0,6],[0,3,7,1,5,4,6,2],[0,4,2,5,1,3,6,7],[0,5,7,4,3,1,6,2],[0,6,0,6,6,6,0,0],[0,7,6,2,7,2,0,6]],\"a\":7,\"b\":1,\"exponent\":7}。請依 [a,b,exponent,ab,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(7, 1, 7, 7, 0, 0, 是)","explanation_zh":"Wave4第二層核對=(7, 1, 7, 7, 0, 0, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-power-product","seed":202610281064,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,6,7,0,4,5],[2,3,6,4,5,1,7,0],[3,6,4,7,0,2,5,1],[4,7,5,0,2,6,1,3],[5,0,1,2,6,7,3,4],[6,4,7,5,1,3,0,2],[7,5,0,1,3,4,2,6]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,7,2,7,0,6],[0,3,7,1,5,4,6,2],[0,4,2,5,1,3,6,7],[0,5,7,4,3,1,6,2],[0,6,0,6,6,6,0,0],[0,7,6,2,7,2,0,6]],"a":7,"b":1,"exponent":7,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-power-product","semantic_template":"qf-mod28-commring-power-product/{\"a\":7,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,7,0,4,5],[2,3,6,4,5,1,7,0],[3,6,4,7,0,2,5,1],[4,7,5,0,2,6,1,3],[5,0,1,2,6,7,3,4],[6,4,7,5,1,3,0,2],[7,5,0,1,3,4,2,6]],\"b\":1,\"exponent\":7,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,7,2,7,0,6],[0,3,7,1,5,4,6,2],[0,4,2,5,1,3,6,7],[0,5,7,4,3,1,6,2],[0,6,0,6,6,6,0,0],[0,7,6,2,7,2,0,6]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0065","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-power-product；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,7,4,5,0],[2,3,6,5,0,7,4,1],[3,6,5,4,1,0,7,2],[4,7,0,1,6,3,2,5],[5,4,7,0,3,2,1,6],[6,5,4,7,2,1,0,3],[7,0,1,2,5,6,3,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,4,6,2,0,4],[0,3,4,1,2,7,6,5],[0,4,6,2,6,4,0,2],[0,5,2,7,4,1,6,3],[0,6,0,6,0,6,0,6],[0,7,4,5,2,3,6,1]],\"a\":7,\"b\":6,\"exponent\":1}。請依 [a,b,exponent,ab,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(7, 6, 1, 6, 6, 6, 是)","explanation_zh":"Wave4第二層核對=(7, 6, 1, 6, 6, 6, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-power-product","seed":202610281065,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,6,7,4,5,0],[2,3,6,5,0,7,4,1],[3,6,5,4,1,0,7,2],[4,7,0,1,6,3,2,5],[5,4,7,0,3,2,1,6],[6,5,4,7,2,1,0,3],[7,0,1,2,5,6,3,4]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,4,6,2,0,4],[0,3,4,1,2,7,6,5],[0,4,6,2,6,4,0,2],[0,5,2,7,4,1,6,3],[0,6,0,6,0,6,0,6],[0,7,4,5,2,3,6,1]],"a":7,"b":6,"exponent":1,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-power-product","semantic_template":"qf-mod28-commring-power-product/{\"a\":7,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,7,4,5,0],[2,3,6,5,0,7,4,1],[3,6,5,4,1,0,7,2],[4,7,0,1,6,3,2,5],[5,4,7,0,3,2,1,6],[6,5,4,7,2,1,0,3],[7,0,1,2,5,6,3,4]],\"b\":6,\"exponent\":1,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,4,6,2,0,4],[0,3,4,1,2,7,6,5],[0,4,6,2,6,4,0,2],[0,5,2,7,4,1,6,3],[0,6,0,6,0,6,0,6],[0,7,4,5,2,3,6,1]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0066","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-power-product；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,0,7,5,4],[2,3,6,5,1,4,7,0],[3,6,5,7,2,0,4,1],[4,0,1,2,7,6,3,5],[5,7,4,0,6,2,1,3],[6,5,7,4,3,1,0,2],[7,4,0,1,5,3,2,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,7,7,2,0,6],[0,3,7,1,5,4,6,2],[0,4,7,5,1,3,6,2],[0,5,2,4,3,1,6,7],[0,6,0,6,6,6,0,0],[0,7,6,2,2,7,0,6]],\"a\":0,\"b\":4,\"exponent\":2}。請依 [a,b,exponent,ab,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(0, 4, 2, 0, 0, 0, 是)","explanation_zh":"Wave4第二層核對=(0, 4, 2, 0, 0, 0, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-power-product","seed":202610281066,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,6,0,7,5,4],[2,3,6,5,1,4,7,0],[3,6,5,7,2,0,4,1],[4,0,1,2,7,6,3,5],[5,7,4,0,6,2,1,3],[6,5,7,4,3,1,0,2],[7,4,0,1,5,3,2,6]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,7,7,2,0,6],[0,3,7,1,5,4,6,2],[0,4,7,5,1,3,6,2],[0,5,2,4,3,1,6,7],[0,6,0,6,6,6,0,0],[0,7,6,2,2,7,0,6]],"a":0,"b":4,"exponent":2,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-power-product","semantic_template":"qf-mod28-commring-power-product/{\"a\":0,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,0,7,5,4],[2,3,6,5,1,4,7,0],[3,6,5,7,2,0,4,1],[4,0,1,2,7,6,3,5],[5,7,4,0,6,2,1,3],[6,5,7,4,3,1,0,2],[7,4,0,1,5,3,2,6]],\"b\":4,\"exponent\":2,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,7,7,2,0,6],[0,3,7,1,5,4,6,2],[0,4,7,5,1,3,6,2],[0,5,2,4,3,1,6,7],[0,6,0,6,6,6,0,0],[0,7,6,2,2,7,0,6]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0067","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-power-product；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,5,0,7,4],[2,3,6,7,0,1,4,5],[3,6,7,4,1,2,5,0],[4,5,0,1,6,7,2,3],[5,0,1,2,7,4,3,6],[6,7,4,5,2,3,0,1],[7,4,5,0,3,6,1,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,4,6,4,0,2],[0,3,4,1,2,7,6,5],[0,4,6,2,6,2,0,4],[0,5,4,7,2,1,6,3],[0,6,0,6,0,6,0,6],[0,7,2,5,4,3,6,1]],\"a\":1,\"b\":2,\"exponent\":3}。請依 [a,b,exponent,ab,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(1, 2, 3, 2, 0, 0, 是)","explanation_zh":"Wave4第二層核對=(1, 2, 3, 2, 0, 0, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-power-product","seed":202610281067,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,6,5,0,7,4],[2,3,6,7,0,1,4,5],[3,6,7,4,1,2,5,0],[4,5,0,1,6,7,2,3],[5,0,1,2,7,4,3,6],[6,7,4,5,2,3,0,1],[7,4,5,0,3,6,1,2]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,4,6,4,0,2],[0,3,4,1,2,7,6,5],[0,4,6,2,6,2,0,4],[0,5,4,7,2,1,6,3],[0,6,0,6,0,6,0,6],[0,7,2,5,4,3,6,1]],"a":1,"b":2,"exponent":3,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-power-product","semantic_template":"qf-mod28-commring-power-product/{\"a\":1,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,5,0,7,4],[2,3,6,7,0,1,4,5],[3,6,7,4,1,2,5,0],[4,5,0,1,6,7,2,3],[5,0,1,2,7,4,3,6],[6,7,4,5,2,3,0,1],[7,4,5,0,3,6,1,2]],\"b\":2,\"exponent\":3,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,4,6,4,0,2],[0,3,4,1,2,7,6,5],[0,4,6,2,6,2,0,4],[0,5,4,7,2,1,6,3],[0,6,0,6,0,6,0,6],[0,7,2,5,4,3,6,1]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0068","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-power-product；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,0,4,7,5],[2,3,6,7,1,0,5,4],[3,6,7,5,2,1,4,0],[4,0,1,2,5,7,3,6],[5,4,0,1,7,6,2,3],[6,7,5,4,3,2,0,1],[7,5,4,0,6,3,1,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,5,5,6,0,2],[0,3,5,1,7,2,6,4],[0,4,5,7,1,2,6,3],[0,5,6,2,2,6,0,5],[0,6,0,6,6,0,0,6],[0,7,2,4,3,5,6,1]],\"a\":2,\"b\":7,\"exponent\":4}。