{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r01-finite-unital-ring-0001","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-qualification；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,4,1,2,4],[0,3,0,3,0,3],[0,4,2,0,4,2],[0,5,4,3,2,1]]}。請依 [n,zero,one,additive_abelian,multiplication_associative,left_distributive,right_distributive,distinct_identities,qualified,first_failure] 欄序回答。","answer":{"kind":"expression","value":"(6, 0, 1, 是, 否, 否, 否, 是, 否, multiplication_associativity)","explanation_zh":"Wave4根節點核對=(6, 0, 1, 是, 否, 否, 否, 是, 否, multiplication_associativity)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-qualification","seed":202610271001,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,4,1,2,4],[0,3,0,3,0,3],[0,4,2,0,4,2],[0,5,4,3,2,1]],"domain":"Complete_finite_unital_ring_tables","operation":"ring-qualification","semantic_template":"qf-mod27-ring-qualification/{\"addition\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,4,1,2,4],[0,3,0,3,0,3],[0,4,2,0,4,2],[0,5,4,3,2,1]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0002","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-qualification；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,2,3,5,0,4],[2,3,5,4,1,0],[3,5,4,0,2,1],[4,0,1,2,5,3],[5,4,0,1,3,2]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,5,0,5,2],[0,3,0,3,3,0],[0,4,5,3,1,2],[0,5,2,0,2,5]]}。請依 [n,zero,one,additive_abelian,multiplication_associative,left_distributive,right_distributive,distinct_identities,qualified,first_failure] 欄序回答。","answer":{"kind":"expression","value":"(6, 0, 1, 是, 是, 是, 是, 是, 是, null)","explanation_zh":"Wave4根節點核對=(6, 0, 1, 是, 是, 是, 是, 是, 是, null)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-qualification","seed":202610271002,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,2,3,5,0,4],[2,3,5,4,1,0],[3,5,4,0,2,1],[4,0,1,2,5,3],[5,4,0,1,3,2]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,5,0,5,2],[0,3,0,3,3,0],[0,4,5,3,1,2],[0,5,2,0,2,5]],"domain":"Complete_finite_unital_ring_tables","operation":"ring-qualification","semantic_template":"qf-mod27-ring-qualification/{\"addition\":[[0,1,2,3,4,5],[1,2,3,5,0,4],[2,3,5,4,1,0],[3,5,4,0,2,1],[4,0,1,2,5,3],[5,4,0,1,3,2]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,5,0,5,2],[0,3,0,3,3,0],[0,4,5,3,1,2],[0,5,2,0,2,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0003","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-qualification；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,2,4,5,3,0],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,3,5,1,0,2],[5,0,1,4,2,3]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,3,2,0,3],[0,3,2,3,0,2],[0,4,0,0,4,4],[0,5,3,2,4,1]]}。請依 [n,zero,one,additive_abelian,multiplication_associative,left_distributive,right_distributive,distinct_identities,qualified,first_failure] 欄序回答。","answer":{"kind":"expression","value":"(6, 0, 1, 是, 是, 是, 是, 是, 是, null)","explanation_zh":"Wave4根節點核對=(6, 0, 1, 是, 是, 是, 是, 是, 是, null)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-qualification","seed":202610271003,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,2,4,5,3,0],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,3,5,1,0,2],[5,0,1,4,2,3]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,3,2,0,3],[0,3,2,3,0,2],[0,4,0,0,4,4],[0,5,3,2,4,1]],"domain":"Complete_finite_unital_ring_tables","operation":"ring-qualification","semantic_template":"qf-mod27-ring-qualification/{\"addition\":[[0,1,2,3,4,5],[1,2,4,5,3,0],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,3,5,1,0,2],[5,0,1,4,2,3]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,3,2,0,3],[0,3,2,3,0,2],[0,4,0,0,4,4],[0,5,3,2,4,1]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0004","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-qualification；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,2,4,0,5,3],[2,4,5,1,3,0],[3,0,1,5,2,4],[4,5,3,2,0,1],[5,3,0,4,1,2]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,5,5,0,2],[0,3,5,1,4,2],[0,4,0,4,4,0],[0,5,2,2,0,5]]}。請依 [n,zero,one,additive_abelian,multiplication_associative,left_distributive,right_distributive,distinct_identities,qualified,first_failure] 欄序回答。","answer":{"kind":"expression","value":"(6, 0, 1, 是, 是, 是, 是, 是, 是, null)","explanation_zh":"Wave4根節點核對=(6, 0, 1, 是, 是, 是, 是, 是, 是, null)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-qualification","seed":202610271004,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,2,4,0,5,3],[2,4,5,1,3,0],[3,0,1,5,2,4],[4,5,3,2,0,1],[5,3,0,4,1,2]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,5,5,0,2],[0,3,5,1,4,2],[0,4,0,4,4,0],[0,5,2,2,0,5]],"domain":"Complete_finite_unital_ring_tables","operation":"ring-qualification","semantic_template":"qf-mod27-ring-qualification/{\"addition\":[[0,1,2,3,4,5],[1,2,4,0,5,3],[2,4,5,1,3,0],[3,0,1,5,2,4],[4,5,3,2,0,1],[5,3,0,4,1,2]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,5,5,0,2],[0,3,5,1,4,2],[0,4,0,4,4,0],[0,5,2,2,0,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0005","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-qualification；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,2,5,4,0,3],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,0,1,5,3,2],[5,3,4,1,2,0]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,3,2,3,0],[0,3,2,3,2,0],[0,4,3,2,1,5],[0,5,0,0,5,5]]}。請依 [n,zero,one,additive_abelian,multiplication_associative,left_distributive,right_distributive,distinct_identities,qualified,first_failure] 欄序回答。","answer":{"kind":"expression","value":"(6, 0, 1, 是, 是, 是, 是, 是, 是, null)","explanation_zh":"Wave4根節點核對=(6, 0, 1, 是, 是, 是, 是, 是, 是, null)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-qualification","seed":202610271005,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,2,5,4,0,3],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,0,1,5,3,2],[5,3,4,1,2,0]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,3,2,3,0],[0,3,2,3,2,0],[0,4,3,2,1,5],[0,5,0,0,5,5]],"domain":"Complete_finite_unital_ring_tables","operation":"ring-qualification","semantic_template":"qf-mod27-ring-qualification/{\"addition\":[[0,1,2,3,4,5],[1,2,5,4,0,3],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,0,1,5,3,2],[5,3,4,1,2,0]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,3,2,3,0],[0,3,2,3,2,0],[0,4,3,2,1,5],[0,5,0,0,5,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0006","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-qualification；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,2,5,0,3,4],[2,5,4,1,0,3],[3,0,1,4,5,2],[4,3,0,5,2,1],[5,4,3,2,1,0]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,4,5,2,0],[0,3,4,1,2,5],[0,4,2,2,4,0],[0,5,0,5,0,5]]}。請依 [n,zero,one,additive_abelian,multiplication_associative,left_distributive,right_distributive,distinct_identities,qualified,first_failure] 欄序回答。","answer":{"kind":"expression","value":"(6, 0, 1, 是, 否, 否, 否, 是, 否, multiplication_associativity)","explanation_zh":"Wave4根節點核對=(6, 0, 1, 是, 否, 否, 否, 是, 否, multiplication_associativity)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-qualification","seed":202610271006,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,2,5,0,3,4],[2,5,4,1,0,3],[3,0,1,4,5,2],[4,3,0,5,2,1],[5,4,3,2,1,0]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,4,5,2,0],[0,3,4,1,2,5],[0,4,2,2,4,0],[0,5,0,5,0,5]],"domain":"Complete_finite_unital_ring_tables","operation":"ring-qualification","semantic_template":"qf-mod27-ring-qualification/{\"addition\":[[0,1,2,3,4,5],[1,2,5,0,3,4],[2,5,4,1,0,3],[3,0,1,4,5,2],[4,3,0,5,2,1],[5,4,3,2,1,0]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,4,5,2,0],[0,3,4,1,2,5],[0,4,2,2,4,0],[0,5,0,5,0,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0007","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-qualification；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,3,4,2,5,0],[2,4,0,5,1,3],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,0,3,1,2,4]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,0,2],[0,3,0,4,3,4],[0,4,0,3,4,3],[0,5,2,4,3,1]]}。請依 [n,zero,one,additive_abelian,multiplication_associative,left_distributive,right_distributive,distinct_identities,qualified,first_failure] 欄序回答。","answer":{"kind":"expression","value":"(6, 0, 1, 是, 是, 是, 是, 是, 是, null)","explanation_zh":"Wave4根節點核對=(6, 0, 1, 是, 是, 是, 是, 是, 是, null)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-qualification","seed":202610271007,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,3,4,2,5,0],[2,4,0,5,1,3],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,0,3,1,2,4]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,0,2],[0,3,0,4,3,4],[0,4,0,3,4,3],[0,5,2,4,3,1]],"domain":"Complete_finite_unital_ring_tables","operation":"ring-qualification","semantic_template":"qf-mod27-ring-qualification/{\"addition\":[[0,1,2,3,4,5],[1,3,4,2,5,0],[2,4,0,5,1,3],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,0,3,1,2,4]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,0,2],[0,3,0,4,3,4],[0,4,0,3,4,3],[0,5,2,4,3,1]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0008","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-qualification；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,3,5,2,0,4],[2,5,0,4,3,1],[3,2,4,5,1,0],[4,0,3,1,5,2],[5,4,1,0,2,3]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,2,0],[0,3,0,5,5,3],[0,4,2,5,1,3],[0,5,0,3,3,5]]}。請依 [n,zero,one,additive_abelian,multiplication_associative,left_distributive,right_distributive,distinct_identities,qualified,first_failure] 欄序回答。","answer":{"kind":"expression","value":"(6, 0, 1, 是, 是, 是, 是, 是, 是, null)","explanation_zh":"Wave4根節點核對=(6, 0, 1, 是, 是, 是, 是, 是, 是, null)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-qualification","seed":202610271008,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,3,5,2,0,4],[2,5,0,4,3,1],[3,2,4,5,1,0],[4,0,3,1,5,2],[5,4,1,0,2,3]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,2,0],[0,3,0,5,5,3],[0,4,2,5,1,3],[0,5,0,3,3,5]],"domain":"Complete_finite_unital_ring_tables","operation":"ring-qualification","semantic_template":"qf-mod27-ring-qualification/{\"addition\":[[0,1,2,3,4,5],[1,3,5,2,0,4],[2,5,0,4,3,1],[3,2,4,5,1,0],[4,0,3,1,5,2],[5,4,1,0,2,3]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,2,0],[0,3,0,5,5,3],[0,4,2,5,1,3],[0,5,0,3,3,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0009","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-qualification；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,3,5,4,2,0],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,2,1,5,0,3],[5,0,4,1,3,2]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,3,0,3],[0,3,3,2,0,2],[0,4,0,0,4,4],[0,5,3,2,4,1]]}。請依 [n,zero,one,additive_abelian,multiplication_associative,left_distributive,right_distributive,distinct_identities,qualified,first_failure] 欄序回答。","answer":{"kind":"expression","value":"(6, 0, 1, 是, 是, 是, 是, 是, 是, null)","explanation_zh":"Wave4根節點核對=(6, 0, 1, 是, 是, 是, 是, 是, 是, null)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-qualification","seed":202610271009,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,3,5,4,2,0],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,2,1,5,0,3],[5,0,4,1,3,2]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,3,0,3],[0,3,3,2,0,2],[0,4,0,0,4,4],[0,5,3,2,4,1]],"domain":"Complete_finite_unital_ring_tables","operation":"ring-qualification","semantic_template":"qf-mod27-ring-qualification/{\"addition\":[[0,1,2,3,4,5],[1,3,5,4,2,0],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,2,1,5,0,3],[5,0,4,1,3,2]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,3,0,3],[0,3,3,2,0,2],[0,4,0,0,4,4],[0,5,3,2,4,1]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0010","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-qualification；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,3,0,4,5,2],[2,0,5,1,3,4],[3,4,1,5,2,0],[4,5,3,2,0,1],[5,2,4,0,1,3]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,5,4,3],[0,3,5,5,0,3],[0,4,4,0,4,0],[0,5,3,3,0,5]]}。請依 [n,zero,one,additive_abelian,multiplication_associative,left_distributive,right_distributive,distinct_identities,qualified,first_failure] 欄序回答。","answer":{"kind":"expression","value":"(6, 0, 1, 是, 是, 是, 是, 是, 是, null)","explanation_zh":"Wave4根節點核對=(6, 0, 1, 是, 是, 是, 是, 是, 是, null)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-qualification","seed":202610271010,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,3,0,4,5,2],[2,0,5,1,3,4],[3,4,1,5,2,0],[4,5,3,2,0,1],[5,2,4,0,1,3]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,5,4,3],[0,3,5,5,0,3],[0,4,4,0,4,0],[0,5,3,3,0,5]],"domain":"Complete_finite_unital_ring_tables","operation":"ring-qualification","semantic_template":"qf-mod27-ring-qualification/{\"addition\":[[0,1,2,3,4,5],[1,3,0,4,5,2],[2,0,5,1,3,4],[3,4,1,5,2,0],[4,5,3,2,0,1],[5,2,4,0,1,3]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,5,4,3],[0,3,5,5,0,3],[0,4,4,0,4,0],[0,5,3,3,0,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0011","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-qualification；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,3,4,5,0,2],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,0,5,1,2,3],[5,2,1,4,3,0]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,4,3,0],[0,3,3,2,2,0],[0,4,3,2,1,5],[0,5,0,0,5,5]]}。請依 [n,zero,one,additive_abelian,multiplication_associative,left_distributive,right_distributive,distinct_identities,qualified,first_failure] 欄序回答。","answer":{"kind":"expression","value":"(6, 0, 1, 是, 否, 否, 否, 是, 否, multiplication_associativity)","explanation_zh":"Wave4根節點核對=(6, 0, 1, 是, 否, 否, 否, 是, 否, multiplication_associativity)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-qualification","seed":202610271011,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,3,4,5,0,2],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,0,5,1,2,3],[5,2,1,4,3,0]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,4,3,0],[0,3,3,2,2,0],[0,4,3,2,1,5],[0,5,0,0,5,5]],"domain":"Complete_finite_unital_ring_tables","operation":"ring-qualification","semantic_template":"qf-mod27-ring-qualification/{\"addition\":[[0,1,2,3,4,5],[1,3,4,5,0,2],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,0,5,1,2,3],[5,2,1,4,3,0]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,4,3,0],[0,3,3,2,2,0],[0,4,3,2,1,5],[0,5,0,0,5,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0012","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-qualification；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,3,0,5,2,4],[2,0,4,1,5,3],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,4,3,2,1,0]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,4,3,5],[0,3,4,4,3,0],[0,4,3,3,4,0],[0,5,5,0,0,5]]}。