請依 [a,b,exponent,ab,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(2, 7, 4, 2, 0, 0, 是)","explanation_zh":"Wave4第二層核對=(2, 7, 4, 2, 0, 0, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-power-product","seed":202610281068,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,6,0,4,7,5],[2,3,6,7,1,0,5,4],[3,6,7,5,2,1,4,0],[4,0,1,2,5,7,3,6],[5,4,0,1,7,6,2,3],[6,7,5,4,3,2,0,1],[7,5,4,0,6,3,1,2]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,5,5,6,0,2],[0,3,5,1,7,2,6,4],[0,4,5,7,1,2,6,3],[0,5,6,2,2,6,0,5],[0,6,0,6,6,0,0,6],[0,7,2,4,3,5,6,1]],"a":2,"b":7,"exponent":4,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-power-product","semantic_template":"qf-mod28-commring-power-product/{\"a\":2,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,0,4,7,5],[2,3,6,7,1,0,5,4],[3,6,7,5,2,1,4,0],[4,0,1,2,5,7,3,6],[5,4,0,1,7,6,2,3],[6,7,5,4,3,2,0,1],[7,5,4,0,6,3,1,2]],\"b\":7,\"exponent\":4,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,5,5,6,0,2],[0,3,5,1,7,2,6,4],[0,4,5,7,1,2,6,3],[0,5,6,2,2,6,0,5],[0,6,0,6,6,0,0,6],[0,7,2,4,3,5,6,1]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0069","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-power-product；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,5,6,0,4],[2,3,7,4,6,0,1,5],[3,7,4,5,0,1,2,6],[4,5,6,0,2,3,7,1],[5,6,0,1,3,7,4,2],[6,0,1,2,7,4,5,3],[7,4,5,6,1,2,3,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,5,2,7,5,0],[0,3,5,1,6,2,4,7],[0,4,2,6,1,5,3,7],[0,5,7,2,5,7,2,0],[0,6,5,4,3,2,1,7],[0,7,0,7,7,0,7,0]],\"a\":3,\"b\":0,\"exponent\":5}。請依 [a,b,exponent,ab,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(3, 0, 5, 0, 0, 0, 是)","explanation_zh":"Wave4第二層核對=(3, 0, 5, 0, 0, 0, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-power-product","seed":202610281069,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,7,5,6,0,4],[2,3,7,4,6,0,1,5],[3,7,4,5,0,1,2,6],[4,5,6,0,2,3,7,1],[5,6,0,1,3,7,4,2],[6,0,1,2,7,4,5,3],[7,4,5,6,1,2,3,0]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,5,2,7,5,0],[0,3,5,1,6,2,4,7],[0,4,2,6,1,5,3,7],[0,5,7,2,5,7,2,0],[0,6,5,4,3,2,1,7],[0,7,0,7,7,0,7,0]],"a":3,"b":0,"exponent":5,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-power-product","semantic_template":"qf-mod28-commring-power-product/{\"a\":3,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,5,6,0,4],[2,3,7,4,6,0,1,5],[3,7,4,5,0,1,2,6],[4,5,6,0,2,3,7,1],[5,6,0,1,3,7,4,2],[6,0,1,2,7,4,5,3],[7,4,5,6,1,2,3,0]],\"b\":0,\"exponent\":5,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,5,2,7,5,0],[0,3,5,1,6,2,4,7],[0,4,2,6,1,5,3,7],[0,5,7,2,5,7,2,0],[0,6,5,4,3,2,1,7],[0,7,0,7,7,0,7,0]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0070","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-power-product；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,6,0,5,4],[2,3,7,4,5,1,0,6],[3,7,4,6,0,2,1,5],[4,6,5,0,2,7,3,1],[5,0,1,2,7,6,4,3],[6,5,0,1,3,4,7,2],[7,4,6,5,1,3,2,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,6,2,6,7,0],[0,3,6,1,5,4,2,7],[0,4,2,5,1,3,6,7],[0,5,6,4,3,1,2,7],[0,6,7,2,6,2,7,0],[0,7,0,7,7,7,0,0]],\"a\":7,\"b\":3,\"exponent\":6}。請依 [a,b,exponent,ab,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(7, 3, 6, 7, 0, 0, 是)","explanation_zh":"Wave4第二層核對=(7, 3, 6, 7, 0, 0, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-power-product","seed":202610281070,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,7,6,0,5,4],[2,3,7,4,5,1,0,6],[3,7,4,6,0,2,1,5],[4,6,5,0,2,7,3,1],[5,0,1,2,7,6,4,3],[6,5,0,1,3,4,7,2],[7,4,6,5,1,3,2,0]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,6,2,6,7,0],[0,3,6,1,5,4,2,7],[0,4,2,5,1,3,6,7],[0,5,6,4,3,1,2,7],[0,6,7,2,6,2,7,0],[0,7,0,7,7,7,0,0]],"a":7,"b":3,"exponent":6,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-power-product","semantic_template":"qf-mod28-commring-power-product/{\"a\":7,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,6,0,5,4],[2,3,7,4,5,1,0,6],[3,7,4,6,0,2,1,5],[4,6,5,0,2,7,3,1],[5,0,1,2,7,6,4,3],[6,5,0,1,3,4,7,2],[7,4,6,5,1,3,2,0]],\"b\":3,\"exponent\":6,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,6,2,6,7,0],[0,3,6,1,5,4,2,7],[0,4,2,5,1,3,6,7],[0,5,6,4,3,1,2,7],[0,6,7,2,6,2,7,0],[0,7,0,7,7,7,0,0]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0071","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-power-product；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,6,4,0,5],[2,3,7,5,0,6,1,4],[3,7,5,4,1,0,2,6],[4,6,0,1,7,3,5,2],[5,4,6,0,3,2,7,1],[6,0,1,2,5,7,4,3],[7,5,4,6,2,1,3,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,4,7,2,4,0],[0,3,4,1,2,6,5,7],[0,4,7,2,7,4,2,0],[0,5,2,6,4,1,3,7],[0,6,4,5,2,3,1,7],[0,7,0,7,0,7,7,0]],\"a\":5,\"b\":4,\"exponent\":7}。請依 [a,b,exponent,ab,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(5, 4, 7, 4, 0, 0, 是)","explanation_zh":"Wave4第二層核對=(5, 4, 7, 4, 0, 0, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-power-product","seed":202610281071,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,7,6,4,0,5],[2,3,7,5,0,6,1,4],[3,7,5,4,1,0,2,6],[4,6,0,1,7,3,5,2],[5,4,6,0,3,2,7,1],[6,0,1,2,5,7,4,3],[7,5,4,6,2,1,3,0]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,4,7,2,4,0],[0,3,4,1,2,6,5,7],[0,4,7,2,7,4,2,0],[0,5,2,6,4,1,3,7],[0,6,4,5,2,3,1,7],[0,7,0,7,0,7,7,0]],"a":5,"b":4,"exponent":7,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-power-product","semantic_template":"qf-mod28-commring-power-product/{\"a\":5,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,6,4,0,5],[2,3,7,5,0,6,1,4],[3,7,5,4,1,0,2,6],[4,6,0,1,7,3,5,2],[5,4,6,0,3,2,7,1],[6,0,1,2,5,7,4,3],[7,5,4,6,2,1,3,0]],\"b\":4,\"exponent\":7,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,4,7,2,4,0],[0,3,4,1,2,6,5,7],[0,4,7,2,7,4,2,0],[0,5,2,6,4,1,3,7],[0,6,4,5,2,3,1,7],[0,7,0,7,0,7,7,0]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0072","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-power-product；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,0,6,4,5],[2,3,7,5,1,4,0,6],[3,7,5,6,2,0,1,4],[4,0,1,2,6,7,5,3],[5,6,4,0,7,2,3,1],[6,4,0,1,5,3,7,2],[7,5,6,4,3,1,2,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,6,6,2,7,0],[0,3,6,1,5,4,2,7],[0,4,6,5,1,3,2,7],[0,5,2,4,3,1,6,7],[0,6,7,2,2,6,7,0],[0,7,0,7,7,7,0,0]],\"a\":6,\"b\":1,\"exponent\":1}。請依 [a,b,exponent,ab,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(6, 1, 1, 6, 6, 6, 是)","explanation_zh":"Wave4第二層核對=(6, 1, 1, 6, 6, 6, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-power-product","seed":202610281072,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,7,0,6,4,5],[2,3,7,5,1,4,0,6],[3,7,5,6,2,0,1,4],[4,0,1,2,6,7,5,3],[5,6,4,0,7,2,3,1],[6,4,0,1,5,3,7,2],[7,5,6,4,3,1,2,0]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,6,6,2,7,0],[0,3,6,1,5,4,2,7],[0,4,6,5,1,3,2,7],[0,5,2,4,3,1,6,7],[0,6,7,2,2,6,7,0],[0,7,0,7,7,7,0,0]],"a":6,"b":1,"exponent":1,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-power-product","semantic_template":"qf-mod28-commring-power-product/{\"a\":6,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,0,6,4,5],[2,3,7,5,1,4,0,6],[3,7,5,6,2,0,1,4],[4,0,1,2,6,7,5,3],[5,6,4,0,7,2,3,1],[6,4,0,1,5,3,7,2],[7,5,6,4,3,1,2,0]],\"b\":1,\"exponent\":1,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,6,6,2,7,0],[0,3,6,1,5,4,2,7],[0,4,6,5,1,3,2,7],[0,5,2,4,3,1,6,7],[0,6,7,2,2,6,7,0],[0,7,0,7,7,7,0,0]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0073","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-power-product；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,5,0,4,6],[2,3,7,6,0,1,5,4],[3,7,6,4,1,2,0,5],[4,5,0,1,7,6,3,2],[5,0,1,2,6,4,7,3],[6,4,5,0,3,7,2,1],[7,6,4,5,2,3,1,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,4,7,4,2,0],[0,3,4,1,2,6,5,7],[0,4,7,2,7,2,4,0],[0,5,4,6,2,1,3,7],[0,6,2,5,4,3,1,7],[0,7,0,7,0,7,7,0]],\"a\":5,\"b\":7,\"exponent\":2}。請依 [a,b,exponent,ab,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(5, 7, 2, 7, 0, 0, 是)","explanation_zh":"Wave4第二層核對=(5, 7, 2, 7, 0, 0, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-power-product","seed":202610281073,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,7,5,0,4,6],[2,3,7,6,0,1,5,4],[3,7,6,4,1,2,0,5],[4,5,0,1,7,6,3,2],[5,0,1,2,6,4,7,3],[6,4,5,0,3,7,2,1],[7,6,4,5,2,3,1,0]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,4,7,4,2,0],[0,3,4,1,2,6,5,7],[0,4,7,2,7,2,4,0],[0,5,4,6,2,1,3,7],[0,6,2,5,4,3,1,7],[0,7,0,7,0,7,7,0]],"a":5,"b":7,"exponent":2,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-power-product","semantic_template":"qf-mod28-commring-power-product/{\"a\":5,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,5,0,4,6],[2,3,7,6,0,1,5,4],[3,7,6,4,1,2,0,5],[4,5,0,1,7,6,3,2],[5,0,1,2,6,4,7,3],[6,4,5,0,3,7,2,1],[7,6,4,5,2,3,1,0]],\"b\":7,\"exponent\":2,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,4,7,4,2,0],[0,3,4,1,2,6,5,7],[0,4,7,2,7,2,4,0],[0,5,4,6,2,1,3,7],[0,6,2,5,4,3,1,7],[0,7,0,7,0,7,7,0]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0074","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-power-product；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,0,4,5,6],[2,3,7,6,1,0,4,5],[3,7,6,5,2,1,0,4],[4,0,1,2,5,6,7,3],[5,4,0,1,6,7,3,2],[6,5,4,0,7,3,2,1],[7,6,5,4,3,2,1,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,5,5,7,2,0],[0,3,5,1,6,2,4,7],[0,4,5,6,1,2,3,7],[0,5,7,2,2,7,5,0],[0,6,2,4,3,5,1,7],[0,7,0,7,7,0,7,0]],\"a\":0,\"b\":4,\"exponent\":3}。請依 [a,b,exponent,ab,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(0, 4, 3, 0, 0, 0, 是)","explanation_zh":"Wave4第二層核對=(0, 4, 3, 0, 0, 0, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-power-product","seed":202610281074,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,7,0,4,5,6],[2,3,7,6,1,0,4,5],[3,7,6,5,2,1,0,4],[4,0,1,2,5,6,7,3],[5,4,0,1,6,7,3,2],[6,5,4,0,7,3,2,1],[7,6,5,4,3,2,1,0]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,5,5,7,2,0],[0,3,5,1,6,2,4,7],[0,4,5,6,1,2,3,7],[0,5,7,2,2,7,5,0],[0,6,2,4,3,5,1,7],[0,7,0,7,7,0,7,0]],"a":0,"b":4,"exponent":3,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-power-product","semantic_template":"qf-mod28-commring-power-product/{\"a\":0,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,0,4,5,6],[2,3,7,6,1,0,4,5],[3,7,6,5,2,1,0,4],[4,0,1,2,5,6,7,3],[5,4,0,1,6,7,3,2],[6,5,4,0,7,3,2,1],[7,6,5,4,3,2,1,0]],\"b\":4,\"exponent\":3,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,5,5,7,2,0],[0,3,5,1,6,2,4,7],[0,4,5,6,1,2,3,7],[0,5,7,2,2,7,5,0],[0,6,2,4,3,5,1,7],[0,7,0,7,7,0,7,0]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0075","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-power-product；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,4,5,3,6,7,0],[2,4,3,6,5,7,0,1],[3,5,6,0,7,1,2,4],[4,3,5,7,6,0,1,2],[5,6,7,1,0,2,4,3],[6,7,0,2,1,4,3,5],[7,0,1,4,2,3,5,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,3,0,6,2,3,6],[0,3,0,0,3,3,0,3],[0,4,6,3,1,7,2,5],[0,5,2,3,7,1,6,4],[0,6,3,0,2,6,3,2],[0,7,6,3,5,4,2,1]],\"a\":1,\"b\":2,\"exponent\":4}。請依 [a,b,exponent,ab,lhs,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(1, 2, 4, 2, 0, 0, 是)","explanation_zh":"Wave4第二層核對=(1, 2, 4, 2, 0, 0, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-power-product","seed":202610281075,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,4,5,3,6,7,0],[2,4,3,6,5,7,0,1],[3,5,6,0,7,1,2,4],[4,3,5,7,6,0,1,2],[5,6,7,1,0,2,4,3],[6,7,0,2,1,4,3,5],[7,0,1,4,2,3,5,6]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,3,0,6,2,3,6],[0,3,0,0,3,3,0,3],[0,4,6,3,1,7,2,5],[0,5,2,3,7,1,6,4],[0,6,3,0,2,6,3,2],[0,7,6,3,5,4,2,1]],"a":1,"b":2,"exponent":4,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-power-product","semantic_template":"qf-mod28-commring-power-product/{\"a\":1,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,4,5,3,6,7,0],[2,4,3,6,5,7,0,1],[3,5,6,0,7,1,2,4],[4,3,5,7,6,0,1,2],[5,6,7,1,0,2,4,3],[6,7,0,2,1,4,3,5],[7,0,1,4,2,3,5,6]],\"b\":2,\"exponent\":4,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,3,0,6,2,3,6],[0,3,0,0,3,3,0,3],[0,4,6,3,1,7,2,5],[0,5,2,3,7,1,6,4],[0,6,3,0,2,6,3,2],[0,7,6,3,5,4,2,1]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0076","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-binomial-square；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,6,0,2,4,6],[0,3,6,1,4,7,2,5],[0,4,0,4,0,4,0,4],[0,5,2,7,4,1,6,3],[0,6,4,2,0,6,4,2],[0,7,6,5,4,3,2,1]],\"a\":1,\"b\":2}。請依 [a,b,sum,lhs,a2,ab,b2,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(1, 2, 3, 1, 1, 2, 4, 1, 是)","explanation_zh":"Wave4第二層核對=(1, 2, 3, 1, 1, 2, 4, 1, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-binomial-square","seed":202610281076,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,6,0,2,4,6],[0,3,6,1,4,7,2,5],[0,4,0,4,0,4,0,4],[0,5,2,7,4,1,6,3],[0,6,4,2,0,6,4,2],[0,7,6,5,4,3,2,1]],"a":1,"b":2,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-binomial-square","semantic_template":"qf-mod28-commring-binomial-square/{\"a\":1,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],\"b\":2,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,6,0,2,4,6],[0,3,6,1,4,7,2,5],[0,4,0,4,0,4,0,4],[0,5,2,7,4,1,6,3],[0,6,4,2,0,6,4,2],[0,7,6,5,4,3,2,1]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0077","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-binomial-square；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,7,0,6],[2,3,4,5,7,6,1,0],[3,4,5,7,6,0,2,1],[4,5,7,6,0,1,3,2],[5,7,6,0,1,2,4,3],[6,0,1,2,3,4,7,5],[7,6,0,1,2,3,5,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,7,0,2,7,4],[0,3,7,1,4,6,5,2],[0,4,0,4,0,4,4,0],[0,5,2,6,4,1,3,7],[0,6,7,5,4,3,1,2],[0,7,4,2,0,7,2,4]],\"a\":2,\"b\":5}。請依 [a,b,sum,lhs,a2,ab,b2,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(2, 5, 6, 1, 4, 2, 1, 1, 是)","explanation_zh":"Wave4第二層核對=(2, 5, 6, 1, 4, 2, 1, 1, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-binomial-square","seed":202610281077,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,7,0,6],[2,3,4,5,7,6,1,0],[3,4,5,7,6,0,2,1],[4,5,7,6,0,1,3,2],[5,7,6,0,1,2,4,3],[6,0,1,2,3,4,7,5],[7,6,0,1,2,3,5,4]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,7,0,2,7,4],[0,3,7,1,4,6,5,2],[0,4,0,4,0,4,4,0],[0,5,2,6,4,1,3,7],[0,6,7,5,4,3,1,2],[0,7,4,2,0,7,2,4]],"a":2,"b":5,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-binomial-square","semantic_template":"qf-mod28-commring-binomial-square/{\"a\":2,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,7,0,6],[2,3,4,5,7,6,1,0],[3,4,5,7,6,0,2,1],[4,5,7,6,0,1,3,2],[5,7,6,0,1,2,4,3],[6,0,1,2,3,4,7,5],[7,6,0,1,2,3,5,4]],\"b\":5,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,7,0,2,7,4],[0,3,7,1,4,6,5,2],[0,4,0,4,0,4,4,0],[0,5,2,6,4,1,3,7],[0,6,7,5,4,3,1,2],[0,7,4,2,0,7,2,4]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0078","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-binomial-square；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,6,7,5,0],[2,3,4,6,5,0,7,1],[3,4,6,5,7,1,0,2],[4,6,5,7,0,2,1,3],[5,7,0,1,2,4,3,6],[6,5,7,0,1,3,2,4],[7,0,1,2,3,6,4,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,5,0,4,2,5],[0,3,5,1,4,2,7,6],[0,4,0,4,0,0,4,4],[0,5,4,2,0,4,5,2],[0,6,2,7,4,5,1,3],[0,7,5,6,4,2,3,1]],\"a\":3,\"b\":0}。