請依 [n,zero,one,additive_abelian,multiplication_associative,left_distributive,right_distributive,distinct_identities,qualified,first_failure] 欄序回答。","answer":{"kind":"expression","value":"(6, 0, 1, 是, 是, 是, 是, 是, 是, null)","explanation_zh":"Wave4根節點核對=(6, 0, 1, 是, 是, 是, 是, 是, 是, null)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-qualification","seed":202610271012,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,3,0,5,2,4],[2,0,4,1,5,3],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,4,3,2,1,0]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,4,3,5],[0,3,4,4,3,0],[0,4,3,3,4,0],[0,5,5,0,0,5]],"domain":"Complete_finite_unital_ring_tables","operation":"ring-qualification","semantic_template":"qf-mod27-ring-qualification/{\"addition\":[[0,1,2,3,4,5],[1,3,0,5,2,4],[2,0,4,1,5,3],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,4,3,2,1,0]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,4,3,5],[0,3,4,4,3,0],[0,4,3,3,4,0],[0,5,5,0,0,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0013","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-qualification；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,4,3,5,2,0],[2,3,0,1,5,4],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,0,4,2,1,3]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,0,2],[0,3,0,3,4,4],[0,4,0,4,3,3],[0,5,2,4,3,1]]}。請依 [n,zero,one,additive_abelian,multiplication_associative,left_distributive,right_distributive,distinct_identities,qualified,first_failure] 欄序回答。","answer":{"kind":"expression","value":"(6, 0, 1, 是, 是, 是, 是, 是, 是, null)","explanation_zh":"Wave4根節點核對=(6, 0, 1, 是, 是, 是, 是, 是, 是, null)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-qualification","seed":202610271013,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,4,3,5,2,0],[2,3,0,1,5,4],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,0,4,2,1,3]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,0,2],[0,3,0,3,4,4],[0,4,0,4,3,3],[0,5,2,4,3,1]],"domain":"Complete_finite_unital_ring_tables","operation":"ring-qualification","semantic_template":"qf-mod27-ring-qualification/{\"addition\":[[0,1,2,3,4,5],[1,4,3,5,2,0],[2,3,0,1,5,4],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,0,4,2,1,3]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,0,2],[0,3,0,3,4,4],[0,4,0,4,3,3],[0,5,2,4,3,1]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0014","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-qualification；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,4,5,0,2,3],[2,5,0,4,3,1],[3,0,4,5,1,2],[4,2,3,1,5,0],[5,3,1,2,0,4]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,2,0,0],[0,3,2,1,5,4],[0,4,0,5,5,4],[0,5,0,4,4,5]]}。請依 [n,zero,one,additive_abelian,multiplication_associative,left_distributive,right_distributive,distinct_identities,qualified,first_failure] 欄序回答。","answer":{"kind":"expression","value":"(6, 0, 1, 是, 是, 是, 是, 是, 是, null)","explanation_zh":"Wave4根節點核對=(6, 0, 1, 是, 是, 是, 是, 是, 是, null)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-qualification","seed":202610271014,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,4,5,0,2,3],[2,5,0,4,3,1],[3,0,4,5,1,2],[4,2,3,1,5,0],[5,3,1,2,0,4]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,2,0,0],[0,3,2,1,5,4],[0,4,0,5,5,4],[0,5,0,4,4,5]],"domain":"Complete_finite_unital_ring_tables","operation":"ring-qualification","semantic_template":"qf-mod27-ring-qualification/{\"addition\":[[0,1,2,3,4,5],[1,4,5,0,2,3],[2,5,0,4,3,1],[3,0,4,5,1,2],[4,2,3,1,5,0],[5,3,1,2,0,4]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,2,0,0],[0,3,2,1,5,4],[0,4,0,5,5,4],[0,5,0,4,4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0015","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-qualification；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,4,5,2,3,0],[2,5,4,1,0,3],[3,2,1,0,5,4],[4,3,0,5,2,1],[5,0,3,4,1,2]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,4,4],[0,3,0,3,0,3],[0,4,4,0,2,2],[0,5,4,3,2,1]]}。請依 [n,zero,one,additive_abelian,multiplication_associative,left_distributive,right_distributive,distinct_identities,qualified,first_failure] 欄序回答。","answer":{"kind":"expression","value":"(6, 0, 1, 是, 是, 是, 是, 是, 是, null)","explanation_zh":"Wave4根節點核對=(6, 0, 1, 是, 是, 是, 是, 是, 是, null)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-qualification","seed":202610271015,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,4,5,2,3,0],[2,5,4,1,0,3],[3,2,1,0,5,4],[4,3,0,5,2,1],[5,0,3,4,1,2]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,4,4],[0,3,0,3,0,3],[0,4,4,0,2,2],[0,5,4,3,2,1]],"domain":"Complete_finite_unital_ring_tables","operation":"ring-qualification","semantic_template":"qf-mod27-ring-qualification/{\"addition\":[[0,1,2,3,4,5],[1,4,5,2,3,0],[2,5,4,1,0,3],[3,2,1,0,5,4],[4,3,0,5,2,1],[5,0,3,4,1,2]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,4,4],[0,3,0,3,0,3],[0,4,4,0,2,2],[0,5,4,3,2,1]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0016","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-qualification；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,4,0,5,3,2],[2,0,5,4,1,3],[3,5,4,0,2,1],[4,3,1,2,5,0],[5,2,3,1,0,4]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,4,5,4],[0,3,3,3,0,0],[0,4,5,0,5,4],[0,5,4,0,4,5]]}。請依 [n,zero,one,additive_abelian,multiplication_associative,left_distributive,right_distributive,distinct_identities,qualified,first_failure] 欄序回答。","answer":{"kind":"expression","value":"(6, 0, 1, 是, 否, 否, 否, 是, 否, multiplication_associativity)","explanation_zh":"Wave4根節點核對=(6, 0, 1, 是, 否, 否, 否, 是, 否, multiplication_associativity)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-qualification","seed":202610271016,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,4,0,5,3,2],[2,0,5,4,1,3],[3,5,4,0,2,1],[4,3,1,2,5,0],[5,2,3,1,0,4]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,4,5,4],[0,3,3,3,0,0],[0,4,5,0,5,4],[0,5,4,0,4,5]],"domain":"Complete_finite_unital_ring_tables","operation":"ring-qualification","semantic_template":"qf-mod27-ring-qualification/{\"addition\":[[0,1,2,3,4,5],[1,4,0,5,3,2],[2,0,5,4,1,3],[3,5,4,0,2,1],[4,3,1,2,5,0],[5,2,3,1,0,4]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,4,5,4],[0,3,3,3,0,0],[0,4,5,0,5,4],[0,5,4,0,4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0017","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-qualification；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,4,3,0,5,2],[2,3,4,5,0,1],[3,0,5,2,1,4],[4,5,0,1,2,3],[5,2,1,4,3,0]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,4,4,0],[0,3,4,1,2,5],[0,4,4,2,2,0],[0,5,0,5,0,5]]}。請依 [n,zero,one,additive_abelian,multiplication_associative,left_distributive,right_distributive,distinct_identities,qualified,first_failure] 欄序回答。","answer":{"kind":"expression","value":"(6, 0, 1, 是, 是, 是, 是, 是, 是, null)","explanation_zh":"Wave4根節點核對=(6, 0, 1, 是, 是, 是, 是, 是, 是, null)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-qualification","seed":202610271017,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,4,3,0,5,2],[2,3,4,5,0,1],[3,0,5,2,1,4],[4,5,0,1,2,3],[5,2,1,4,3,0]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,4,4,0],[0,3,4,1,2,5],[0,4,4,2,2,0],[0,5,0,5,0,5]],"domain":"Complete_finite_unital_ring_tables","operation":"ring-qualification","semantic_template":"qf-mod27-ring-qualification/{\"addition\":[[0,1,2,3,4,5],[1,4,3,0,5,2],[2,3,4,5,0,1],[3,0,5,2,1,4],[4,5,0,1,2,3],[5,2,1,4,3,0]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,4,4,0],[0,3,4,1,2,5],[0,4,4,2,2,0],[0,5,0,5,0,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0018","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-qualification；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,4,0,2,5,3],[2,0,3,5,1,4],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,3,4,1,2,0]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,4,3,5],[0,3,4,3,4,0],[0,4,3,4,3,0],[0,5,5,0,0,5]]}。請依 [n,zero,one,additive_abelian,multiplication_associative,left_distributive,right_distributive,distinct_identities,qualified,first_failure] 欄序回答。","answer":{"kind":"expression","value":"(6, 0, 1, 是, 是, 是, 是, 是, 是, null)","explanation_zh":"Wave4根節點核對=(6, 0, 1, 是, 是, 是, 是, 是, 是, null)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-qualification","seed":202610271018,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,4,0,2,5,3],[2,0,3,5,1,4],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,3,4,1,2,0]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,4,3,5],[0,3,4,3,4,0],[0,4,3,4,3,0],[0,5,5,0,0,5]],"domain":"Complete_finite_unital_ring_tables","operation":"ring-qualification","semantic_template":"qf-mod27-ring-qualification/{\"addition\":[[0,1,2,3,4,5],[1,4,0,2,5,3],[2,0,3,5,1,4],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,3,4,1,2,0]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,4,3,5],[0,3,4,3,4,0],[0,4,3,4,3,0],[0,5,5,0,0,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0019","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-qualification；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,5,3,4,0,2],[2,3,0,1,5,4],[3,4,1,5,2,0],[4,0,5,2,3,1],[5,2,4,0,1,3]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,2,0],[0,3,0,3,5,5],[0,4,2,5,1,3],[0,5,0,5,3,3]]}。請依 [n,zero,one,additive_abelian,multiplication_associative,left_distributive,right_distributive,distinct_identities,qualified,first_failure] 欄序回答。","answer":{"kind":"expression","value":"(6, 0, 1, 是, 是, 是, 是, 是, 是, null)","explanation_zh":"Wave4根節點核對=(6, 0, 1, 是, 是, 是, 是, 是, 是, null)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-qualification","seed":202610271019,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,5,3,4,0,2],[2,3,0,1,5,4],[3,4,1,5,2,0],[4,0,5,2,3,1],[5,2,4,0,1,3]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,2,0],[0,3,0,3,5,5],[0,4,2,5,1,3],[0,5,0,5,3,3]],"domain":"Complete_finite_unital_ring_tables","operation":"ring-qualification","semantic_template":"qf-mod27-ring-qualification/{\"addition\":[[0,1,2,3,4,5],[1,5,3,4,0,2],[2,3,0,1,5,4],[3,4,1,5,2,0],[4,0,5,2,3,1],[5,2,4,0,1,3]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,2,0],[0,3,0,3,5,5],[0,4,2,5,1,3],[0,5,0,5,3,3]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0020","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-qualification；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,5,4,0,3,2],[2,4,0,5,1,3],[3,0,5,4,2,1],[4,3,1,2,5,0],[5,2,3,1,0,4]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,2,0,0],[0,3,2,1,5,4],[0,4,0,5,4,5],[0,5,0,4,5,4]]}。請依 [n,zero,one,additive_abelian,multiplication_associative,left_distributive,right_distributive,distinct_identities,qualified,first_failure] 欄序回答。","answer":{"kind":"expression","value":"(6, 0, 1, 是, 是, 是, 是, 是, 是, null)","explanation_zh":"Wave4根節點核對=(6, 0, 1, 是, 是, 是, 是, 是, 是, null)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-qualification","seed":202610271020,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,5,4,0,3,2],[2,4,0,5,1,3],[3,0,5,4,2,1],[4,3,1,2,5,0],[5,2,3,1,0,4]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,2,0,0],[0,3,2,1,5,4],[0,4,0,5,4,5],[0,5,0,4,5,4]],"domain":"Complete_finite_unital_ring_tables","operation":"ring-qualification","semantic_template":"qf-mod27-ring-qualification/{\"addition\":[[0,1,2,3,4,5],[1,5,4,0,3,2],[2,4,0,5,1,3],[3,0,5,4,2,1],[4,3,1,2,5,0],[5,2,3,1,0,4]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,2,0,0],[0,3,2,1,5,4],[0,4,0,5,4,5],[0,5,0,4,5,4]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0021","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-qualification；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,5,4,2,0,3],[2,4,5,1,3,0],[3,2,1,0,5,4],[4,0,3,5,2,1],[5,3,0,4,1,2]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,1,5,5],[0,3,0,3,3,0],[0,4,5,3,1,2],[0,5,5,0,2,2]]}。請依 [n,zero,one,additive_abelian,multiplication_associative,left_distributive,right_distributive,distinct_identities,qualified,first_failure] 欄序回答。","answer":{"kind":"expression","value":"(6, 0, 1, 是, 否, 否, 否, 是, 否, multiplication_associativity)","explanation_zh":"Wave4根節點核對=(6, 0, 1, 是, 否, 否, 否, 是, 否, multiplication_associativity)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-qualification","seed":202610271021,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,5,4,2,0,3],[2,4,5,1,3,0],[3,2,1,0,5,4],[4,0,3,5,2,1],[5,3,0,4,1,2]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,1,5,5],[0,3,0,3,3,0],[0,4,5,3,1,2],[0,5,5,0,2,2]],"domain":"Complete_finite_unital_ring_tables","operation":"ring-qualification","semantic_template":"qf-mod27-ring-qualification/{\"addition\":[[0,1,2,3,4,5],[1,5,4,2,0,3],[2,4,5,1,3,0],[3,2,1,0,5,4],[4,0,3,5,2,1],[5,3,0,4,1,2]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,1,5,5],[0,3,0,3,3,0],[0,4,5,3,1,2],[0,5,5,0,2,2]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0022","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-qualification；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,5,0,4,2,3],[2,0,4,5,3,1],[3,4,5,0,1,2],[4,2,3,1,5,0],[5,3,1,2,0,4]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,3,5,4],[0,3,3,3,0,0],[0,4,5,0,4,5],[0,5,4,0,5,4]]}。請依 [n,zero,one,additive_abelian,multiplication_associative,left_distributive,right_distributive,distinct_identities,qualified,first_failure] 欄序回答。","answer":{"kind":"expression","value":"(6, 0, 1, 是, 是, 是, 是, 是, 是, null)","explanation_zh":"Wave4根節點核對=(6, 0, 1, 是, 是, 是, 是, 是, 是, null)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-qualification","seed":202610271022,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,5,0,4,2,3],[2,0,4,5,3,1],[3,4,5,0,1,2],[4,2,3,1,5,0],[5,3,1,2,0,4]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,3,5,4],[0,3,3,3,0,0],[0,4,5,0,4,5],[0,5,4,0,5,4]],"domain":"Complete_finite_unital_ring_tables","operation":"ring-qualification","semantic_template":"qf-mod27-ring-qualification/{\"addition\":[[0,1,2,3,4,5],[1,5,0,4,2,3],[2,0,4,5,3,1],[3,4,5,0,1,2],[4,2,3,1,5,0],[5,3,1,2,0,4]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,3,5,4],[0,3,3,3,0,0],[0,4,5,0,4,5],[0,5,4,0,5,4]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0023","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-qualification；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,5,3,0,2,4],[2,3,5,4,1,0],[3,0,4,2,5,1],[4,2,1,5,0,3],[5,4,0,1,3,2]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,5,0,5],[0,3,5,1,4,2],[0,4,0,4,4,0],[0,5,5,2,0,2]]}。