請依 [a,b,sum,lhs,a2,ab,b2,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(3, 0, 3, 1, 1, 0, 0, 1, 是)","explanation_zh":"Wave4第二層核對=(3, 0, 3, 1, 1, 0, 0, 1, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-binomial-square","seed":202610281078,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,4,6,7,5,0],[2,3,4,6,5,0,7,1],[3,4,6,5,7,1,0,2],[4,6,5,7,0,2,1,3],[5,7,0,1,2,4,3,6],[6,5,7,0,1,3,2,4],[7,0,1,2,3,6,4,5]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,5,0,4,2,5],[0,3,5,1,4,2,7,6],[0,4,0,4,0,0,4,4],[0,5,4,2,0,4,5,2],[0,6,2,7,4,5,1,3],[0,7,5,6,4,2,3,1]],"a":3,"b":0,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-binomial-square","semantic_template":"qf-mod28-commring-binomial-square/{\"a\":3,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,6,7,5,0],[2,3,4,6,5,0,7,1],[3,4,6,5,7,1,0,2],[4,6,5,7,0,2,1,3],[5,7,0,1,2,4,3,6],[6,5,7,0,1,3,2,4],[7,0,1,2,3,6,4,5]],\"b\":0,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,5,0,4,2,5],[0,3,5,1,4,2,7,6],[0,4,0,4,0,0,4,4],[0,5,4,2,0,4,5,2],[0,6,2,7,4,5,1,3],[0,7,5,6,4,2,3,1]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0079","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-binomial-square；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,6,0,7,5],[2,3,4,6,7,1,5,0],[3,4,6,7,5,2,0,1],[4,6,7,5,0,3,1,2],[5,0,1,2,3,7,4,6],[6,7,5,0,1,4,2,3],[7,5,0,1,2,6,3,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,7,0,7,2,4],[0,3,7,1,4,6,5,2],[0,4,0,4,0,4,4,0],[0,5,7,6,4,1,3,2],[0,6,2,5,4,3,1,7],[0,7,4,2,0,2,7,4]],\"a\":4,\"b\":3}。請依 [a,b,sum,lhs,a2,ab,b2,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(4, 3, 5, 1, 0, 4, 1, 1, 是)","explanation_zh":"Wave4第二層核對=(4, 3, 5, 1, 0, 4, 1, 1, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-binomial-square","seed":202610281079,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,4,6,0,7,5],[2,3,4,6,7,1,5,0],[3,4,6,7,5,2,0,1],[4,6,7,5,0,3,1,2],[5,0,1,2,3,7,4,6],[6,7,5,0,1,4,2,3],[7,5,0,1,2,6,3,4]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,7,0,7,2,4],[0,3,7,1,4,6,5,2],[0,4,0,4,0,4,4,0],[0,5,7,6,4,1,3,2],[0,6,2,5,4,3,1,7],[0,7,4,2,0,2,7,4]],"a":4,"b":3,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-binomial-square","semantic_template":"qf-mod28-commring-binomial-square/{\"a\":4,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,6,0,7,5],[2,3,4,6,7,1,5,0],[3,4,6,7,5,2,0,1],[4,6,7,5,0,3,1,2],[5,0,1,2,3,7,4,6],[6,7,5,0,1,4,2,3],[7,5,0,1,2,6,3,4]],\"b\":3,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,7,0,7,2,4],[0,3,7,1,4,6,5,2],[0,4,0,4,0,4,4,0],[0,5,7,6,4,1,3,2],[0,6,2,5,4,3,1,7],[0,7,4,2,0,2,7,4]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0080","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-binomial-square；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,7,6,0,5],[2,3,4,7,5,0,1,6],[3,4,7,5,6,1,2,0],[4,7,5,6,0,2,3,1],[5,6,0,1,2,4,7,3],[6,0,1,2,3,7,5,4],[7,5,6,0,1,3,4,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,5,0,4,5,2],[0,3,5,1,4,2,7,6],[0,4,0,4,0,0,4,4],[0,5,4,2,0,4,2,5],[0,6,5,7,4,2,1,3],[0,7,2,6,4,5,3,1]],\"a\":7,\"b\":5}。請依 [a,b,sum,lhs,a2,ab,b2,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(7, 5, 3, 1, 1, 5, 4, 1, 是)","explanation_zh":"Wave4第二層核對=(7, 5, 3, 1, 1, 5, 4, 1, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-binomial-square","seed":202610281080,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,4,7,6,0,5],[2,3,4,7,5,0,1,6],[3,4,7,5,6,1,2,0],[4,7,5,6,0,2,3,1],[5,6,0,1,2,4,7,3],[6,0,1,2,3,7,5,4],[7,5,6,0,1,3,4,2]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,5,0,4,5,2],[0,3,5,1,4,2,7,6],[0,4,0,4,0,0,4,4],[0,5,4,2,0,4,2,5],[0,6,5,7,4,2,1,3],[0,7,2,6,4,5,3,1]],"a":7,"b":5,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-binomial-square","semantic_template":"qf-mod28-commring-binomial-square/{\"a\":7,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,7,6,0,5],[2,3,4,7,5,0,1,6],[3,4,7,5,6,1,2,0],[4,7,5,6,0,2,3,1],[5,6,0,1,2,4,7,3],[6,0,1,2,3,7,5,4],[7,5,6,0,1,3,4,2]],\"b\":5,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,5,0,4,5,2],[0,3,5,1,4,2,7,6],[0,4,0,4,0,0,4,4],[0,5,4,2,0,4,2,5],[0,6,5,7,4,2,1,3],[0,7,2,6,4,5,3,1]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0081","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-binomial-square；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,7,0,5,6],[2,3,4,7,6,1,0,5],[3,4,7,6,5,2,1,0],[4,7,6,5,0,3,2,1],[5,0,1,2,3,6,7,4],[6,5,0,1,2,7,4,3],[7,6,5,0,1,4,3,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,6,0,6,4,2],[0,3,6,1,4,7,2,5],[0,4,0,4,0,4,0,4],[0,5,6,7,4,1,2,3],[0,6,4,2,0,2,4,6],[0,7,2,5,4,3,6,1]],\"a\":6,\"b\":1}。請依 [a,b,sum,lhs,a2,ab,b2,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(6, 1, 5, 1, 4, 6, 1, 1, 是)","explanation_zh":"Wave4第二層核對=(6, 1, 5, 1, 4, 6, 1, 1, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-binomial-square","seed":202610281081,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,4,7,0,5,6],[2,3,4,7,6,1,0,5],[3,4,7,6,5,2,1,0],[4,7,6,5,0,3,2,1],[5,0,1,2,3,6,7,4],[6,5,0,1,2,7,4,3],[7,6,5,0,1,4,3,2]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,6,0,6,4,2],[0,3,6,1,4,7,2,5],[0,4,0,4,0,4,0,4],[0,5,6,7,4,1,2,3],[0,6,4,2,0,2,4,6],[0,7,2,5,4,3,6,1]],"a":6,"b":1,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-binomial-square","semantic_template":"qf-mod28-commring-binomial-square/{\"a\":6,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,7,0,5,6],[2,3,4,7,6,1,0,5],[3,4,7,6,5,2,1,0],[4,7,6,5,0,3,2,1],[5,0,1,2,3,6,7,4],[6,5,0,1,2,7,4,3],[7,6,5,0,1,4,3,2]],\"b\":1,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,6,0,6,4,2],[0,3,6,1,4,7,2,5],[0,4,0,4,0,4,0,4],[0,5,6,7,4,1,2,3],[0,6,4,2,0,2,4,6],[0,7,2,5,4,3,6,1]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0082","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-binomial-square；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,6,4,7,0],[2,3,5,4,7,6,0,1],[3,5,4,6,0,7,1,2],[4,6,7,0,2,1,3,5],[5,4,6,7,1,0,2,3],[6,7,0,1,3,2,5,4],[7,0,1,2,5,3,4,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,6,2,0,5,6],[0,3,6,1,7,5,2,4],[0,4,2,7,1,5,6,3],[0,5,0,5,5,0,0,5],[0,6,5,2,6,0,5,2],[0,7,6,4,3,5,2,1]],\"a\":7,\"b\":5}。請依 [a,b,sum,lhs,a2,ab,b2,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(7, 5, 3, 1, 1, 5, 0, 1, 是)","explanation_zh":"Wave4第二層核對=(7, 5, 3, 1, 1, 5, 0, 1, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-binomial-square","seed":202610281082,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,5,6,4,7,0],[2,3,5,4,7,6,0,1],[3,5,4,6,0,7,1,2],[4,6,7,0,2,1,3,5],[5,4,6,7,1,0,2,3],[6,7,0,1,3,2,5,4],[7,0,1,2,5,3,4,6]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,6,2,0,5,6],[0,3,6,1,7,5,2,4],[0,4,2,7,1,5,6,3],[0,5,0,5,5,0,0,5],[0,6,5,2,6,0,5,2],[0,7,6,4,3,5,2,1]],"a":7,"b":5,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-binomial-square","semantic_template":"qf-mod28-commring-binomial-square/{\"a\":7,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,6,4,7,0],[2,3,5,4,7,6,0,1],[3,5,4,6,0,7,1,2],[4,6,7,0,2,1,3,5],[5,4,6,7,1,0,2,3],[6,7,0,1,3,2,5,4],[7,0,1,2,5,3,4,6]],\"b\":5,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,6,2,0,5,6],[0,3,6,1,7,5,2,4],[0,4,2,7,1,5,6,3],[0,5,0,5,5,0,0,5],[0,6,5,2,6,0,5,2],[0,7,6,4,3,5,2,1]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0083","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-binomial-square；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,7,4,0,6],[2,3,5,4,6,7,1,0],[3,5,4,7,0,6,2,1],[4,7,6,0,2,1,5,3],[5,4,7,6,1,0,3,2],[6,0,1,2,5,3,7,4],[7,6,0,1,3,2,4,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,7,2,0,7,5],[0,3,7,1,6,5,4,2],[0,4,2,6,1,5,3,7],[0,5,0,5,5,0,5,0],[0,6,7,4,3,5,1,2],[0,7,5,2,7,0,2,5]],\"a\":0,\"b\":6}。請依 [a,b,sum,lhs,a2,ab,b2,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(0, 6, 6, 1, 0, 0, 1, 1, 是)","explanation_zh":"Wave4第二層核對=(0, 6, 6, 1, 0, 0, 1, 1, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-binomial-square","seed":202610281083,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,5,7,4,0,6],[2,3,5,4,6,7,1,0],[3,5,4,7,0,6,2,1],[4,7,6,0,2,1,5,3],[5,4,7,6,1,0,3,2],[6,0,1,2,5,3,7,4],[7,6,0,1,3,2,4,5]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,7,2,0,7,5],[0,3,7,1,6,5,4,2],[0,4,2,6,1,5,3,7],[0,5,0,5,5,0,5,0],[0,6,7,4,3,5,1,2],[0,7,5,2,7,0,2,5]],"a":0,"b":6,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-binomial-square","semantic_template":"qf-mod28-commring-binomial-square/{\"a\":0,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,7,4,0,6],[2,3,5,4,6,7,1,0],[3,5,4,7,0,6,2,1],[4,7,6,0,2,1,5,3],[5,4,7,6,1,0,3,2],[6,0,1,2,5,3,7,4],[7,6,0,1,3,2,4,5]],\"b\":6,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,7,2,0,7,5],[0,3,7,1,6,5,4,2],[0,4,2,6,1,5,3,7],[0,5,0,5,5,0,5,0],[0,6,7,4,3,5,1,2],[0,7,5,2,7,0,2,5]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0084","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-binomial-square；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,7,6,4,0],[2,3,5,6,0,4,7,1],[3,5,6,4,1,7,0,2],[4,7,0,1,5,2,3,6],[5,6,4,7,2,0,1,3],[6,4,7,0,3,1,2,5],[7,0,1,2,6,3,5,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,4,5,0,2,4],[0,3,4,1,2,5,7,6],[0,4,5,2,5,0,4,2],[0,5,0,5,0,0,5,5],[0,6,2,7,4,5,1,3],[0,7,4,6,2,5,3,1]],\"a\":1,\"b\":2}。請依 [a,b,sum,lhs,a2,ab,b2,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(1, 2, 3, 1, 1, 2, 5, 1, 是)","explanation_zh":"Wave4第二層核對=(1, 2, 3, 1, 1, 2, 5, 1, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-binomial-square","seed":202610281084,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,5,7,6,4,0],[2,3,5,6,0,4,7,1],[3,5,6,4,1,7,0,2],[4,7,0,1,5,2,3,6],[5,6,4,7,2,0,1,3],[6,4,7,0,3,1,2,5],[7,0,1,2,6,3,5,4]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,4,5,0,2,4],[0,3,4,1,2,5,7,6],[0,4,5,2,5,0,4,2],[0,5,0,5,0,0,5,5],[0,6,2,7,4,5,1,3],[0,7,4,6,2,5,3,1]],"a":1,"b":2,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-binomial-square","semantic_template":"qf-mod28-commring-binomial-square/{\"a\":1,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,7,6,4,0],[2,3,5,6,0,4,7,1],[3,5,6,4,1,7,0,2],[4,7,0,1,5,2,3,6],[5,6,4,7,2,0,1,3],[6,4,7,0,3,1,2,5],[7,0,1,2,6,3,5,4]],\"b\":2,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,4,5,0,2,4],[0,3,4,1,2,5,7,6],[0,4,5,2,5,0,4,2],[0,5,0,5,0,0,5,5],[0,6,2,7,4,5,1,3],[0,7,4,6,2,5,3,1]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0085","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-binomial-square；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,0,6,7,4],[2,3,5,6,1,7,4,0],[3,5,6,7,2,4,0,1],[4,0,1,2,7,3,5,6],[5,6,7,4,3,0,1,2],[6,7,4,0,5,1,2,3],[7,4,0,1,6,2,3,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,7,7,0,2,5],[0,3,7,1,6,5,4,2],[0,4,7,6,1,5,3,2],[0,5,0,5,5,0,5,0],[0,6,2,4,3,5,1,7],[0,7,5,2,2,0,7,5]],\"a\":2,\"b\":6}。請依 [a,b,sum,lhs,a2,ab,b2,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(2, 6, 4, 1, 5, 2, 1, 1, 是)","explanation_zh":"Wave4第二層核對=(2, 6, 4, 1, 5, 2, 1, 1, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-binomial-square","seed":202610281085,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,5,0,6,7,4],[2,3,5,6,1,7,4,0],[3,5,6,7,2,4,0,1],[4,0,1,2,7,3,5,6],[5,6,7,4,3,0,1,2],[6,7,4,0,5,1,2,3],[7,4,0,1,6,2,3,5]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,7,7,0,2,5],[0,3,7,1,6,5,4,2],[0,4,7,6,1,5,3,2],[0,5,0,5,5,0,5,0],[0,6,2,4,3,5,1,7],[0,7,5,2,2,0,7,5]],"a":2,"b":6,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-binomial-square","semantic_template":"qf-mod28-commring-binomial-square/{\"a\":2,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,0,6,7,4],[2,3,5,6,1,7,4,0],[3,5,6,7,2,4,0,1],[4,0,1,2,7,3,5,6],[5,6,7,4,3,0,1,2],[6,7,4,0,5,1,2,3],[7,4,0,1,6,2,3,5]],\"b\":6,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,7,7,0,2,5],[0,3,7,1,6,5,4,2],[0,4,7,6,1,5,3,2],[0,5,0,5,5,0,5,0],[0,6,2,4,3,5,1,7],[0,7,5,2,2,0,7,5]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0086","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-binomial-square；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,6,7,0,4],[2,3,5,7,0,4,1,6],[3,5,7,4,1,6,2,0],[4,6,0,1,5,2,7,3],[5,7,4,6,2,0,3,1],[6,0,1,2,7,3,4,5],[7,4,6,0,3,1,5,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,4,5,0,4,2],[0,3,4,1,2,5,7,6],[0,4,5,2,5,0,2,4],[0,5,0,5,0,0,5,5],[0,6,4,7,2,5,1,3],[0,7,2,6,4,5,3,1]],\"a\":3,\"b\":0}。請依 [a,b,sum,lhs,a2,ab,b2,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(3, 0, 3, 1, 1, 0, 0, 1, 是)","explanation_zh":"Wave4第二層核對=(3, 0, 3, 1, 1, 0, 0, 1, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-binomial-square","seed":202610281086,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,5,6,7,0,4],[2,3,5,7,0,4,1,6],[3,5,7,4,1,6,2,0],[4,6,0,1,5,2,7,3],[5,7,4,6,2,0,3,1],[6,0,1,2,7,3,4,5],[7,4,6,0,3,1,5,2]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,4,5,0,4,2],[0,3,4,1,2,5,7,6],[0,4,5,2,5,0,2,4],[0,5,0,5,0,0,5,5],[0,6,4,7,2,5,1,3],[0,7,2,6,4,5,3,1]],"a":3,"b":0,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-binomial-square","semantic_template":"qf-mod28-commring-binomial-square/{\"a\":3,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,6,7,0,4],[2,3,5,7,0,4,1,6],[3,5,7,4,1,6,2,0],[4,6,0,1,5,2,7,3],[5,7,4,6,2,0,3,1],[6,0,1,2,7,3,4,5],[7,4,6,0,3,1,5,2]],\"b\":0,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,4,5,0,4,2],[0,3,4,1,2,5,7,6],[0,4,5,2,5,0,2,4],[0,5,0,5,0,0,5,5],[0,6,4,7,2,5,1,3],[0,7,2,6,4,5,3,1]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0087","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-binomial-square；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,0,7,4,6],[2,3,5,7,1,6,0,4],[3,5,7,6,2,4,1,0],[4,0,1,2,6,3,7,5],[5,7,6,4,3,0,2,1],[6,4,0,1,7,2,5,3],[7,6,4,0,5,1,3,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,6,6,0,5,2],[0,3,6,1,7,5,2,4],[0,4,6,7,1,5,2,3],[0,5,0,5,5,0,0,5],[0,6,5,2,2,0,5,6],[0,7,2,4,3,5,6,1]],\"a\":5,\"b\":3}。