請依 [n,zero,one,additive_abelian,multiplication_associative,left_distributive,right_distributive,distinct_identities,qualified,first_failure] 欄序回答。","answer":{"kind":"expression","value":"(6, 0, 1, 是, 是, 是, 是, 是, 是, null)","explanation_zh":"Wave4根節點核對=(6, 0, 1, 是, 是, 是, 是, 是, 是, null)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-qualification","seed":202610271023,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,5,3,0,2,4],[2,3,5,4,1,0],[3,0,4,2,5,1],[4,2,1,5,0,3],[5,4,0,1,3,2]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,5,0,5],[0,3,5,1,4,2],[0,4,0,4,4,0],[0,5,5,2,0,2]],"domain":"Complete_finite_unital_ring_tables","operation":"ring-qualification","semantic_template":"qf-mod27-ring-qualification/{\"addition\":[[0,1,2,3,4,5],[1,5,3,0,2,4],[2,3,5,4,1,0],[3,0,4,2,5,1],[4,2,1,5,0,3],[5,4,0,1,3,2]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,5,0,5],[0,3,5,1,4,2],[0,4,0,4,4,0],[0,5,5,2,0,2]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0024","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-qualification；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,5,0,2,3,4],[2,0,3,4,5,1],[3,2,4,5,1,0],[4,3,5,1,0,2],[5,4,1,0,2,3]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,5,4,3],[0,3,5,3,0,5],[0,4,4,0,4,0],[0,5,3,5,0,3]]}。請依 [n,zero,one,additive_abelian,multiplication_associative,left_distributive,right_distributive,distinct_identities,qualified,first_failure] 欄序回答。","answer":{"kind":"expression","value":"(6, 0, 1, 是, 是, 是, 是, 是, 是, null)","explanation_zh":"Wave4根節點核對=(6, 0, 1, 是, 是, 是, 是, 是, 是, null)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-qualification","seed":202610271024,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,5,0,2,3,4],[2,0,3,4,5,1],[3,2,4,5,1,0],[4,3,5,1,0,2],[5,4,1,0,2,3]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,5,4,3],[0,3,5,3,0,5],[0,4,4,0,4,0],[0,5,3,5,0,3]],"domain":"Complete_finite_unital_ring_tables","operation":"ring-qualification","semantic_template":"qf-mod27-ring-qualification/{\"addition\":[[0,1,2,3,4,5],[1,5,0,2,3,4],[2,0,3,4,5,1],[3,2,4,5,1,0],[4,3,5,1,0,2],[5,4,1,0,2,3]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,5,4,3],[0,3,5,3,0,5],[0,4,4,0,4,0],[0,5,3,5,0,3]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0025","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-qualification；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,4,3,5,0,2],[2,3,1,4,5,0],[3,5,4,0,2,1],[4,0,5,2,1,3],[5,2,0,1,3,4]],\"multiplication\":[[0,0,0,0,0,0],[0,4,1,0,1,4],[0,1,2,3,4,5],[0,0,3,3,0,3],[0,1,4,0,4,1],[0,4,5,3,1,2]]}。請依 [n,zero,one,additive_abelian,multiplication_associative,left_distributive,right_distributive,distinct_identities,qualified,first_failure] 欄序回答。","answer":{"kind":"expression","value":"(6, 0, 2, 是, 是, 是, 是, 是, 是, null)","explanation_zh":"Wave4根節點核對=(6, 0, 2, 是, 是, 是, 是, 是, 是, null)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-qualification","seed":202610271025,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,4,3,5,0,2],[2,3,1,4,5,0],[3,5,4,0,2,1],[4,0,5,2,1,3],[5,2,0,1,3,4]],"multiplication":[[0,0,0,0,0,0],[0,4,1,0,1,4],[0,1,2,3,4,5],[0,0,3,3,0,3],[0,1,4,0,4,1],[0,4,5,3,1,2]],"domain":"Complete_finite_unital_ring_tables","operation":"ring-qualification","semantic_template":"qf-mod27-ring-qualification/{\"addition\":[[0,1,2,3,4,5],[1,4,3,5,0,2],[2,3,1,4,5,0],[3,5,4,0,2,1],[4,0,5,2,1,3],[5,2,0,1,3,4]],\"multiplication\":[[0,0,0,0,0,0],[0,4,1,0,1,4],[0,1,2,3,4,5],[0,0,3,3,0,3],[0,1,4,0,4,1],[0,4,5,3,1,2]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0026","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-additive-inverse；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,4,0,2,4],[0,3,0,3,0,3],[0,4,2,0,4,2],[0,5,4,3,2,1]],\"element\":1}。請依 [element,zero,inverse,left_sum,right_sum] 欄序回答。","answer":{"kind":"expression","value":"(1, 0, 5, 0, 0)","explanation_zh":"Wave4根節點核對=(1, 0, 5, 0, 0)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-additive-inverse","seed":202610271026,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,4,0,2,4],[0,3,0,3,0,3],[0,4,2,0,4,2],[0,5,4,3,2,1]],"element":1,"domain":"Complete_finite_unital_ring_tables","operation":"ring-additive-inverse","semantic_template":"qf-mod27-ring-additive-inverse/{\"addition\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"element\":1,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,4,0,2,4],[0,3,0,3,0,3],[0,4,2,0,4,2],[0,5,4,3,2,1]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0027","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-additive-inverse；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,2,3,5,0,4],[2,3,5,4,1,0],[3,5,4,0,2,1],[4,0,1,2,5,3],[5,4,0,1,3,2]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,5,0,5,2],[0,3,0,3,3,0],[0,4,5,3,1,2],[0,5,2,0,2,5]],\"element\":0}。請依 [element,zero,inverse,left_sum,right_sum] 欄序回答。","answer":{"kind":"expression","value":"(0, 0, 0, 0, 0)","explanation_zh":"Wave4根節點核對=(0, 0, 0, 0, 0)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-additive-inverse","seed":202610271027,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,2,3,5,0,4],[2,3,5,4,1,0],[3,5,4,0,2,1],[4,0,1,2,5,3],[5,4,0,1,3,2]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,5,0,5,2],[0,3,0,3,3,0],[0,4,5,3,1,2],[0,5,2,0,2,5]],"element":0,"domain":"Complete_finite_unital_ring_tables","operation":"ring-additive-inverse","semantic_template":"qf-mod27-ring-additive-inverse/{\"addition\":[[0,1,2,3,4,5],[1,2,3,5,0,4],[2,3,5,4,1,0],[3,5,4,0,2,1],[4,0,1,2,5,3],[5,4,0,1,3,2]],\"element\":0,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,5,0,5,2],[0,3,0,3,3,0],[0,4,5,3,1,2],[0,5,2,0,2,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0028","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-additive-inverse；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,2,4,5,3,0],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,3,5,1,0,2],[5,0,1,4,2,3]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,3,2,0,3],[0,3,2,3,0,2],[0,4,0,0,4,4],[0,5,3,2,4,1]],\"element\":5}。請依 [element,zero,inverse,left_sum,right_sum] 欄序回答。","answer":{"kind":"expression","value":"(5, 0, 1, 0, 0)","explanation_zh":"Wave4根節點核對=(5, 0, 1, 0, 0)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-additive-inverse","seed":202610271028,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,2,4,5,3,0],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,3,5,1,0,2],[5,0,1,4,2,3]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,3,2,0,3],[0,3,2,3,0,2],[0,4,0,0,4,4],[0,5,3,2,4,1]],"element":5,"domain":"Complete_finite_unital_ring_tables","operation":"ring-additive-inverse","semantic_template":"qf-mod27-ring-additive-inverse/{\"addition\":[[0,1,2,3,4,5],[1,2,4,5,3,0],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,3,5,1,0,2],[5,0,1,4,2,3]],\"element\":5,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,3,2,0,3],[0,3,2,3,0,2],[0,4,0,0,4,4],[0,5,3,2,4,1]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0029","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-additive-inverse；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,2,4,0,5,3],[2,4,5,1,3,0],[3,0,1,5,2,4],[4,5,3,2,0,1],[5,3,0,4,1,2]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,5,5,0,2],[0,3,5,1,4,2],[0,4,0,4,4,0],[0,5,2,2,0,5]],\"element\":4}。請依 [element,zero,inverse,left_sum,right_sum] 欄序回答。","answer":{"kind":"expression","value":"(4, 0, 4, 0, 0)","explanation_zh":"Wave4根節點核對=(4, 0, 4, 0, 0)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-additive-inverse","seed":202610271029,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,2,4,0,5,3],[2,4,5,1,3,0],[3,0,1,5,2,4],[4,5,3,2,0,1],[5,3,0,4,1,2]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,5,5,0,2],[0,3,5,1,4,2],[0,4,0,4,4,0],[0,5,2,2,0,5]],"element":4,"domain":"Complete_finite_unital_ring_tables","operation":"ring-additive-inverse","semantic_template":"qf-mod27-ring-additive-inverse/{\"addition\":[[0,1,2,3,4,5],[1,2,4,0,5,3],[2,4,5,1,3,0],[3,0,1,5,2,4],[4,5,3,2,0,1],[5,3,0,4,1,2]],\"element\":4,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,5,5,0,2],[0,3,5,1,4,2],[0,4,0,4,4,0],[0,5,2,2,0,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0030","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-additive-inverse；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,2,5,4,0,3],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,0,1,5,3,2],[5,3,4,1,2,0]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,3,2,3,0],[0,3,2,3,2,0],[0,4,3,2,1,5],[0,5,0,0,5,5]],\"element\":3}。請依 [element,zero,inverse,left_sum,right_sum] 欄序回答。","answer":{"kind":"expression","value":"(3, 0, 2, 0, 0)","explanation_zh":"Wave4根節點核對=(3, 0, 2, 0, 0)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-additive-inverse","seed":202610271030,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,2,5,4,0,3],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,0,1,5,3,2],[5,3,4,1,2,0]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,3,2,3,0],[0,3,2,3,2,0],[0,4,3,2,1,5],[0,5,0,0,5,5]],"element":3,"domain":"Complete_finite_unital_ring_tables","operation":"ring-additive-inverse","semantic_template":"qf-mod27-ring-additive-inverse/{\"addition\":[[0,1,2,3,4,5],[1,2,5,4,0,3],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,0,1,5,3,2],[5,3,4,1,2,0]],\"element\":3,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,3,2,3,0],[0,3,2,3,2,0],[0,4,3,2,1,5],[0,5,0,0,5,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0031","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-additive-inverse；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,2,5,0,3,4],[2,5,4,1,0,3],[3,0,1,4,5,2],[4,3,0,5,2,1],[5,4,3,2,1,0]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,4,4,2,0],[0,3,4,1,2,5],[0,4,2,2,4,0],[0,5,0,5,0,5]],\"element\":2}。請依 [element,zero,inverse,left_sum,right_sum] 欄序回答。","answer":{"kind":"expression","value":"(2, 0, 4, 0, 0)","explanation_zh":"Wave4根節點核對=(2, 0, 4, 0, 0)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-additive-inverse","seed":202610271031,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,2,5,0,3,4],[2,5,4,1,0,3],[3,0,1,4,5,2],[4,3,0,5,2,1],[5,4,3,2,1,0]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,4,4,2,0],[0,3,4,1,2,5],[0,4,2,2,4,0],[0,5,0,5,0,5]],"element":2,"domain":"Complete_finite_unital_ring_tables","operation":"ring-additive-inverse","semantic_template":"qf-mod27-ring-additive-inverse/{\"addition\":[[0,1,2,3,4,5],[1,2,5,0,3,4],[2,5,4,1,0,3],[3,0,1,4,5,2],[4,3,0,5,2,1],[5,4,3,2,1,0]],\"element\":2,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,4,4,2,0],[0,3,4,1,2,5],[0,4,2,2,4,0],[0,5,0,5,0,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0032","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-additive-inverse；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,3,4,2,5,0],[2,4,0,5,1,3],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,0,3,1,2,4]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,0,2],[0,3,0,4,3,4],[0,4,0,3,4,3],[0,5,2,4,3,1]],\"element\":1}。請依 [element,zero,inverse,left_sum,right_sum] 欄序回答。","answer":{"kind":"expression","value":"(1, 0, 5, 0, 0)","explanation_zh":"Wave4根節點核對=(1, 0, 5, 0, 0)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-additive-inverse","seed":202610271032,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,3,4,2,5,0],[2,4,0,5,1,3],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,0,3,1,2,4]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,0,2],[0,3,0,4,3,4],[0,4,0,3,4,3],[0,5,2,4,3,1]],"element":1,"domain":"Complete_finite_unital_ring_tables","operation":"ring-additive-inverse","semantic_template":"qf-mod27-ring-additive-inverse/{\"addition\":[[0,1,2,3,4,5],[1,3,4,2,5,0],[2,4,0,5,1,3],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,0,3,1,2,4]],\"element\":1,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,0,2],[0,3,0,4,3,4],[0,4,0,3,4,3],[0,5,2,4,3,1]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0033","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-additive-inverse；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,3,5,2,0,4],[2,5,0,4,3,1],[3,2,4,5,1,0],[4,0,3,1,5,2],[5,4,1,0,2,3]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,2,0],[0,3,0,5,5,3],[0,4,2,5,1,3],[0,5,0,3,3,5]],\"element\":0}。請依 [element,zero,inverse,left_sum,right_sum] 欄序回答。","answer":{"kind":"expression","value":"(0, 0, 0, 0, 0)","explanation_zh":"Wave4根節點核對=(0, 0, 0, 0, 0)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-additive-inverse","seed":202610271033,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,3,5,2,0,4],[2,5,0,4,3,1],[3,2,4,5,1,0],[4,0,3,1,5,2],[5,4,1,0,2,3]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,2,0],[0,3,0,5,5,3],[0,4,2,5,1,3],[0,5,0,3,3,5]],"element":0,"domain":"Complete_finite_unital_ring_tables","operation":"ring-additive-inverse","semantic_template":"qf-mod27-ring-additive-inverse/{\"addition\":[[0,1,2,3,4,5],[1,3,5,2,0,4],[2,5,0,4,3,1],[3,2,4,5,1,0],[4,0,3,1,5,2],[5,4,1,0,2,3]],\"element\":0,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,2,0],[0,3,0,5,5,3],[0,4,2,5,1,3],[0,5,0,3,3,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0034","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-additive-inverse；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,3,5,4,2,0],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,2,1,5,0,3],[5,0,4,1,3,2]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,3,0,3],[0,3,3,2,0,2],[0,4,0,0,4,4],[0,5,3,2,4,1]],\"element\":5}。請依 [element,zero,inverse,left_sum,right_sum] 欄序回答。","answer":{"kind":"expression","value":"(5, 0, 1, 0, 0)","explanation_zh":"Wave4根節點核對=(5, 0, 1, 0, 0)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-additive-inverse","seed":202610271034,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,3,5,4,2,0],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,2,1,5,0,3],[5,0,4,1,3,2]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,3,0,3],[0,3,3,2,0,2],[0,4,0,0,4,4],[0,5,3,2,4,1]],"element":5,"domain":"Complete_finite_unital_ring_tables","operation":"ring-additive-inverse","semantic_template":"qf-mod27-ring-additive-inverse/{\"addition\":[[0,1,2,3,4,5],[1,3,5,4,2,0],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,2,1,5,0,3],[5,0,4,1,3,2]],\"element\":5,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,3,0,3],[0,3,3,2,0,2],[0,4,0,0,4,4],[0,5,3,2,4,1]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0035","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-additive-inverse；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,3,0,4,5,2],[2,0,5,1,3,4],[3,4,1,5,2,0],[4,5,3,2,0,1],[5,2,4,0,1,3]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,5,4,3],[0,3,5,5,0,3],[0,4,4,0,4,0],[0,5,3,3,0,5]],\"element\":4}。請依 [element,zero,inverse,left_sum,right_sum] 欄序回答。","answer":{"kind":"expression","value":"(4, 0, 4, 0, 0)","explanation_zh":"Wave4根節點核對=(4, 0, 4, 0, 0)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-additive-inverse","seed":202610271035,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,3,0,4,5,2],[2,0,5,1,3,4],[3,4,1,5,2,0],[4,5,3,2,0,1],[5,2,4,0,1,3]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,5,4,3],[0,3,5,5,0,3],[0,4,4,0,4,0],[0,5,3,3,0,5]],"element":4,"domain":"Complete_finite_unital_ring_tables","operation":"ring-additive-inverse","semantic_template":"qf-mod27-ring-additive-inverse/{\"addition\":[[0,1,2,3,4,5],[1,3,0,4,5,2],[2,0,5,1,3,4],[3,4,1,5,2,0],[4,5,3,2,0,1],[5,2,4,0,1,3]],\"element\":4,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,5,4,3],[0,3,5,5,0,3],[0,4,4,0,4,0],[0,5,3,3,0,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0036","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-additive-inverse；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,3,4,5,0,2],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,0,5,1,2,3],[5,2,1,4,3,0]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,3,3,0],[0,3,3,2,2,0],[0,4,3,2,1,5],[0,5,0,0,5,5]],\"element\":3}。請依 [element,zero,inverse,left_sum,right_sum] 欄序回答。","answer":{"kind":"expression","value":"(3, 0, 2, 0, 0)","explanation_zh":"Wave4根節點核對=(3, 0, 2, 0, 0)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-additive-inverse","seed":202610271036,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,3,4,5,0,2],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,0,5,1,2,3],[5,2,1,4,3,0]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,3,3,0],[0,3,3,2,2,0],[0,4,3,2,1,5],[0,5,0,0,5,5]],"element":3,"domain":"Complete_finite_unital_ring_tables","operation":"ring-additive-inverse","semantic_template":"qf-mod27-ring-additive-inverse/{\"addition\":[[0,1,2,3,4,5],[1,3,4,5,0,2],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,0,5,1,2,3],[5,2,1,4,3,0]],\"element\":3,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,3,3,0],[0,3,3,2,2,0],[0,4,3,2,1,5],[0,5,0,0,5,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0037","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-additive-inverse；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,3,0,5,2,4],[2,0,4,1,5,3],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,4,3,2,1,0]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,4,3,5],[0,3,4,4,3,0],[0,4,3,3,4,0],[0,5,5,0,0,5]],\"element\":2}。請依 [element,zero,inverse,left_sum,right_sum] 欄序回答。","answer":{"kind":"expression","value":"(2, 0, 1, 0, 0)","explanation_zh":"Wave4根節點核對=(2, 0, 1, 0, 0)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-additive-inverse","seed":202610271037,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,3,0,5,2,4],[2,0,4,1,5,3],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,4,3,2,1,0]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,4,3,5],[0,3,4,4,3,0],[0,4,3,3,4,0],[0,5,5,0,0,5]],"element":2,"domain":"Complete_finite_unital_ring_tables","operation":"ring-additive-inverse","semantic_template":"qf-mod27-ring-additive-inverse/{\"addition\":[[0,1,2,3,4,5],[1,3,0,5,2,4],[2,0,4,1,5,3],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,4,3,2,1,0]],\"element\":2,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,4,3,5],[0,3,4,4,3,0],[0,4,3,3,4,0],[0,5,5,0,0,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0038","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-additive-inverse；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,4,3,5,2,0],[2,3,0,1,5,4],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,0,4,2,1,3]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,0,2],[0,3,0,3,4,4],[0,4,0,4,3,3],[0,5,2,4,3,1]],\"element\":1}。請依 [element,zero,inverse,left_sum,right_sum] 欄序回答。","answer":{"kind":"expression","value":"(1, 0, 5, 0, 0)","explanation_zh":"Wave4根節點核對=(1, 0, 5, 0, 0)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-additive-inverse","seed":202610271038,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,4,3,5,2,0],[2,3,0,1,5,4],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,0,4,2,1,3]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,0,2],[0,3,0,3,4,4],[0,4,0,4,3,3],[0,5,2,4,3,1]],"element":1,"domain":"Complete_finite_unital_ring_tables","operation":"ring-additive-inverse","semantic_template":"qf-mod27-ring-additive-inverse/{\"addition\":[[0,1,2,3,4,5],[1,4,3,5,2,0],[2,3,0,1,5,4],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,0,4,2,1,3]],\"element\":1,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,0,2],[0,3,0,3,4,4],[0,4,0,4,3,3],[0,5,2,4,3,1]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0039","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-additive-inverse；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,4,5,0,2,3],[2,5,0,4,3,1],[3,0,4,5,1,2],[4,2,3,1,5,0],[5,3,1,2,0,4]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,2,0,0],[0,3,2,1,5,4],[0,4,0,5,5,4],[0,5,0,4,4,5]],\"element\":0}。請依 [element,zero,inverse,left_sum,right_sum] 欄序回答。","answer":{"kind":"expression","value":"(0, 0, 0, 0, 0)","explanation_zh":"Wave4根節點核對=(0, 0, 0, 0, 0)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-additive-inverse","seed":202610271039,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,4,5,0,2,3],[2,5,0,4,3,1],[3,0,4,5,1,2],[4,2,3,1,5,0],[5,3,1,2,0,4]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,2,0,0],[0,3,2,1,5,4],[0,4,0,5,5,4],[0,5,0,4,4,5]],"element":0,"domain":"Complete_finite_unital_ring_tables","operation":"ring-additive-inverse","semantic_template":"qf-mod27-ring-additive-inverse/{\"addition\":[[0,1,2,3,4,5],[1,4,5,0,2,3],[2,5,0,4,3,1],[3,0,4,5,1,2],[4,2,3,1,5,0],[5,3,1,2,0,4]],\"element\":0,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,2,0,0],[0,3,2,1,5,4],[0,4,0,5,5,4],[0,5,0,4,4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0040","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-additive-inverse；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,4,5,2,3,0],[2,5,4,1,0,3],[3,2,1,0,5,4],[4,3,0,5,2,1],[5,0,3,4,1,2]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,4,4],[0,3,0,3,0,3],[0,4,4,0,2,2],[0,5,4,3,2,1]],\"element\":5}。請依 [element,zero,inverse,left_sum,right_sum] 欄序回答。","answer":{"kind":"expression","value":"(5, 0, 1, 0, 0)","explanation_zh":"Wave4根節點核對=(5, 0, 1, 0, 0)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-additive-inverse","seed":202610271040,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,4,5,2,3,0],[2,5,4,1,0,3],[3,2,1,0,5,4],[4,3,0,5,2,1],[5,0,3,4,1,2]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,4,4],[0,3,0,3,0,3],[0,4,4,0,2,2],[0,5,4,3,2,1]],"element":5,"domain":"Complete_finite_unital_ring_tables","operation":"ring-additive-inverse","semantic_template":"qf-mod27-ring-additive-inverse/{\"addition\":[[0,1,2,3,4,5],[1,4,5,2,3,0],[2,5,4,1,0,3],[3,2,1,0,5,4],[4,3,0,5,2,1],[5,0,3,4,1,2]],\"element\":5,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,4,4],[0,3,0,3,0,3],[0,4,4,0,2,2],[0,5,4,3,2,1]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0041","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-additive-inverse；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,4,0,5,3,2],[2,0,5,4,1,3],[3,5,4,0,2,1],[4,3,1,2,5,0],[5,2,3,1,0,4]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,3,5,4],[0,3,3,3,0,0],[0,4,5,0,5,4],[0,5,4,0,4,5]],\"element\":4}。請依 [element,zero,inverse,left_sum,right_sum] 欄序回答。","answer":{"kind":"expression","value":"(4, 0, 5, 0, 0)","explanation_zh":"Wave4根節點核對=(4, 0, 5, 0, 0)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-additive-inverse","seed":202610271041,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,4,0,5,3,2],[2,0,5,4,1,3],[3,5,4,0,2,1],[4,3,1,2,5,0],[5,2,3,1,0,4]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,3,5,4],[0,3,3,3,0,0],[0,4,5,0,5,4],[0,5,4,0,4,5]],"element":4,"domain":"Complete_finite_unital_ring_tables","operation":"ring-additive-inverse","semantic_template":"qf-mod27-ring-additive-inverse/{\"addition\":[[0,1,2,3,4,5],[1,4,0,5,3,2],[2,0,5,4,1,3],[3,5,4,0,2,1],[4,3,1,2,5,0],[5,2,3,1,0,4]],\"element\":4,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,3,5,4],[0,3,3,3,0,0],[0,4,5,0,5,4],[0,5,4,0,4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0042","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-additive-inverse；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,4,3,0,5,2],[2,3,4,5,0,1],[3,0,5,2,1,4],[4,5,0,1,2,3],[5,2,1,4,3,0]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,4,4,0],[0,3,4,1,2,5],[0,4,4,2,2,0],[0,5,0,5,0,5]],\"element\":3}。請依 [element,zero,inverse,left_sum,right_sum] 欄序回答。","answer":{"kind":"expression","value":"(3, 0, 1, 0, 0)","explanation_zh":"Wave4根節點核對=(3, 0, 1, 0, 0)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-additive-inverse","seed":202610271042,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,4,3,0,5,2],[2,3,4,5,0,1],[3,0,5,2,1,4],[4,5,0,1,2,3],[5,2,1,4,3,0]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,4,4,0],[0,3,4,1,2,5],[0,4,4,2,2,0],[0,5,0,5,0,5]],"element":3,"domain":"Complete_finite_unital_ring_tables","operation":"ring-additive-inverse","semantic_template":"qf-mod27-ring-additive-inverse/{\"addition\":[[0,1,2,3,4,5],[1,4,3,0,5,2],[2,3,4,5,0,1],[3,0,5,2,1,4],[4,5,0,1,2,3],[5,2,1,4,3,0]],\"element\":3,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,4,4,0],[0,3,4,1,2,5],[0,4,4,2,2,0],[0,5,0,5,0,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0043","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-additive-inverse；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,4,0,2,5,3],[2,0,3,5,1,4],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,3,4,1,2,0]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,4,3,5],[0,3,4,3,4,0],[0,4,3,4,3,0],[0,5,5,0,0,5]],\"element\":2}。請依 [element,zero,inverse,left_sum,right_sum] 欄序回答。","answer":{"kind":"expression","value":"(2, 0, 1, 0, 0)","explanation_zh":"Wave4根節點核對=(2, 0, 1, 0, 0)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-additive-inverse","seed":202610271043,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,4,0,2,5,3],[2,0,3,5,1,4],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,3,4,1,2,0]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,4,3,5],[0,3,4,3,4,0],[0,4,3,4,3,0],[0,5,5,0,0,5]],"element":2,"domain":"Complete_finite_unital_ring_tables","operation":"ring-additive-inverse","semantic_template":"qf-mod27-ring-additive-inverse/{\"addition\":[[0,1,2,3,4,5],[1,4,0,2,5,3],[2,0,3,5,1,4],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,3,4,1,2,0]],\"element\":2,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,4,3,5],[0,3,4,3,4,0],[0,4,3,4,3,0],[0,5,5,0,0,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0044","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-additive-inverse；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,5,3,4,0,2],[2,3,0,1,5,4],[3,4,1,5,2,0],[4,0,5,2,3,1],[5,2,4,0,1,3]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,2,0],[0,3,0,3,5,5],[0,4,2,5,1,3],[0,5,0,5,3,3]],\"element\":1}。請依 [element,zero,inverse,left_sum,right_sum] 欄序回答。","answer":{"kind":"expression","value":"(1, 0, 4, 0, 0)","explanation_zh":"Wave4根節點核對=(1, 0, 4, 0, 0)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-additive-inverse","seed":202610271044,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,5,3,4,0,2],[2,3,0,1,5,4],[3,4,1,5,2,0],[4,0,5,2,3,1],[5,2,4,0,1,3]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,2,0],[0,3,0,3,5,5],[0,4,2,5,1,3],[0,5,0,5,3,3]],"element":1,"domain":"Complete_finite_unital_ring_tables","operation":"ring-additive-inverse","semantic_template":"qf-mod27-ring-additive-inverse/{\"addition\":[[0,1,2,3,4,5],[1,5,3,4,0,2],[2,3,0,1,5,4],[3,4,1,5,2,0],[4,0,5,2,3,1],[5,2,4,0,1,3]],\"element\":1,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,2,0],[0,3,0,3,5,5],[0,4,2,5,1,3],[0,5,0,5,3,3]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0045","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-additive-inverse；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,5,4,0,3,2],[2,4,0,5,1,3],[3,0,5,4,2,1],[4,3,1,2,5,0],[5,2,3,1,0,4]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,2,0,0],[0,3,2,1,5,4],[0,4,0,5,4,5],[0,5,0,4,5,4]],\"element\":0}。請依 [element,zero,inverse,left_sum,right_sum] 欄序回答。","answer":{"kind":"expression","value":"(0, 0, 0, 0, 0)","explanation_zh":"Wave4根節點核對=(0, 0, 0, 0, 0)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-additive-inverse","seed":202610271045,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,5,4,0,3,2],[2,4,0,5,1,3],[3,0,5,4,2,1],[4,3,1,2,5,0],[5,2,3,1,0,4]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,2,0,0],[0,3,2,1,5,4],[0,4,0,5,4,5],[0,5,0,4,5,4]],"element":0,"domain":"Complete_finite_unital_ring_tables","operation":"ring-additive-inverse","semantic_template":"qf-mod27-ring-additive-inverse/{\"addition\":[[0,1,2,3,4,5],[1,5,4,0,3,2],[2,4,0,5,1,3],[3,0,5,4,2,1],[4,3,1,2,5,0],[5,2,3,1,0,4]],\"element\":0,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,2,0,0],[0,3,2,1,5,4],[0,4,0,5,4,5],[0,5,0,4,5,4]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0046","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-additive-inverse；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,5,4,2,0,3],[2,4,5,1,3,0],[3,2,1,0,5,4],[4,0,3,5,2,1],[5,3,0,4,1,2]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,5,5],[0,3,0,3,3,0],[0,4,5,3,1,2],[0,5,5,0,2,2]],\"element\":5}。