請依 [a,b,sum,lhs,a2,ab,b2,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(5, 3, 4, 1, 0, 5, 1, 1, 是)","explanation_zh":"Wave4第二層核對=(5, 3, 4, 1, 0, 5, 1, 1, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-binomial-square","seed":202610281087,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,5,0,7,4,6],[2,3,5,7,1,6,0,4],[3,5,7,6,2,4,1,0],[4,0,1,2,6,3,7,5],[5,7,6,4,3,0,2,1],[6,4,0,1,7,2,5,3],[7,6,4,0,5,1,3,2]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,6,6,0,5,2],[0,3,6,1,7,5,2,4],[0,4,6,7,1,5,2,3],[0,5,0,5,5,0,0,5],[0,6,5,2,2,0,5,6],[0,7,2,4,3,5,6,1]],"a":5,"b":3,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-binomial-square","semantic_template":"qf-mod28-commring-binomial-square/{\"a\":5,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,5,0,7,4,6],[2,3,5,7,1,6,0,4],[3,5,7,6,2,4,1,0],[4,0,1,2,6,3,7,5],[5,7,6,4,3,0,2,1],[6,4,0,1,7,2,5,3],[7,6,4,0,5,1,3,2]],\"b\":3,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,5,6,6,0,5,2],[0,3,6,1,7,5,2,4],[0,4,6,7,1,5,2,3],[0,5,0,5,5,0,0,5],[0,6,5,2,2,0,5,6],[0,7,2,4,3,5,6,1]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0088","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-binomial-square；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,5,7,4,0],[2,3,6,4,7,0,5,1],[3,6,4,5,0,1,7,2],[4,5,7,0,2,3,1,6],[5,7,0,1,3,6,2,4],[6,4,5,7,1,2,0,3],[7,0,1,2,6,4,3,5]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,5,2,6,0,5],[0,3,5,1,7,2,6,4],[0,4,2,7,1,5,6,3],[0,5,6,2,5,6,0,2],[0,6,0,6,6,0,0,6],[0,7,5,4,3,2,6,1]],\"a\":4,\"b\":5}。請依 [a,b,sum,lhs,a2,ab,b2,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(4, 5, 3, 1, 1, 5, 6, 1, 是)","explanation_zh":"Wave4第二層核對=(4, 5, 3, 1, 1, 5, 6, 1, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-binomial-square","seed":202610281088,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,6,5,7,4,0],[2,3,6,4,7,0,5,1],[3,6,4,5,0,1,7,2],[4,5,7,0,2,3,1,6],[5,7,0,1,3,6,2,4],[6,4,5,7,1,2,0,3],[7,0,1,2,6,4,3,5]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,5,2,6,0,5],[0,3,5,1,7,2,6,4],[0,4,2,7,1,5,6,3],[0,5,6,2,5,6,0,2],[0,6,0,6,6,0,0,6],[0,7,5,4,3,2,6,1]],"a":4,"b":5,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-binomial-square","semantic_template":"qf-mod28-commring-binomial-square/{\"a\":4,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,5,7,4,0],[2,3,6,4,7,0,5,1],[3,6,4,5,0,1,7,2],[4,5,7,0,2,3,1,6],[5,7,0,1,3,6,2,4],[6,4,5,7,1,2,0,3],[7,0,1,2,6,4,3,5]],\"b\":5,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,5,2,6,0,5],[0,3,5,1,7,2,6,4],[0,4,2,7,1,5,6,3],[0,5,6,2,5,6,0,2],[0,6,0,6,6,0,0,6],[0,7,5,4,3,2,6,1]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0089","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-binomial-square；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,7,0,4,5],[2,3,6,4,5,1,7,0],[3,6,4,7,0,2,5,1],[4,7,5,0,2,6,1,3],[5,0,1,2,6,7,3,4],[6,4,7,5,1,3,0,2],[7,5,0,1,3,4,2,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,7,2,7,0,6],[0,3,7,1,5,4,6,2],[0,4,2,5,1,3,6,7],[0,5,7,4,3,1,6,2],[0,6,0,6,6,6,0,0],[0,7,6,2,7,2,0,6]],\"a\":7,\"b\":1}。請依 [a,b,sum,lhs,a2,ab,b2,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(7, 1, 5, 1, 6, 7, 1, 1, 是)","explanation_zh":"Wave4第二層核對=(7, 1, 5, 1, 6, 7, 1, 1, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-binomial-square","seed":202610281089,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,6,7,0,4,5],[2,3,6,4,5,1,7,0],[3,6,4,7,0,2,5,1],[4,7,5,0,2,6,1,3],[5,0,1,2,6,7,3,4],[6,4,7,5,1,3,0,2],[7,5,0,1,3,4,2,6]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,7,2,7,0,6],[0,3,7,1,5,4,6,2],[0,4,2,5,1,3,6,7],[0,5,7,4,3,1,6,2],[0,6,0,6,6,6,0,0],[0,7,6,2,7,2,0,6]],"a":7,"b":1,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-binomial-square","semantic_template":"qf-mod28-commring-binomial-square/{\"a\":7,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,7,0,4,5],[2,3,6,4,5,1,7,0],[3,6,4,7,0,2,5,1],[4,7,5,0,2,6,1,3],[5,0,1,2,6,7,3,4],[6,4,7,5,1,3,0,2],[7,5,0,1,3,4,2,6]],\"b\":1,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,7,2,7,0,6],[0,3,7,1,5,4,6,2],[0,4,2,5,1,3,6,7],[0,5,7,4,3,1,6,2],[0,6,0,6,6,6,0,0],[0,7,6,2,7,2,0,6]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0090","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-binomial-square；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,7,4,5,0],[2,3,6,5,0,7,4,1],[3,6,5,4,1,0,7,2],[4,7,0,1,6,3,2,5],[5,4,7,0,3,2,1,6],[6,5,4,7,2,1,0,3],[7,0,1,2,5,6,3,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,4,6,2,0,4],[0,3,4,1,2,7,6,5],[0,4,6,2,6,4,0,2],[0,5,2,7,4,1,6,3],[0,6,0,6,0,6,0,6],[0,7,4,5,2,3,6,1]],\"a\":7,\"b\":6}。請依 [a,b,sum,lhs,a2,ab,b2,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(7, 6, 3, 1, 1, 6, 0, 1, 是)","explanation_zh":"Wave4第二層核對=(7, 6, 3, 1, 1, 6, 0, 1, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-binomial-square","seed":202610281090,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,6,7,4,5,0],[2,3,6,5,0,7,4,1],[3,6,5,4,1,0,7,2],[4,7,0,1,6,3,2,5],[5,4,7,0,3,2,1,6],[6,5,4,7,2,1,0,3],[7,0,1,2,5,6,3,4]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,4,6,2,0,4],[0,3,4,1,2,7,6,5],[0,4,6,2,6,4,0,2],[0,5,2,7,4,1,6,3],[0,6,0,6,0,6,0,6],[0,7,4,5,2,3,6,1]],"a":7,"b":6,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-binomial-square","semantic_template":"qf-mod28-commring-binomial-square/{\"a\":7,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,7,4,5,0],[2,3,6,5,0,7,4,1],[3,6,5,4,1,0,7,2],[4,7,0,1,6,3,2,5],[5,4,7,0,3,2,1,6],[6,5,4,7,2,1,0,3],[7,0,1,2,5,6,3,4]],\"b\":6,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,4,6,2,0,4],[0,3,4,1,2,7,6,5],[0,4,6,2,6,4,0,2],[0,5,2,7,4,1,6,3],[0,6,0,6,0,6,0,6],[0,7,4,5,2,3,6,1]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0091","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-binomial-square；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,0,7,5,4],[2,3,6,5,1,4,7,0],[3,6,5,7,2,0,4,1],[4,0,1,2,7,6,3,5],[5,7,4,0,6,2,1,3],[6,5,7,4,3,1,0,2],[7,4,0,1,5,3,2,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,7,7,2,0,6],[0,3,7,1,5,4,6,2],[0,4,7,5,1,3,6,2],[0,5,2,4,3,1,6,7],[0,6,0,6,6,6,0,0],[0,7,6,2,2,7,0,6]],\"a\":0,\"b\":4}。請依 [a,b,sum,lhs,a2,ab,b2,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(0, 4, 4, 1, 0, 0, 1, 1, 是)","explanation_zh":"Wave4第二層核對=(0, 4, 4, 1, 0, 0, 1, 1, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-binomial-square","seed":202610281091,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,6,0,7,5,4],[2,3,6,5,1,4,7,0],[3,6,5,7,2,0,4,1],[4,0,1,2,7,6,3,5],[5,7,4,0,6,2,1,3],[6,5,7,4,3,1,0,2],[7,4,0,1,5,3,2,6]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,7,7,2,0,6],[0,3,7,1,5,4,6,2],[0,4,7,5,1,3,6,2],[0,5,2,4,3,1,6,7],[0,6,0,6,6,6,0,0],[0,7,6,2,2,7,0,6]],"a":0,"b":4,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-binomial-square","semantic_template":"qf-mod28-commring-binomial-square/{\"a\":0,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,0,7,5,4],[2,3,6,5,1,4,7,0],[3,6,5,7,2,0,4,1],[4,0,1,2,7,6,3,5],[5,7,4,0,6,2,1,3],[6,5,7,4,3,1,0,2],[7,4,0,1,5,3,2,6]],\"b\":4,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,7,7,2,0,6],[0,3,7,1,5,4,6,2],[0,4,7,5,1,3,6,2],[0,5,2,4,3,1,6,7],[0,6,0,6,6,6,0,0],[0,7,6,2,2,7,0,6]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0092","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-binomial-square；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,5,0,7,4],[2,3,6,7,0,1,4,5],[3,6,7,4,1,2,5,0],[4,5,0,1,6,7,2,3],[5,0,1,2,7,4,3,6],[6,7,4,5,2,3,0,1],[7,4,5,0,3,6,1,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,4,6,4,0,2],[0,3,4,1,2,7,6,5],[0,4,6,2,6,2,0,4],[0,5,4,7,2,1,6,3],[0,6,0,6,0,6,0,6],[0,7,2,5,4,3,6,1]],\"a\":1,\"b\":2}。