請依 [element,zero,inverse,left_sum,right_sum] 欄序回答。","answer":{"kind":"expression","value":"(5, 0, 2, 0, 0)","explanation_zh":"Wave4根節點核對=(5, 0, 2, 0, 0)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-additive-inverse","seed":202610271046,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,5,4,2,0,3],[2,4,5,1,3,0],[3,2,1,0,5,4],[4,0,3,5,2,1],[5,3,0,4,1,2]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,5,5],[0,3,0,3,3,0],[0,4,5,3,1,2],[0,5,5,0,2,2]],"element":5,"domain":"Complete_finite_unital_ring_tables","operation":"ring-additive-inverse","semantic_template":"qf-mod27-ring-additive-inverse/{\"addition\":[[0,1,2,3,4,5],[1,5,4,2,0,3],[2,4,5,1,3,0],[3,2,1,0,5,4],[4,0,3,5,2,1],[5,3,0,4,1,2]],\"element\":5,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,5,5],[0,3,0,3,3,0],[0,4,5,3,1,2],[0,5,5,0,2,2]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0047","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-additive-inverse；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,5,0,4,2,3],[2,0,4,5,3,1],[3,4,5,0,1,2],[4,2,3,1,5,0],[5,3,1,2,0,4]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,3,5,4],[0,3,3,3,0,0],[0,4,5,0,4,5],[0,5,4,0,5,4]],\"element\":4}。請依 [element,zero,inverse,left_sum,right_sum] 欄序回答。","answer":{"kind":"expression","value":"(4, 0, 5, 0, 0)","explanation_zh":"Wave4根節點核對=(4, 0, 5, 0, 0)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-additive-inverse","seed":202610271047,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,5,0,4,2,3],[2,0,4,5,3,1],[3,4,5,0,1,2],[4,2,3,1,5,0],[5,3,1,2,0,4]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,3,5,4],[0,3,3,3,0,0],[0,4,5,0,4,5],[0,5,4,0,5,4]],"element":4,"domain":"Complete_finite_unital_ring_tables","operation":"ring-additive-inverse","semantic_template":"qf-mod27-ring-additive-inverse/{\"addition\":[[0,1,2,3,4,5],[1,5,0,4,2,3],[2,0,4,5,3,1],[3,4,5,0,1,2],[4,2,3,1,5,0],[5,3,1,2,0,4]],\"element\":4,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,3,5,4],[0,3,3,3,0,0],[0,4,5,0,4,5],[0,5,4,0,5,4]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0048","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-additive-inverse；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,5,3,0,2,4],[2,3,5,4,1,0],[3,0,4,2,5,1],[4,2,1,5,0,3],[5,4,0,1,3,2]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,5,0,5],[0,3,5,1,4,2],[0,4,0,4,4,0],[0,5,5,2,0,2]],\"element\":3}。請依 [element,zero,inverse,left_sum,right_sum] 欄序回答。","answer":{"kind":"expression","value":"(3, 0, 1, 0, 0)","explanation_zh":"Wave4根節點核對=(3, 0, 1, 0, 0)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-additive-inverse","seed":202610271048,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,5,3,0,2,4],[2,3,5,4,1,0],[3,0,4,2,5,1],[4,2,1,5,0,3],[5,4,0,1,3,2]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,5,0,5],[0,3,5,1,4,2],[0,4,0,4,4,0],[0,5,5,2,0,2]],"element":3,"domain":"Complete_finite_unital_ring_tables","operation":"ring-additive-inverse","semantic_template":"qf-mod27-ring-additive-inverse/{\"addition\":[[0,1,2,3,4,5],[1,5,3,0,2,4],[2,3,5,4,1,0],[3,0,4,2,5,1],[4,2,1,5,0,3],[5,4,0,1,3,2]],\"element\":3,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,5,0,5],[0,3,5,1,4,2],[0,4,0,4,4,0],[0,5,5,2,0,2]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0049","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-additive-inverse；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,5,0,2,3,4],[2,0,3,4,5,1],[3,2,4,5,1,0],[4,3,5,1,0,2],[5,4,1,0,2,3]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,5,4,3],[0,3,5,3,0,5],[0,4,4,0,4,0],[0,5,3,5,0,3]],\"element\":2}。請依 [element,zero,inverse,left_sum,right_sum] 欄序回答。","answer":{"kind":"expression","value":"(2, 0, 1, 0, 0)","explanation_zh":"Wave4根節點核對=(2, 0, 1, 0, 0)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-additive-inverse","seed":202610271049,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,5,0,2,3,4],[2,0,3,4,5,1],[3,2,4,5,1,0],[4,3,5,1,0,2],[5,4,1,0,2,3]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,5,4,3],[0,3,5,3,0,5],[0,4,4,0,4,0],[0,5,3,5,0,3]],"element":2,"domain":"Complete_finite_unital_ring_tables","operation":"ring-additive-inverse","semantic_template":"qf-mod27-ring-additive-inverse/{\"addition\":[[0,1,2,3,4,5],[1,5,0,2,3,4],[2,0,3,4,5,1],[3,2,4,5,1,0],[4,3,5,1,0,2],[5,4,1,0,2,3]],\"element\":2,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,5,4,3],[0,3,5,3,0,5],[0,4,4,0,4,0],[0,5,3,5,0,3]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0050","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-additive-inverse；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,4,3,5,0,2],[2,3,1,4,5,0],[3,5,4,0,2,1],[4,0,5,2,1,3],[5,2,0,1,3,4]],\"multiplication\":[[0,0,0,0,0,0],[0,4,1,0,1,4],[0,1,2,3,4,5],[0,0,3,3,0,3],[0,1,4,0,4,1],[0,4,5,3,1,2]],\"element\":1}。請依 [element,zero,inverse,left_sum,right_sum] 欄序回答。","answer":{"kind":"expression","value":"(1, 0, 4, 0, 0)","explanation_zh":"Wave4根節點核對=(1, 0, 4, 0, 0)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-additive-inverse","seed":202610271050,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,4,3,5,0,2],[2,3,1,4,5,0],[3,5,4,0,2,1],[4,0,5,2,1,3],[5,2,0,1,3,4]],"multiplication":[[0,0,0,0,0,0],[0,4,1,0,1,4],[0,1,2,3,4,5],[0,0,3,3,0,3],[0,1,4,0,4,1],[0,4,5,3,1,2]],"element":1,"domain":"Complete_finite_unital_ring_tables","operation":"ring-additive-inverse","semantic_template":"qf-mod27-ring-additive-inverse/{\"addition\":[[0,1,2,3,4,5],[1,4,3,5,0,2],[2,3,1,4,5,0],[3,5,4,0,2,1],[4,0,5,2,1,3],[5,2,0,1,3,4]],\"element\":1,\"multiplication\":[[0,0,0,0,0,0],[0,4,1,0,1,4],[0,1,2,3,4,5],[0,0,3,3,0,3],[0,1,4,0,4,1],[0,4,5,3,1,2]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0051","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-power；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,4,0,2,4],[0,3,0,3,0,3],[0,4,2,0,4,2],[0,5,4,3,2,1]],\"element\":1,\"exponent\":1}。請依 [element,exponent,one,result] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, 1, 1)","explanation_zh":"Wave4根節點核對=(1, 1, 1, 1)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-power","seed":202610271051,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,4,0,2,4],[0,3,0,3,0,3],[0,4,2,0,4,2],[0,5,4,3,2,1]],"element":1,"exponent":1,"domain":"Complete_finite_unital_ring_tables","operation":"ring-power","semantic_template":"qf-mod27-ring-power/{\"addition\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"element\":1,\"exponent\":1,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,4,0,2,4],[0,3,0,3,0,3],[0,4,2,0,4,2],[0,5,4,3,2,1]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0052","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-power；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,2,3,5,0,4],[2,3,5,4,1,0],[3,5,4,0,2,1],[4,0,1,2,5,3],[5,4,0,1,3,2]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,5,0,5,2],[0,3,0,3,3,0],[0,4,5,3,1,2],[0,5,2,0,2,5]],\"element\":0,\"exponent\":2}。請依 [element,exponent,one,result] 欄序回答。","answer":{"kind":"expression","value":"(0, 2, 1, 0)","explanation_zh":"Wave4根節點核對=(0, 2, 1, 0)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-power","seed":202610271052,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,2,3,5,0,4],[2,3,5,4,1,0],[3,5,4,0,2,1],[4,0,1,2,5,3],[5,4,0,1,3,2]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,5,0,5,2],[0,3,0,3,3,0],[0,4,5,3,1,2],[0,5,2,0,2,5]],"element":0,"exponent":2,"domain":"Complete_finite_unital_ring_tables","operation":"ring-power","semantic_template":"qf-mod27-ring-power/{\"addition\":[[0,1,2,3,4,5],[1,2,3,5,0,4],[2,3,5,4,1,0],[3,5,4,0,2,1],[4,0,1,2,5,3],[5,4,0,1,3,2]],\"element\":0,\"exponent\":2,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,5,0,5,2],[0,3,0,3,3,0],[0,4,5,3,1,2],[0,5,2,0,2,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0053","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-power；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,2,4,5,3,0],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,3,5,1,0,2],[5,0,1,4,2,3]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,3,2,0,3],[0,3,2,3,0,2],[0,4,0,0,4,4],[0,5,3,2,4,1]],\"element\":5,\"exponent\":3}。請依 [element,exponent,one,result] 欄序回答。","answer":{"kind":"expression","value":"(5, 3, 1, 5)","explanation_zh":"Wave4根節點核對=(5, 3, 1, 5)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-power","seed":202610271053,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,2,4,5,3,0],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,3,5,1,0,2],[5,0,1,4,2,3]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,3,2,0,3],[0,3,2,3,0,2],[0,4,0,0,4,4],[0,5,3,2,4,1]],"element":5,"exponent":3,"domain":"Complete_finite_unital_ring_tables","operation":"ring-power","semantic_template":"qf-mod27-ring-power/{\"addition\":[[0,1,2,3,4,5],[1,2,4,5,3,0],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,3,5,1,0,2],[5,0,1,4,2,3]],\"element\":5,\"exponent\":3,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,3,2,0,3],[0,3,2,3,0,2],[0,4,0,0,4,4],[0,5,3,2,4,1]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0054","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-power；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,2,4,0,5,3],[2,4,5,1,3,0],[3,0,1,5,2,4],[4,5,3,2,0,1],[5,3,0,4,1,2]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,5,5,0,2],[0,3,5,1,4,2],[0,4,0,4,4,0],[0,5,2,2,0,5]],\"element\":4,\"exponent\":4}。請依 [element,exponent,one,result] 欄序回答。","answer":{"kind":"expression","value":"(4, 4, 1, 4)","explanation_zh":"Wave4根節點核對=(4, 4, 1, 4)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-power","seed":202610271054,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,2,4,0,5,3],[2,4,5,1,3,0],[3,0,1,5,2,4],[4,5,3,2,0,1],[5,3,0,4,1,2]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,5,5,0,2],[0,3,5,1,4,2],[0,4,0,4,4,0],[0,5,2,2,0,5]],"element":4,"exponent":4,"domain":"Complete_finite_unital_ring_tables","operation":"ring-power","semantic_template":"qf-mod27-ring-power/{\"addition\":[[0,1,2,3,4,5],[1,2,4,0,5,3],[2,4,5,1,3,0],[3,0,1,5,2,4],[4,5,3,2,0,1],[5,3,0,4,1,2]],\"element\":4,\"exponent\":4,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,5,5,0,2],[0,3,5,1,4,2],[0,4,0,4,4,0],[0,5,2,2,0,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0055","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-power；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,2,5,4,0,3],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,0,1,5,3,2],[5,3,4,1,2,0]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,3,2,3,0],[0,3,2,3,2,0],[0,4,3,2,1,5],[0,5,0,0,5,5]],\"element\":3,\"exponent\":5}。請依 [element,exponent,one,result] 欄序回答。","answer":{"kind":"expression","value":"(3, 5, 1, 3)","explanation_zh":"Wave4根節點核對=(3, 5, 1, 3)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-power","seed":202610271055,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,2,5,4,0,3],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,0,1,5,3,2],[5,3,4,1,2,0]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,3,2,3,0],[0,3,2,3,2,0],[0,4,3,2,1,5],[0,5,0,0,5,5]],"element":3,"exponent":5,"domain":"Complete_finite_unital_ring_tables","operation":"ring-power","semantic_template":"qf-mod27-ring-power/{\"addition\":[[0,1,2,3,4,5],[1,2,5,4,0,3],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,0,1,5,3,2],[5,3,4,1,2,0]],\"element\":3,\"exponent\":5,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,3,2,3,0],[0,3,2,3,2,0],[0,4,3,2,1,5],[0,5,0,0,5,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0056","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-power；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,2,5,0,3,4],[2,5,4,1,0,3],[3,0,1,4,5,2],[4,3,0,5,2,1],[5,4,3,2,1,0]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,4,4,2,0],[0,3,4,1,2,5],[0,4,2,2,4,0],[0,5,0,5,0,5]],\"element\":2,\"exponent\":6}。請依 [element,exponent,one,result] 欄序回答。","answer":{"kind":"expression","value":"(2, 6, 1, 4)","explanation_zh":"Wave4根節點核對=(2, 6, 1, 4)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-power","seed":202610271056,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,2,5,0,3,4],[2,5,4,1,0,3],[3,0,1,4,5,2],[4,3,0,5,2,1],[5,4,3,2,1,0]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,4,4,2,0],[0,3,4,1,2,5],[0,4,2,2,4,0],[0,5,0,5,0,5]],"element":2,"exponent":6,"domain":"Complete_finite_unital_ring_tables","operation":"ring-power","semantic_template":"qf-mod27-ring-power/{\"addition\":[[0,1,2,3,4,5],[1,2,5,0,3,4],[2,5,4,1,0,3],[3,0,1,4,5,2],[4,3,0,5,2,1],[5,4,3,2,1,0]],\"element\":2,\"exponent\":6,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,4,4,2,0],[0,3,4,1,2,5],[0,4,2,2,4,0],[0,5,0,5,0,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0057","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-power；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,3,4,2,5,0],[2,4,0,5,1,3],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,0,3,1,2,4]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,0,2],[0,3,0,4,3,4],[0,4,0,3,4,3],[0,5,2,4,3,1]],\"element\":1,\"exponent\":7}。請依 [element,exponent,one,result] 欄序回答。","answer":{"kind":"expression","value":"(1, 7, 1, 1)","explanation_zh":"Wave4根節點核對=(1, 7, 1, 1)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-power","seed":202610271057,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,3,4,2,5,0],[2,4,0,5,1,3],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,0,3,1,2,4]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,0,2],[0,3,0,4,3,4],[0,4,0,3,4,3],[0,5,2,4,3,1]],"element":1,"exponent":7,"domain":"Complete_finite_unital_ring_tables","operation":"ring-power","semantic_template":"qf-mod27-ring-power/{\"addition\":[[0,1,2,3,4,5],[1,3,4,2,5,0],[2,4,0,5,1,3],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,0,3,1,2,4]],\"element\":1,\"exponent\":7,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,0,2],[0,3,0,4,3,4],[0,4,0,3,4,3],[0,5,2,4,3,1]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0058","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-power；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,3,5,2,0,4],[2,5,0,4,3,1],[3,2,4,5,1,0],[4,0,3,1,5,2],[5,4,1,0,2,3]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,2,0],[0,3,0,5,5,3],[0,4,2,5,1,3],[0,5,0,3,3,5]],\"element\":0,\"exponent\":8}。請依 [element,exponent,one,result] 欄序回答。","answer":{"kind":"expression","value":"(0, 8, 1, 0)","explanation_zh":"Wave4根節點核對=(0, 8, 1, 0)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-power","seed":202610271058,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,3,5,2,0,4],[2,5,0,4,3,1],[3,2,4,5,1,0],[4,0,3,1,5,2],[5,4,1,0,2,3]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,2,0],[0,3,0,5,5,3],[0,4,2,5,1,3],[0,5,0,3,3,5]],"element":0,"exponent":8,"domain":"Complete_finite_unital_ring_tables","operation":"ring-power","semantic_template":"qf-mod27-ring-power/{\"addition\":[[0,1,2,3,4,5],[1,3,5,2,0,4],[2,5,0,4,3,1],[3,2,4,5,1,0],[4,0,3,1,5,2],[5,4,1,0,2,3]],\"element\":0,\"exponent\":8,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,2,0],[0,3,0,5,5,3],[0,4,2,5,1,3],[0,5,0,3,3,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0059","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-power；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,3,5,4,2,0],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,2,1,5,0,3],[5,0,4,1,3,2]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,3,0,3],[0,3,3,2,0,2],[0,4,0,0,4,4],[0,5,3,2,4,1]],\"element\":5,\"exponent\":9}。請依 [element,exponent,one,result] 欄序回答。","answer":{"kind":"expression","value":"(5, 9, 1, 5)","explanation_zh":"Wave4根節點核對=(5, 9, 1, 5)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-power","seed":202610271059,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,3,5,4,2,0],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,2,1,5,0,3],[5,0,4,1,3,2]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,3,0,3],[0,3,3,2,0,2],[0,4,0,0,4,4],[0,5,3,2,4,1]],"element":5,"exponent":9,"domain":"Complete_finite_unital_ring_tables","operation":"ring-power","semantic_template":"qf-mod27-ring-power/{\"addition\":[[0,1,2,3,4,5],[1,3,5,4,2,0],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,2,1,5,0,3],[5,0,4,1,3,2]],\"element\":5,\"exponent\":9,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,3,0,3],[0,3,3,2,0,2],[0,4,0,0,4,4],[0,5,3,2,4,1]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0060","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-power；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,3,0,4,5,2],[2,0,5,1,3,4],[3,4,1,5,2,0],[4,5,3,2,0,1],[5,2,4,0,1,3]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,5,4,3],[0,3,5,5,0,3],[0,4,4,0,4,0],[0,5,3,3,0,5]],\"element\":4,\"exponent\":10}。請依 [element,exponent,one,result] 欄序回答。","answer":{"kind":"expression","value":"(4, 10, 1, 4)","explanation_zh":"Wave4根節點核對=(4, 10, 1, 4)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-power","seed":202610271060,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,3,0,4,5,2],[2,0,5,1,3,4],[3,4,1,5,2,0],[4,5,3,2,0,1],[5,2,4,0,1,3]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,5,4,3],[0,3,5,5,0,3],[0,4,4,0,4,0],[0,5,3,3,0,5]],"element":4,"exponent":10,"domain":"Complete_finite_unital_ring_tables","operation":"ring-power","semantic_template":"qf-mod27-ring-power/{\"addition\":[[0,1,2,3,4,5],[1,3,0,4,5,2],[2,0,5,1,3,4],[3,4,1,5,2,0],[4,5,3,2,0,1],[5,2,4,0,1,3]],\"element\":4,\"exponent\":10,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,5,4,3],[0,3,5,5,0,3],[0,4,4,0,4,0],[0,5,3,3,0,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0061","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-power；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,3,4,5,0,2],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,0,5,1,2,3],[5,2,1,4,3,0]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,3,3,0],[0,3,3,2,2,0],[0,4,3,2,1,5],[0,5,0,0,5,5]],\"element\":3,\"exponent\":11}。請依 [element,exponent,one,result] 欄序回答。","answer":{"kind":"expression","value":"(3, 11, 1, 3)","explanation_zh":"Wave4根節點核對=(3, 11, 1, 3)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-power","seed":202610271061,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,3,4,5,0,2],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,0,5,1,2,3],[5,2,1,4,3,0]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,3,3,0],[0,3,3,2,2,0],[0,4,3,2,1,5],[0,5,0,0,5,5]],"element":3,"exponent":11,"domain":"Complete_finite_unital_ring_tables","operation":"ring-power","semantic_template":"qf-mod27-ring-power/{\"addition\":[[0,1,2,3,4,5],[1,3,4,5,0,2],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,0,5,1,2,3],[5,2,1,4,3,0]],\"element\":3,\"exponent\":11,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,3,3,0],[0,3,3,2,2,0],[0,4,3,2,1,5],[0,5,0,0,5,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0062","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-power；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,3,0,5,2,4],[2,0,4,1,5,3],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,4,3,2,1,0]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,4,3,5],[0,3,4,4,3,0],[0,4,3,3,4,0],[0,5,5,0,0,5]],\"element\":2,\"exponent\":12}。請依 [element,exponent,one,result] 欄序回答。","answer":{"kind":"expression","value":"(2, 12, 1, 1)","explanation_zh":"Wave4根節點核對=(2, 12, 1, 1)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-power","seed":202610271062,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,3,0,5,2,4],[2,0,4,1,5,3],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,4,3,2,1,0]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,4,3,5],[0,3,4,4,3,0],[0,4,3,3,4,0],[0,5,5,0,0,5]],"element":2,"exponent":12,"domain":"Complete_finite_unital_ring_tables","operation":"ring-power","semantic_template":"qf-mod27-ring-power/{\"addition\":[[0,1,2,3,4,5],[1,3,0,5,2,4],[2,0,4,1,5,3],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,4,3,2,1,0]],\"element\":2,\"exponent\":12,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,4,3,5],[0,3,4,4,3,0],[0,4,3,3,4,0],[0,5,5,0,0,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0063","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-power；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,4,3,5,2,0],[2,3,0,1,5,4],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,0,4,2,1,3]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,0,2],[0,3,0,3,4,4],[0,4,0,4,3,3],[0,5,2,4,3,1]],\"element\":1,\"exponent\":13}。請依 [element,exponent,one,result] 欄序回答。","answer":{"kind":"expression","value":"(1, 13, 1, 1)","explanation_zh":"Wave4根節點核對=(1, 13, 1, 1)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-power","seed":202610271063,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,4,3,5,2,0],[2,3,0,1,5,4],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,0,4,2,1,3]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,0,2],[0,3,0,3,4,4],[0,4,0,4,3,3],[0,5,2,4,3,1]],"element":1,"exponent":13,"domain":"Complete_finite_unital_ring_tables","operation":"ring-power","semantic_template":"qf-mod27-ring-power/{\"addition\":[[0,1,2,3,4,5],[1,4,3,5,2,0],[2,3,0,1,5,4],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,0,4,2,1,3]],\"element\":1,\"exponent\":13,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,0,2],[0,3,0,3,4,4],[0,4,0,4,3,3],[0,5,2,4,3,1]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0064","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-power；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,4,5,0,2,3],[2,5,0,4,3,1],[3,0,4,5,1,2],[4,2,3,1,5,0],[5,3,1,2,0,4]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,2,0,0],[0,3,2,1,5,4],[0,4,0,5,5,4],[0,5,0,4,4,5]],\"element\":0,\"exponent\":14}。請依 [element,exponent,one,result] 欄序回答。","answer":{"kind":"expression","value":"(0, 14, 1, 0)","explanation_zh":"Wave4根節點核對=(0, 14, 1, 0)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-power","seed":202610271064,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,4,5,0,2,3],[2,5,0,4,3,1],[3,0,4,5,1,2],[4,2,3,1,5,0],[5,3,1,2,0,4]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,2,0,0],[0,3,2,1,5,4],[0,4,0,5,5,4],[0,5,0,4,4,5]],"element":0,"exponent":14,"domain":"Complete_finite_unital_ring_tables","operation":"ring-power","semantic_template":"qf-mod27-ring-power/{\"addition\":[[0,1,2,3,4,5],[1,4,5,0,2,3],[2,5,0,4,3,1],[3,0,4,5,1,2],[4,2,3,1,5,0],[5,3,1,2,0,4]],\"element\":0,\"exponent\":14,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,2,0,0],[0,3,2,1,5,4],[0,4,0,5,5,4],[0,5,0,4,4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0065","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-power；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,4,5,2,3,0],[2,5,4,1,0,3],[3,2,1,0,5,4],[4,3,0,5,2,1],[5,0,3,4,1,2]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,4,4],[0,3,0,3,0,3],[0,4,4,0,2,2],[0,5,4,3,2,1]],\"element\":5,\"exponent\":15}。請依 [element,exponent,one,result] 欄序回答。","answer":{"kind":"expression","value":"(5, 15, 1, 5)","explanation_zh":"Wave4根節點核對=(5, 15, 1, 5)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-power","seed":202610271065,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,4,5,2,3,0],[2,5,4,1,0,3],[3,2,1,0,5,4],[4,3,0,5,2,1],[5,0,3,4,1,2]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,4,4],[0,3,0,3,0,3],[0,4,4,0,2,2],[0,5,4,3,2,1]],"element":5,"exponent":15,"domain":"Complete_finite_unital_ring_tables","operation":"ring-power","semantic_template":"qf-mod27-ring-power/{\"addition\":[[0,1,2,3,4,5],[1,4,5,2,3,0],[2,5,4,1,0,3],[3,2,1,0,5,4],[4,3,0,5,2,1],[5,0,3,4,1,2]],\"element\":5,\"exponent\":15,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,4,4],[0,3,0,3,0,3],[0,4,4,0,2,2],[0,5,4,3,2,1]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0066","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-power；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,4,0,5,3,2],[2,0,5,4,1,3],[3,5,4,0,2,1],[4,3,1,2,5,0],[5,2,3,1,0,4]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,3,5,4],[0,3,3,3,0,0],[0,4,5,0,5,4],[0,5,4,0,4,5]],\"element\":4,\"exponent\":16}。請依 [element,exponent,one,result] 欄序回答。","answer":{"kind":"expression","value":"(4, 16, 1, 5)","explanation_zh":"Wave4根節點核對=(4, 16, 1, 5)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-power","seed":202610271066,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,4,0,5,3,2],[2,0,5,4,1,3],[3,5,4,0,2,1],[4,3,1,2,5,0],[5,2,3,1,0,4]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,3,5,4],[0,3,3,3,0,0],[0,4,5,0,5,4],[0,5,4,0,4,5]],"element":4,"exponent":16,"domain":"Complete_finite_unital_ring_tables","operation":"ring-power","semantic_template":"qf-mod27-ring-power/{\"addition\":[[0,1,2,3,4,5],[1,4,0,5,3,2],[2,0,5,4,1,3],[3,5,4,0,2,1],[4,3,1,2,5,0],[5,2,3,1,0,4]],\"element\":4,\"exponent\":16,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,3,5,4],[0,3,3,3,0,0],[0,4,5,0,5,4],[0,5,4,0,4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0067","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-power；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,4,3,0,5,2],[2,3,4,5,0,1],[3,0,5,2,1,4],[4,5,0,1,2,3],[5,2,1,4,3,0]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,4,4,0],[0,3,4,1,2,5],[0,4,4,2,2,0],[0,5,0,5,0,5]],\"element\":3,\"exponent\":17}。請依 [element,exponent,one,result] 欄序回答。","answer":{"kind":"expression","value":"(3, 17, 1, 3)","explanation_zh":"Wave4根節點核對=(3, 17, 1, 3)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-power","seed":202610271067,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,4,3,0,5,2],[2,3,4,5,0,1],[3,0,5,2,1,4],[4,5,0,1,2,3],[5,2,1,4,3,0]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,4,4,0],[0,3,4,1,2,5],[0,4,4,2,2,0],[0,5,0,5,0,5]],"element":3,"exponent":17,"domain":"Complete_finite_unital_ring_tables","operation":"ring-power","semantic_template":"qf-mod27-ring-power/{\"addition\":[[0,1,2,3,4,5],[1,4,3,0,5,2],[2,3,4,5,0,1],[3,0,5,2,1,4],[4,5,0,1,2,3],[5,2,1,4,3,0]],\"element\":3,\"exponent\":17,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,4,4,0],[0,3,4,1,2,5],[0,4,4,2,2,0],[0,5,0,5,0,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0068","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-power；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,4,0,2,5,3],[2,0,3,5,1,4],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,3,4,1,2,0]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,4,3,5],[0,3,4,3,4,0],[0,4,3,4,3,0],[0,5,5,0,0,5]],\"element\":2,\"exponent\":18}。請依 [element,exponent,one,result] 欄序回答。","answer":{"kind":"expression","value":"(2, 18, 1, 1)","explanation_zh":"Wave4根節點核對=(2, 18, 1, 1)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-power","seed":202610271068,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,4,0,2,5,3],[2,0,3,5,1,4],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,3,4,1,2,0]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,4,3,5],[0,3,4,3,4,0],[0,4,3,4,3,0],[0,5,5,0,0,5]],"element":2,"exponent":18,"domain":"Complete_finite_unital_ring_tables","operation":"ring-power","semantic_template":"qf-mod27-ring-power/{\"addition\":[[0,1,2,3,4,5],[1,4,0,2,5,3],[2,0,3,5,1,4],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,3,4,1,2,0]],\"element\":2,\"exponent\":18,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,4,3,5],[0,3,4,3,4,0],[0,4,3,4,3,0],[0,5,5,0,0,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0069","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-power；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,5,3,4,0,2],[2,3,0,1,5,4],[3,4,1,5,2,0],[4,0,5,2,3,1],[5,2,4,0,1,3]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,2,0],[0,3,0,3,5,5],[0,4,2,5,1,3],[0,5,0,5,3,3]],\"element\":1,\"exponent\":19}。請依 [element,exponent,one,result] 欄序回答。","answer":{"kind":"expression","value":"(1, 19, 1, 1)","explanation_zh":"Wave4根節點核對=(1, 19, 1, 1)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-power","seed":202610271069,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,5,3,4,0,2],[2,3,0,1,5,4],[3,4,1,5,2,0],[4,0,5,2,3,1],[5,2,4,0,1,3]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,2,0],[0,3,0,3,5,5],[0,4,2,5,1,3],[0,5,0,5,3,3]],"element":1,"exponent":19,"domain":"Complete_finite_unital_ring_tables","operation":"ring-power","semantic_template":"qf-mod27-ring-power/{\"addition\":[[0,1,2,3,4,5],[1,5,3,4,0,2],[2,3,0,1,5,4],[3,4,1,5,2,0],[4,0,5,2,3,1],[5,2,4,0,1,3]],\"element\":1,\"exponent\":19,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,2,0],[0,3,0,3,5,5],[0,4,2,5,1,3],[0,5,0,5,3,3]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0070","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-power；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,5,4,0,3,2],[2,4,0,5,1,3],[3,0,5,4,2,1],[4,3,1,2,5,0],[5,2,3,1,0,4]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,2,0,0],[0,3,2,1,5,4],[0,4,0,5,4,5],[0,5,0,4,5,4]],\"element\":0,\"exponent\":20}。