請依 [a,b,sum,lhs,a2,ab,b2,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(1, 2, 3, 1, 1, 2, 6, 1, 是)","explanation_zh":"Wave4第二層核對=(1, 2, 3, 1, 1, 2, 6, 1, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-binomial-square","seed":202610281092,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,6,5,0,7,4],[2,3,6,7,0,1,4,5],[3,6,7,4,1,2,5,0],[4,5,0,1,6,7,2,3],[5,0,1,2,7,4,3,6],[6,7,4,5,2,3,0,1],[7,4,5,0,3,6,1,2]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,4,6,4,0,2],[0,3,4,1,2,7,6,5],[0,4,6,2,6,2,0,4],[0,5,4,7,2,1,6,3],[0,6,0,6,0,6,0,6],[0,7,2,5,4,3,6,1]],"a":1,"b":2,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-binomial-square","semantic_template":"qf-mod28-commring-binomial-square/{\"a\":1,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,5,0,7,4],[2,3,6,7,0,1,4,5],[3,6,7,4,1,2,5,0],[4,5,0,1,6,7,2,3],[5,0,1,2,7,4,3,6],[6,7,4,5,2,3,0,1],[7,4,5,0,3,6,1,2]],\"b\":2,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,4,6,4,0,2],[0,3,4,1,2,7,6,5],[0,4,6,2,6,2,0,4],[0,5,4,7,2,1,6,3],[0,6,0,6,0,6,0,6],[0,7,2,5,4,3,6,1]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0093","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-binomial-square；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,0,4,7,5],[2,3,6,7,1,0,5,4],[3,6,7,5,2,1,4,0],[4,0,1,2,5,7,3,6],[5,4,0,1,7,6,2,3],[6,7,5,4,3,2,0,1],[7,5,4,0,6,3,1,2]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,5,5,6,0,2],[0,3,5,1,7,2,6,4],[0,4,5,7,1,2,6,3],[0,5,6,2,2,6,0,5],[0,6,0,6,6,0,0,6],[0,7,2,4,3,5,6,1]],\"a\":2,\"b\":7}。請依 [a,b,sum,lhs,a2,ab,b2,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(2, 7, 4, 1, 6, 2, 1, 1, 是)","explanation_zh":"Wave4第二層核對=(2, 7, 4, 1, 6, 2, 1, 1, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-binomial-square","seed":202610281093,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,6,0,4,7,5],[2,3,6,7,1,0,5,4],[3,6,7,5,2,1,4,0],[4,0,1,2,5,7,3,6],[5,4,0,1,7,6,2,3],[6,7,5,4,3,2,0,1],[7,5,4,0,6,3,1,2]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,5,5,6,0,2],[0,3,5,1,7,2,6,4],[0,4,5,7,1,2,6,3],[0,5,6,2,2,6,0,5],[0,6,0,6,6,0,0,6],[0,7,2,4,3,5,6,1]],"a":2,"b":7,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-binomial-square","semantic_template":"qf-mod28-commring-binomial-square/{\"a\":2,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,6,0,4,7,5],[2,3,6,7,1,0,5,4],[3,6,7,5,2,1,4,0],[4,0,1,2,5,7,3,6],[5,4,0,1,7,6,2,3],[6,7,5,4,3,2,0,1],[7,5,4,0,6,3,1,2]],\"b\":7,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,6,5,5,6,0,2],[0,3,5,1,7,2,6,4],[0,4,5,7,1,2,6,3],[0,5,6,2,2,6,0,5],[0,6,0,6,6,0,0,6],[0,7,2,4,3,5,6,1]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0094","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-binomial-square；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,5,6,0,4],[2,3,7,4,6,0,1,5],[3,7,4,5,0,1,2,6],[4,5,6,0,2,3,7,1],[5,6,0,1,3,7,4,2],[6,0,1,2,7,4,5,3],[7,4,5,6,1,2,3,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,5,2,7,5,0],[0,3,5,1,6,2,4,7],[0,4,2,6,1,5,3,7],[0,5,7,2,5,7,2,0],[0,6,5,4,3,2,1,7],[0,7,0,7,7,0,7,0]],\"a\":3,\"b\":0}。請依 [a,b,sum,lhs,a2,ab,b2,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(3, 0, 3, 1, 1, 0, 0, 1, 是)","explanation_zh":"Wave4第二層核對=(3, 0, 3, 1, 1, 0, 0, 1, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-binomial-square","seed":202610281094,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,7,5,6,0,4],[2,3,7,4,6,0,1,5],[3,7,4,5,0,1,2,6],[4,5,6,0,2,3,7,1],[5,6,0,1,3,7,4,2],[6,0,1,2,7,4,5,3],[7,4,5,6,1,2,3,0]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,5,2,7,5,0],[0,3,5,1,6,2,4,7],[0,4,2,6,1,5,3,7],[0,5,7,2,5,7,2,0],[0,6,5,4,3,2,1,7],[0,7,0,7,7,0,7,0]],"a":3,"b":0,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-binomial-square","semantic_template":"qf-mod28-commring-binomial-square/{\"a\":3,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,5,6,0,4],[2,3,7,4,6,0,1,5],[3,7,4,5,0,1,2,6],[4,5,6,0,2,3,7,1],[5,6,0,1,3,7,4,2],[6,0,1,2,7,4,5,3],[7,4,5,6,1,2,3,0]],\"b\":0,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,5,2,7,5,0],[0,3,5,1,6,2,4,7],[0,4,2,6,1,5,3,7],[0,5,7,2,5,7,2,0],[0,6,5,4,3,2,1,7],[0,7,0,7,7,0,7,0]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0095","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-binomial-square；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,6,0,5,4],[2,3,7,4,5,1,0,6],[3,7,4,6,0,2,1,5],[4,6,5,0,2,7,3,1],[5,0,1,2,7,6,4,3],[6,5,0,1,3,4,7,2],[7,4,6,5,1,3,2,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,6,2,6,7,0],[0,3,6,1,5,4,2,7],[0,4,2,5,1,3,6,7],[0,5,6,4,3,1,2,7],[0,6,7,2,6,2,7,0],[0,7,0,7,7,7,0,0]],\"a\":7,\"b\":3}。請依 [a,b,sum,lhs,a2,ab,b2,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(7, 3, 5, 1, 0, 7, 1, 1, 是)","explanation_zh":"Wave4第二層核對=(7, 3, 5, 1, 0, 7, 1, 1, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-binomial-square","seed":202610281095,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,7,6,0,5,4],[2,3,7,4,5,1,0,6],[3,7,4,6,0,2,1,5],[4,6,5,0,2,7,3,1],[5,0,1,2,7,6,4,3],[6,5,0,1,3,4,7,2],[7,4,6,5,1,3,2,0]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,6,2,6,7,0],[0,3,6,1,5,4,2,7],[0,4,2,5,1,3,6,7],[0,5,6,4,3,1,2,7],[0,6,7,2,6,2,7,0],[0,7,0,7,7,7,0,0]],"a":7,"b":3,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-binomial-square","semantic_template":"qf-mod28-commring-binomial-square/{\"a\":7,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,6,0,5,4],[2,3,7,4,5,1,0,6],[3,7,4,6,0,2,1,5],[4,6,5,0,2,7,3,1],[5,0,1,2,7,6,4,3],[6,5,0,1,3,4,7,2],[7,4,6,5,1,3,2,0]],\"b\":3,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,6,2,6,7,0],[0,3,6,1,5,4,2,7],[0,4,2,5,1,3,6,7],[0,5,6,4,3,1,2,7],[0,6,7,2,6,2,7,0],[0,7,0,7,7,7,0,0]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0096","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-binomial-square；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,6,4,0,5],[2,3,7,5,0,6,1,4],[3,7,5,4,1,0,2,6],[4,6,0,1,7,3,5,2],[5,4,6,0,3,2,7,1],[6,0,1,2,5,7,4,3],[7,5,4,6,2,1,3,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,4,7,2,4,0],[0,3,4,1,2,6,5,7],[0,4,7,2,7,4,2,0],[0,5,2,6,4,1,3,7],[0,6,4,5,2,3,1,7],[0,7,0,7,0,7,7,0]],\"a\":5,\"b\":4}。