請依 [element,exponent,one,result] 欄序回答。","answer":{"kind":"expression","value":"(0, 20, 1, 0)","explanation_zh":"Wave4根節點核對=(0, 20, 1, 0)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-power","seed":202610271070,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,5,4,0,3,2],[2,4,0,5,1,3],[3,0,5,4,2,1],[4,3,1,2,5,0],[5,2,3,1,0,4]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,2,0,0],[0,3,2,1,5,4],[0,4,0,5,4,5],[0,5,0,4,5,4]],"element":0,"exponent":20,"domain":"Complete_finite_unital_ring_tables","operation":"ring-power","semantic_template":"qf-mod27-ring-power/{\"addition\":[[0,1,2,3,4,5],[1,5,4,0,3,2],[2,4,0,5,1,3],[3,0,5,4,2,1],[4,3,1,2,5,0],[5,2,3,1,0,4]],\"element\":0,\"exponent\":20,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,2,0,0],[0,3,2,1,5,4],[0,4,0,5,4,5],[0,5,0,4,5,4]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0071","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-power；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,5,4,2,0,3],[2,4,5,1,3,0],[3,2,1,0,5,4],[4,0,3,5,2,1],[5,3,0,4,1,2]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,5,5],[0,3,0,3,3,0],[0,4,5,3,1,2],[0,5,5,0,2,2]],\"element\":5,\"exponent\":21}。請依 [element,exponent,one,result] 欄序回答。","answer":{"kind":"expression","value":"(5, 21, 1, 5)","explanation_zh":"Wave4根節點核對=(5, 21, 1, 5)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-power","seed":202610271071,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,5,4,2,0,3],[2,4,5,1,3,0],[3,2,1,0,5,4],[4,0,3,5,2,1],[5,3,0,4,1,2]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,5,5],[0,3,0,3,3,0],[0,4,5,3,1,2],[0,5,5,0,2,2]],"element":5,"exponent":21,"domain":"Complete_finite_unital_ring_tables","operation":"ring-power","semantic_template":"qf-mod27-ring-power/{\"addition\":[[0,1,2,3,4,5],[1,5,4,2,0,3],[2,4,5,1,3,0],[3,2,1,0,5,4],[4,0,3,5,2,1],[5,3,0,4,1,2]],\"element\":5,\"exponent\":21,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,5,5],[0,3,0,3,3,0],[0,4,5,3,1,2],[0,5,5,0,2,2]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0072","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-power；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,5,0,4,2,3],[2,0,4,5,3,1],[3,4,5,0,1,2],[4,2,3,1,5,0],[5,3,1,2,0,4]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,3,5,4],[0,3,3,3,0,0],[0,4,5,0,4,5],[0,5,4,0,5,4]],\"element\":4,\"exponent\":22}。請依 [element,exponent,one,result] 欄序回答。","answer":{"kind":"expression","value":"(4, 22, 1, 4)","explanation_zh":"Wave4根節點核對=(4, 22, 1, 4)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-power","seed":202610271072,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,5,0,4,2,3],[2,0,4,5,3,1],[3,4,5,0,1,2],[4,2,3,1,5,0],[5,3,1,2,0,4]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,3,5,4],[0,3,3,3,0,0],[0,4,5,0,4,5],[0,5,4,0,5,4]],"element":4,"exponent":22,"domain":"Complete_finite_unital_ring_tables","operation":"ring-power","semantic_template":"qf-mod27-ring-power/{\"addition\":[[0,1,2,3,4,5],[1,5,0,4,2,3],[2,0,4,5,3,1],[3,4,5,0,1,2],[4,2,3,1,5,0],[5,3,1,2,0,4]],\"element\":4,\"exponent\":22,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,3,5,4],[0,3,3,3,0,0],[0,4,5,0,4,5],[0,5,4,0,5,4]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0073","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-power；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,5,3,0,2,4],[2,3,5,4,1,0],[3,0,4,2,5,1],[4,2,1,5,0,3],[5,4,0,1,3,2]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,5,0,5],[0,3,5,1,4,2],[0,4,0,4,4,0],[0,5,5,2,0,2]],\"element\":3,\"exponent\":23}。請依 [element,exponent,one,result] 欄序回答。","answer":{"kind":"expression","value":"(3, 23, 1, 3)","explanation_zh":"Wave4根節點核對=(3, 23, 1, 3)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-power","seed":202610271073,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,5,3,0,2,4],[2,3,5,4,1,0],[3,0,4,2,5,1],[4,2,1,5,0,3],[5,4,0,1,3,2]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,5,0,5],[0,3,5,1,4,2],[0,4,0,4,4,0],[0,5,5,2,0,2]],"element":3,"exponent":23,"domain":"Complete_finite_unital_ring_tables","operation":"ring-power","semantic_template":"qf-mod27-ring-power/{\"addition\":[[0,1,2,3,4,5],[1,5,3,0,2,4],[2,3,5,4,1,0],[3,0,4,2,5,1],[4,2,1,5,0,3],[5,4,0,1,3,2]],\"element\":3,\"exponent\":23,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,5,0,5],[0,3,5,1,4,2],[0,4,0,4,4,0],[0,5,5,2,0,2]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0074","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-power；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,5,0,2,3,4],[2,0,3,4,5,1],[3,2,4,5,1,0],[4,3,5,1,0,2],[5,4,1,0,2,3]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,5,4,3],[0,3,5,3,0,5],[0,4,4,0,4,0],[0,5,3,5,0,3]],\"element\":2,\"exponent\":24}。請依 [element,exponent,one,result] 欄序回答。","answer":{"kind":"expression","value":"(2, 24, 1, 1)","explanation_zh":"Wave4根節點核對=(2, 24, 1, 1)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-power","seed":202610271074,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,5,0,2,3,4],[2,0,3,4,5,1],[3,2,4,5,1,0],[4,3,5,1,0,2],[5,4,1,0,2,3]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,5,4,3],[0,3,5,3,0,5],[0,4,4,0,4,0],[0,5,3,5,0,3]],"element":2,"exponent":24,"domain":"Complete_finite_unital_ring_tables","operation":"ring-power","semantic_template":"qf-mod27-ring-power/{\"addition\":[[0,1,2,3,4,5],[1,5,0,2,3,4],[2,0,3,4,5,1],[3,2,4,5,1,0],[4,3,5,1,0,2],[5,4,1,0,2,3]],\"element\":2,\"exponent\":24,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,5,4,3],[0,3,5,3,0,5],[0,4,4,0,4,0],[0,5,3,5,0,3]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0075","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-power；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,4,3,5,0,2],[2,3,1,4,5,0],[3,5,4,0,2,1],[4,0,5,2,1,3],[5,2,0,1,3,4]],\"multiplication\":[[0,0,0,0,0,0],[0,4,1,0,1,4],[0,1,2,3,4,5],[0,0,3,3,0,3],[0,1,4,0,4,1],[0,4,5,3,1,2]],\"element\":1,\"exponent\":25}。請依 [element,exponent,one,result] 欄序回答。","answer":{"kind":"expression","value":"(1, 25, 2, 1)","explanation_zh":"Wave4根節點核對=(1, 25, 2, 1)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-power","seed":202610271075,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,4,3,5,0,2],[2,3,1,4,5,0],[3,5,4,0,2,1],[4,0,5,2,1,3],[5,2,0,1,3,4]],"multiplication":[[0,0,0,0,0,0],[0,4,1,0,1,4],[0,1,2,3,4,5],[0,0,3,3,0,3],[0,1,4,0,4,1],[0,4,5,3,1,2]],"element":1,"exponent":25,"domain":"Complete_finite_unital_ring_tables","operation":"ring-power","semantic_template":"qf-mod27-ring-power/{\"addition\":[[0,1,2,3,4,5],[1,4,3,5,0,2],[2,3,1,4,5,0],[3,5,4,0,2,1],[4,0,5,2,1,3],[5,2,0,1,3,4]],\"element\":1,\"exponent\":25,\"multiplication\":[[0,0,0,0,0,0],[0,4,1,0,1,4],[0,1,2,3,4,5],[0,0,3,3,0,3],[0,1,4,0,4,1],[0,4,5,3,1,2]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0076","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-identities；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,4,0,2,4],[0,3,0,3,0,3],[0,4,2,0,4,2],[0,5,4,3,2,1]]}。請依 [zero,one,distinct] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 是)","explanation_zh":"Wave4根節點核對=(0, 1, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-identities","seed":202610271076,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,4,0,2,4],[0,3,0,3,0,3],[0,4,2,0,4,2],[0,5,4,3,2,1]],"domain":"Complete_finite_unital_ring_tables","operation":"ring-identities","semantic_template":"qf-mod27-ring-identities/{\"addition\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,4,0,2,4],[0,3,0,3,0,3],[0,4,2,0,4,2],[0,5,4,3,2,1]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0077","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-identities；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,2,3,5,0,4],[2,3,5,4,1,0],[3,5,4,0,2,1],[4,0,1,2,5,3],[5,4,0,1,3,2]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,5,0,5,2],[0,3,0,3,3,0],[0,4,5,3,1,2],[0,5,2,0,2,5]]}。請依 [zero,one,distinct] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 是)","explanation_zh":"Wave4根節點核對=(0, 1, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-identities","seed":202610271077,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,2,3,5,0,4],[2,3,5,4,1,0],[3,5,4,0,2,1],[4,0,1,2,5,3],[5,4,0,1,3,2]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,5,0,5,2],[0,3,0,3,3,0],[0,4,5,3,1,2],[0,5,2,0,2,5]],"domain":"Complete_finite_unital_ring_tables","operation":"ring-identities","semantic_template":"qf-mod27-ring-identities/{\"addition\":[[0,1,2,3,4,5],[1,2,3,5,0,4],[2,3,5,4,1,0],[3,5,4,0,2,1],[4,0,1,2,5,3],[5,4,0,1,3,2]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,5,0,5,2],[0,3,0,3,3,0],[0,4,5,3,1,2],[0,5,2,0,2,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0078","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-identities；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,2,4,5,3,0],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,3,5,1,0,2],[5,0,1,4,2,3]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,3,2,0,3],[0,3,2,3,0,2],[0,4,0,0,4,4],[0,5,3,2,4,1]]}。請依 [zero,one,distinct] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 是)","explanation_zh":"Wave4根節點核對=(0, 1, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-identities","seed":202610271078,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,2,4,5,3,0],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,3,5,1,0,2],[5,0,1,4,2,3]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,3,2,0,3],[0,3,2,3,0,2],[0,4,0,0,4,4],[0,5,3,2,4,1]],"domain":"Complete_finite_unital_ring_tables","operation":"ring-identities","semantic_template":"qf-mod27-ring-identities/{\"addition\":[[0,1,2,3,4,5],[1,2,4,5,3,0],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,3,5,1,0,2],[5,0,1,4,2,3]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,3,2,0,3],[0,3,2,3,0,2],[0,4,0,0,4,4],[0,5,3,2,4,1]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0079","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-identities；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,2,4,0,5,3],[2,4,5,1,3,0],[3,0,1,5,2,4],[4,5,3,2,0,1],[5,3,0,4,1,2]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,5,5,0,2],[0,3,5,1,4,2],[0,4,0,4,4,0],[0,5,2,2,0,5]]}。請依 [zero,one,distinct] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 是)","explanation_zh":"Wave4根節點核對=(0, 1, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-identities","seed":202610271079,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,2,4,0,5,3],[2,4,5,1,3,0],[3,0,1,5,2,4],[4,5,3,2,0,1],[5,3,0,4,1,2]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,5,5,0,2],[0,3,5,1,4,2],[0,4,0,4,4,0],[0,5,2,2,0,5]],"domain":"Complete_finite_unital_ring_tables","operation":"ring-identities","semantic_template":"qf-mod27-ring-identities/{\"addition\":[[0,1,2,3,4,5],[1,2,4,0,5,3],[2,4,5,1,3,0],[3,0,1,5,2,4],[4,5,3,2,0,1],[5,3,0,4,1,2]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,5,5,0,2],[0,3,5,1,4,2],[0,4,0,4,4,0],[0,5,2,2,0,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0080","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-identities；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,2,5,4,0,3],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,0,1,5,3,2],[5,3,4,1,2,0]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,3,2,3,0],[0,3,2,3,2,0],[0,4,3,2,1,5],[0,5,0,0,5,5]]}。請依 [zero,one,distinct] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 是)","explanation_zh":"Wave4根節點核對=(0, 1, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-identities","seed":202610271080,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,2,5,4,0,3],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,0,1,5,3,2],[5,3,4,1,2,0]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,3,2,3,0],[0,3,2,3,2,0],[0,4,3,2,1,5],[0,5,0,0,5,5]],"domain":"Complete_finite_unital_ring_tables","operation":"ring-identities","semantic_template":"qf-mod27-ring-identities/{\"addition\":[[0,1,2,3,4,5],[1,2,5,4,0,3],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,0,1,5,3,2],[5,3,4,1,2,0]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,3,2,3,0],[0,3,2,3,2,0],[0,4,3,2,1,5],[0,5,0,0,5,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0081","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-identities；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,2,5,0,3,4],[2,5,4,1,0,3],[3,0,1,4,5,2],[4,3,0,5,2,1],[5,4,3,2,1,0]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,4,4,2,0],[0,3,4,1,2,5],[0,4,2,2,4,0],[0,5,0,5,0,5]]}。請依 [zero,one,distinct] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 是)","explanation_zh":"Wave4根節點核對=(0, 1, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-identities","seed":202610271081,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,2,5,0,3,4],[2,5,4,1,0,3],[3,0,1,4,5,2],[4,3,0,5,2,1],[5,4,3,2,1,0]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,4,4,2,0],[0,3,4,1,2,5],[0,4,2,2,4,0],[0,5,0,5,0,5]],"domain":"Complete_finite_unital_ring_tables","operation":"ring-identities","semantic_template":"qf-mod27-ring-identities/{\"addition\":[[0,1,2,3,4,5],[1,2,5,0,3,4],[2,5,4,1,0,3],[3,0,1,4,5,2],[4,3,0,5,2,1],[5,4,3,2,1,0]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,4,4,2,0],[0,3,4,1,2,5],[0,4,2,2,4,0],[0,5,0,5,0,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0082","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-identities；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,3,4,2,5,0],[2,4,0,5,1,3],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,0,3,1,2,4]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,0,2],[0,3,0,4,3,4],[0,4,0,3,4,3],[0,5,2,4,3,1]]}。請依 [zero,one,distinct] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 是)","explanation_zh":"Wave4根節點核對=(0, 1, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-identities","seed":202610271082,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,3,4,2,5,0],[2,4,0,5,1,3],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,0,3,1,2,4]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,0,2],[0,3,0,4,3,4],[0,4,0,3,4,3],[0,5,2,4,3,1]],"domain":"Complete_finite_unital_ring_tables","operation":"ring-identities","semantic_template":"qf-mod27-ring-identities/{\"addition\":[[0,1,2,3,4,5],[1,3,4,2,5,0],[2,4,0,5,1,3],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,0,3,1,2,4]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,0,2],[0,3,0,4,3,4],[0,4,0,3,4,3],[0,5,2,4,3,1]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0083","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-identities；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,3,5,2,0,4],[2,5,0,4,3,1],[3,2,4,5,1,0],[4,0,3,1,5,2],[5,4,1,0,2,3]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,2,0],[0,3,0,5,5,3],[0,4,2,5,1,3],[0,5,0,3,3,5]]}。