請依 [a,b,sum,lhs,a2,ab,b2,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(5, 4, 3, 1, 1, 4, 7, 1, 是)","explanation_zh":"Wave4第二層核對=(5, 4, 3, 1, 1, 4, 7, 1, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-binomial-square","seed":202610281096,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,7,6,4,0,5],[2,3,7,5,0,6,1,4],[3,7,5,4,1,0,2,6],[4,6,0,1,7,3,5,2],[5,4,6,0,3,2,7,1],[6,0,1,2,5,7,4,3],[7,5,4,6,2,1,3,0]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,4,7,2,4,0],[0,3,4,1,2,6,5,7],[0,4,7,2,7,4,2,0],[0,5,2,6,4,1,3,7],[0,6,4,5,2,3,1,7],[0,7,0,7,0,7,7,0]],"a":5,"b":4,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-binomial-square","semantic_template":"qf-mod28-commring-binomial-square/{\"a\":5,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,6,4,0,5],[2,3,7,5,0,6,1,4],[3,7,5,4,1,0,2,6],[4,6,0,1,7,3,5,2],[5,4,6,0,3,2,7,1],[6,0,1,2,5,7,4,3],[7,5,4,6,2,1,3,0]],\"b\":4,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,4,7,2,4,0],[0,3,4,1,2,6,5,7],[0,4,7,2,7,4,2,0],[0,5,2,6,4,1,3,7],[0,6,4,5,2,3,1,7],[0,7,0,7,0,7,7,0]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0097","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-binomial-square；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,0,6,4,5],[2,3,7,5,1,4,0,6],[3,7,5,6,2,0,1,4],[4,0,1,2,6,7,5,3],[5,6,4,0,7,2,3,1],[6,4,0,1,5,3,7,2],[7,5,6,4,3,1,2,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,6,6,2,7,0],[0,3,6,1,5,4,2,7],[0,4,6,5,1,3,2,7],[0,5,2,4,3,1,6,7],[0,6,7,2,2,6,7,0],[0,7,0,7,7,7,0,0]],\"a\":6,\"b\":1}。請依 [a,b,sum,lhs,a2,ab,b2,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(6, 1, 4, 1, 7, 6, 1, 1, 是)","explanation_zh":"Wave4第二層核對=(6, 1, 4, 1, 7, 6, 1, 1, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-binomial-square","seed":202610281097,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,7,0,6,4,5],[2,3,7,5,1,4,0,6],[3,7,5,6,2,0,1,4],[4,0,1,2,6,7,5,3],[5,6,4,0,7,2,3,1],[6,4,0,1,5,3,7,2],[7,5,6,4,3,1,2,0]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,6,6,2,7,0],[0,3,6,1,5,4,2,7],[0,4,6,5,1,3,2,7],[0,5,2,4,3,1,6,7],[0,6,7,2,2,6,7,0],[0,7,0,7,7,7,0,0]],"a":6,"b":1,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-binomial-square","semantic_template":"qf-mod28-commring-binomial-square/{\"a\":6,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,0,6,4,5],[2,3,7,5,1,4,0,6],[3,7,5,6,2,0,1,4],[4,0,1,2,6,7,5,3],[5,6,4,0,7,2,3,1],[6,4,0,1,5,3,7,2],[7,5,6,4,3,1,2,0]],\"b\":1,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,6,6,2,7,0],[0,3,6,1,5,4,2,7],[0,4,6,5,1,3,2,7],[0,5,2,4,3,1,6,7],[0,6,7,2,2,6,7,0],[0,7,0,7,7,7,0,0]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0098","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-binomial-square；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,5,0,4,6],[2,3,7,6,0,1,5,4],[3,7,6,4,1,2,0,5],[4,5,0,1,7,6,3,2],[5,0,1,2,6,4,7,3],[6,4,5,0,3,7,2,1],[7,6,4,5,2,3,1,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,4,7,4,2,0],[0,3,4,1,2,6,5,7],[0,4,7,2,7,2,4,0],[0,5,4,6,2,1,3,7],[0,6,2,5,4,3,1,7],[0,7,0,7,0,7,7,0]],\"a\":5,\"b\":7}。請依 [a,b,sum,lhs,a2,ab,b2,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(5, 7, 3, 1, 1, 7, 0, 1, 是)","explanation_zh":"Wave4第二層核對=(5, 7, 3, 1, 1, 7, 0, 1, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-binomial-square","seed":202610281098,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,7,5,0,4,6],[2,3,7,6,0,1,5,4],[3,7,6,4,1,2,0,5],[4,5,0,1,7,6,3,2],[5,0,1,2,6,4,7,3],[6,4,5,0,3,7,2,1],[7,6,4,5,2,3,1,0]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,4,7,4,2,0],[0,3,4,1,2,6,5,7],[0,4,7,2,7,2,4,0],[0,5,4,6,2,1,3,7],[0,6,2,5,4,3,1,7],[0,7,0,7,0,7,7,0]],"a":5,"b":7,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-binomial-square","semantic_template":"qf-mod28-commring-binomial-square/{\"a\":5,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,5,0,4,6],[2,3,7,6,0,1,5,4],[3,7,6,4,1,2,0,5],[4,5,0,1,7,6,3,2],[5,0,1,2,6,4,7,3],[6,4,5,0,3,7,2,1],[7,6,4,5,2,3,1,0]],\"b\":7,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,4,7,4,2,0],[0,3,4,1,2,6,5,7],[0,4,7,2,7,2,4,0],[0,5,4,6,2,1,3,7],[0,6,2,5,4,3,1,7],[0,7,0,7,0,7,7,0]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0099","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-binomial-square；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,0,4,5,6],[2,3,7,6,1,0,4,5],[3,7,6,5,2,1,0,4],[4,0,1,2,5,6,7,3],[5,4,0,1,6,7,3,2],[6,5,4,0,7,3,2,1],[7,6,5,4,3,2,1,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,5,5,7,2,0],[0,3,5,1,6,2,4,7],[0,4,5,6,1,2,3,7],[0,5,7,2,2,7,5,0],[0,6,2,4,3,5,1,7],[0,7,0,7,7,0,7,0]],\"a\":0,\"b\":4}。請依 [a,b,sum,lhs,a2,ab,b2,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(0, 4, 4, 1, 0, 0, 1, 1, 是)","explanation_zh":"Wave4第二層核對=(0, 4, 4, 1, 0, 0, 1, 1, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-binomial-square","seed":202610281099,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,3,7,0,4,5,6],[2,3,7,6,1,0,4,5],[3,7,6,5,2,1,0,4],[4,0,1,2,5,6,7,3],[5,4,0,1,6,7,3,2],[6,5,4,0,7,3,2,1],[7,6,5,4,3,2,1,0]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,5,5,7,2,0],[0,3,5,1,6,2,4,7],[0,4,5,6,1,2,3,7],[0,5,7,2,2,7,5,0],[0,6,2,4,3,5,1,7],[0,7,0,7,7,0,7,0]],"a":0,"b":4,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-binomial-square","semantic_template":"qf-mod28-commring-binomial-square/{\"a\":0,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,0,4,5,6],[2,3,7,6,1,0,4,5],[3,7,6,5,2,1,0,4],[4,0,1,2,5,6,7,3],[5,4,0,1,6,7,3,2],[6,5,4,0,7,3,2,1],[7,6,5,4,3,2,1,0]],\"b\":4,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,5,5,7,2,0],[0,3,5,1,6,2,4,7],[0,4,5,6,1,2,3,7],[0,5,7,2,2,7,5,0],[0,6,2,4,3,5,1,7],[0,7,0,7,7,0,7,0]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r01-finite-commutative-ring-0100","batch_id":"batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-commutative-ring"],"title_zh":"有限交換環","stem_zh":"同一完整有限含一環的加法表與乘法表。操作=commring-binomial-square；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,4,5,3,6,7,0],[2,4,3,6,5,7,0,1],[3,5,6,0,7,1,2,4],[4,3,5,7,6,0,1,2],[5,6,7,1,0,2,4,3],[6,7,0,2,1,4,3,5],[7,0,1,4,2,3,5,6]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,3,0,6,2,3,6],[0,3,0,0,3,3,0,3],[0,4,6,3,1,7,2,5],[0,5,2,3,7,1,6,4],[0,6,3,0,2,6,3,2],[0,7,6,3,5,4,2,1]],\"a\":1,\"b\":2}。請依 [a,b,sum,lhs,a2,ab,b2,rhs,equal] 欄序回答。","answer":{"kind":"expression","value":"(1, 2, 4, 1, 1, 2, 3, 1, 是)","explanation_zh":"Wave4第二層核對=(1, 2, 4, 1, 1, 2, 3, 1, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環"]},"variant":{"family_id":"qf-mod28-commring-binomial-square","seed":202610281100,"parameters":{"n":8,"addition":[[0,1,2,3,4,5,6,7],[1,2,4,5,3,6,7,0],[2,4,3,6,5,7,0,1],[3,5,6,0,7,1,2,4],[4,3,5,7,6,0,1,2],[5,6,7,1,0,2,4,3],[6,7,0,2,1,4,3,5],[7,0,1,4,2,3,5,6]],"multiplication":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,3,0,6,2,3,6],[0,3,0,0,3,3,0,3],[0,4,6,3,1,7,2,5],[0,5,2,3,7,1,6,4],[0,6,3,0,2,6,3,2],[0,7,6,3,5,4,2,1]],"a":1,"b":2,"domain":"Complete_finite_commutative_ring_tables","operation":"commring-binomial-square","semantic_template":"qf-mod28-commring-binomial-square/{\"a\":1,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,4,5,3,6,7,0],[2,4,3,6,5,7,0,1],[3,5,6,0,7,1,2,4],[4,3,5,7,6,0,1,2],[5,6,7,1,0,2,4,3],[6,7,0,2,1,4,3,5],[7,0,1,4,2,3,5,6]],\"b\":2,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,3,0,6,2,3,6],[0,3,0,0,3,3,0,3],[0,4,6,3,1,7,2,5],[0,5,2,3,7,1,6,4],[0,6,3,0,2,6,3,2],[0,7,6,3,5,4,2,1]],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}