請依 [zero,one,distinct] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 是)","explanation_zh":"Wave4根節點核對=(0, 1, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-identities","seed":202610271083,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,3,5,2,0,4],[2,5,0,4,3,1],[3,2,4,5,1,0],[4,0,3,1,5,2],[5,4,1,0,2,3]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,2,0],[0,3,0,5,5,3],[0,4,2,5,1,3],[0,5,0,3,3,5]],"domain":"Complete_finite_unital_ring_tables","operation":"ring-identities","semantic_template":"qf-mod27-ring-identities/{\"addition\":[[0,1,2,3,4,5],[1,3,5,2,0,4],[2,5,0,4,3,1],[3,2,4,5,1,0],[4,0,3,1,5,2],[5,4,1,0,2,3]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,2,0],[0,3,0,5,5,3],[0,4,2,5,1,3],[0,5,0,3,3,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0084","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-identities；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,3,5,4,2,0],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,2,1,5,0,3],[5,0,4,1,3,2]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,3,0,3],[0,3,3,2,0,2],[0,4,0,0,4,4],[0,5,3,2,4,1]]}。請依 [zero,one,distinct] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 是)","explanation_zh":"Wave4根節點核對=(0, 1, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-identities","seed":202610271084,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,3,5,4,2,0],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,2,1,5,0,3],[5,0,4,1,3,2]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,3,0,3],[0,3,3,2,0,2],[0,4,0,0,4,4],[0,5,3,2,4,1]],"domain":"Complete_finite_unital_ring_tables","operation":"ring-identities","semantic_template":"qf-mod27-ring-identities/{\"addition\":[[0,1,2,3,4,5],[1,3,5,4,2,0],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,2,1,5,0,3],[5,0,4,1,3,2]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,3,0,3],[0,3,3,2,0,2],[0,4,0,0,4,4],[0,5,3,2,4,1]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0085","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-identities；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,3,0,4,5,2],[2,0,5,1,3,4],[3,4,1,5,2,0],[4,5,3,2,0,1],[5,2,4,0,1,3]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,5,4,3],[0,3,5,5,0,3],[0,4,4,0,4,0],[0,5,3,3,0,5]]}。請依 [zero,one,distinct] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 是)","explanation_zh":"Wave4根節點核對=(0, 1, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-identities","seed":202610271085,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,3,0,4,5,2],[2,0,5,1,3,4],[3,4,1,5,2,0],[4,5,3,2,0,1],[5,2,4,0,1,3]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,5,4,3],[0,3,5,5,0,3],[0,4,4,0,4,0],[0,5,3,3,0,5]],"domain":"Complete_finite_unital_ring_tables","operation":"ring-identities","semantic_template":"qf-mod27-ring-identities/{\"addition\":[[0,1,2,3,4,5],[1,3,0,4,5,2],[2,0,5,1,3,4],[3,4,1,5,2,0],[4,5,3,2,0,1],[5,2,4,0,1,3]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,5,4,3],[0,3,5,5,0,3],[0,4,4,0,4,0],[0,5,3,3,0,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0086","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-identities；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,3,4,5,0,2],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,0,5,1,2,3],[5,2,1,4,3,0]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,3,3,0],[0,3,3,2,2,0],[0,4,3,2,1,5],[0,5,0,0,5,5]]}。請依 [zero,one,distinct] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 是)","explanation_zh":"Wave4根節點核對=(0, 1, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-identities","seed":202610271086,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,3,4,5,0,2],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,0,5,1,2,3],[5,2,1,4,3,0]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,3,3,0],[0,3,3,2,2,0],[0,4,3,2,1,5],[0,5,0,0,5,5]],"domain":"Complete_finite_unital_ring_tables","operation":"ring-identities","semantic_template":"qf-mod27-ring-identities/{\"addition\":[[0,1,2,3,4,5],[1,3,4,5,0,2],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,0,5,1,2,3],[5,2,1,4,3,0]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,3,3,0],[0,3,3,2,2,0],[0,4,3,2,1,5],[0,5,0,0,5,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0087","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-identities；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,3,0,5,2,4],[2,0,4,1,5,3],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,4,3,2,1,0]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,4,3,5],[0,3,4,4,3,0],[0,4,3,3,4,0],[0,5,5,0,0,5]]}。請依 [zero,one,distinct] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 是)","explanation_zh":"Wave4根節點核對=(0, 1, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-identities","seed":202610271087,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,3,0,5,2,4],[2,0,4,1,5,3],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,4,3,2,1,0]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,4,3,5],[0,3,4,4,3,0],[0,4,3,3,4,0],[0,5,5,0,0,5]],"domain":"Complete_finite_unital_ring_tables","operation":"ring-identities","semantic_template":"qf-mod27-ring-identities/{\"addition\":[[0,1,2,3,4,5],[1,3,0,5,2,4],[2,0,4,1,5,3],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,4,3,2,1,0]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,4,3,5],[0,3,4,4,3,0],[0,4,3,3,4,0],[0,5,5,0,0,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0088","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-identities；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,4,3,5,2,0],[2,3,0,1,5,4],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,0,4,2,1,3]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,0,2],[0,3,0,3,4,4],[0,4,0,4,3,3],[0,5,2,4,3,1]]}。請依 [zero,one,distinct] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 是)","explanation_zh":"Wave4根節點核對=(0, 1, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-identities","seed":202610271088,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,4,3,5,2,0],[2,3,0,1,5,4],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,0,4,2,1,3]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,0,2],[0,3,0,3,4,4],[0,4,0,4,3,3],[0,5,2,4,3,1]],"domain":"Complete_finite_unital_ring_tables","operation":"ring-identities","semantic_template":"qf-mod27-ring-identities/{\"addition\":[[0,1,2,3,4,5],[1,4,3,5,2,0],[2,3,0,1,5,4],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,0,4,2,1,3]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,0,2],[0,3,0,3,4,4],[0,4,0,4,3,3],[0,5,2,4,3,1]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0089","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-identities；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,4,5,0,2,3],[2,5,0,4,3,1],[3,0,4,5,1,2],[4,2,3,1,5,0],[5,3,1,2,0,4]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,2,0,0],[0,3,2,1,5,4],[0,4,0,5,5,4],[0,5,0,4,4,5]]}。請依 [zero,one,distinct] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 是)","explanation_zh":"Wave4根節點核對=(0, 1, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-identities","seed":202610271089,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,4,5,0,2,3],[2,5,0,4,3,1],[3,0,4,5,1,2],[4,2,3,1,5,0],[5,3,1,2,0,4]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,2,0,0],[0,3,2,1,5,4],[0,4,0,5,5,4],[0,5,0,4,4,5]],"domain":"Complete_finite_unital_ring_tables","operation":"ring-identities","semantic_template":"qf-mod27-ring-identities/{\"addition\":[[0,1,2,3,4,5],[1,4,5,0,2,3],[2,5,0,4,3,1],[3,0,4,5,1,2],[4,2,3,1,5,0],[5,3,1,2,0,4]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,2,0,0],[0,3,2,1,5,4],[0,4,0,5,5,4],[0,5,0,4,4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0090","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-identities；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,4,5,2,3,0],[2,5,4,1,0,3],[3,2,1,0,5,4],[4,3,0,5,2,1],[5,0,3,4,1,2]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,4,4],[0,3,0,3,0,3],[0,4,4,0,2,2],[0,5,4,3,2,1]]}。請依 [zero,one,distinct] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 是)","explanation_zh":"Wave4根節點核對=(0, 1, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-identities","seed":202610271090,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,4,5,2,3,0],[2,5,4,1,0,3],[3,2,1,0,5,4],[4,3,0,5,2,1],[5,0,3,4,1,2]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,4,4],[0,3,0,3,0,3],[0,4,4,0,2,2],[0,5,4,3,2,1]],"domain":"Complete_finite_unital_ring_tables","operation":"ring-identities","semantic_template":"qf-mod27-ring-identities/{\"addition\":[[0,1,2,3,4,5],[1,4,5,2,3,0],[2,5,4,1,0,3],[3,2,1,0,5,4],[4,3,0,5,2,1],[5,0,3,4,1,2]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,4,4],[0,3,0,3,0,3],[0,4,4,0,2,2],[0,5,4,3,2,1]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0091","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-identities；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,4,0,5,3,2],[2,0,5,4,1,3],[3,5,4,0,2,1],[4,3,1,2,5,0],[5,2,3,1,0,4]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,3,5,4],[0,3,3,3,0,0],[0,4,5,0,5,4],[0,5,4,0,4,5]]}。請依 [zero,one,distinct] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 是)","explanation_zh":"Wave4根節點核對=(0, 1, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-identities","seed":202610271091,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,4,0,5,3,2],[2,0,5,4,1,3],[3,5,4,0,2,1],[4,3,1,2,5,0],[5,2,3,1,0,4]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,3,5,4],[0,3,3,3,0,0],[0,4,5,0,5,4],[0,5,4,0,4,5]],"domain":"Complete_finite_unital_ring_tables","operation":"ring-identities","semantic_template":"qf-mod27-ring-identities/{\"addition\":[[0,1,2,3,4,5],[1,4,0,5,3,2],[2,0,5,4,1,3],[3,5,4,0,2,1],[4,3,1,2,5,0],[5,2,3,1,0,4]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,3,5,4],[0,3,3,3,0,0],[0,4,5,0,5,4],[0,5,4,0,4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0092","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-identities；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,4,3,0,5,2],[2,3,4,5,0,1],[3,0,5,2,1,4],[4,5,0,1,2,3],[5,2,1,4,3,0]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,4,4,0],[0,3,4,1,2,5],[0,4,4,2,2,0],[0,5,0,5,0,5]]}。請依 [zero,one,distinct] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 是)","explanation_zh":"Wave4根節點核對=(0, 1, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-identities","seed":202610271092,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,4,3,0,5,2],[2,3,4,5,0,1],[3,0,5,2,1,4],[4,5,0,1,2,3],[5,2,1,4,3,0]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,4,4,0],[0,3,4,1,2,5],[0,4,4,2,2,0],[0,5,0,5,0,5]],"domain":"Complete_finite_unital_ring_tables","operation":"ring-identities","semantic_template":"qf-mod27-ring-identities/{\"addition\":[[0,1,2,3,4,5],[1,4,3,0,5,2],[2,3,4,5,0,1],[3,0,5,2,1,4],[4,5,0,1,2,3],[5,2,1,4,3,0]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,4,4,0],[0,3,4,1,2,5],[0,4,4,2,2,0],[0,5,0,5,0,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0093","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-identities；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,4,0,2,5,3],[2,0,3,5,1,4],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,3,4,1,2,0]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,4,3,5],[0,3,4,3,4,0],[0,4,3,4,3,0],[0,5,5,0,0,5]]}。請依 [zero,one,distinct] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 是)","explanation_zh":"Wave4根節點核對=(0, 1, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-identities","seed":202610271093,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,4,0,2,5,3],[2,0,3,5,1,4],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,3,4,1,2,0]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,4,3,5],[0,3,4,3,4,0],[0,4,3,4,3,0],[0,5,5,0,0,5]],"domain":"Complete_finite_unital_ring_tables","operation":"ring-identities","semantic_template":"qf-mod27-ring-identities/{\"addition\":[[0,1,2,3,4,5],[1,4,0,2,5,3],[2,0,3,5,1,4],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,3,4,1,2,0]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,4,3,5],[0,3,4,3,4,0],[0,4,3,4,3,0],[0,5,5,0,0,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0094","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-identities；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,5,3,4,0,2],[2,3,0,1,5,4],[3,4,1,5,2,0],[4,0,5,2,3,1],[5,2,4,0,1,3]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,2,0],[0,3,0,3,5,5],[0,4,2,5,1,3],[0,5,0,5,3,3]]}。請依 [zero,one,distinct] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 是)","explanation_zh":"Wave4根節點核對=(0, 1, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-identities","seed":202610271094,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,5,3,4,0,2],[2,3,0,1,5,4],[3,4,1,5,2,0],[4,0,5,2,3,1],[5,2,4,0,1,3]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,2,0],[0,3,0,3,5,5],[0,4,2,5,1,3],[0,5,0,5,3,3]],"domain":"Complete_finite_unital_ring_tables","operation":"ring-identities","semantic_template":"qf-mod27-ring-identities/{\"addition\":[[0,1,2,3,4,5],[1,5,3,4,0,2],[2,3,0,1,5,4],[3,4,1,5,2,0],[4,0,5,2,3,1],[5,2,4,0,1,3]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,2,0],[0,3,0,3,5,5],[0,4,2,5,1,3],[0,5,0,5,3,3]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0095","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-identities；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,5,4,0,3,2],[2,4,0,5,1,3],[3,0,5,4,2,1],[4,3,1,2,5,0],[5,2,3,1,0,4]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,2,0,0],[0,3,2,1,5,4],[0,4,0,5,4,5],[0,5,0,4,5,4]]}。請依 [zero,one,distinct] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 是)","explanation_zh":"Wave4根節點核對=(0, 1, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-identities","seed":202610271095,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,5,4,0,3,2],[2,4,0,5,1,3],[3,0,5,4,2,1],[4,3,1,2,5,0],[5,2,3,1,0,4]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,2,0,0],[0,3,2,1,5,4],[0,4,0,5,4,5],[0,5,0,4,5,4]],"domain":"Complete_finite_unital_ring_tables","operation":"ring-identities","semantic_template":"qf-mod27-ring-identities/{\"addition\":[[0,1,2,3,4,5],[1,5,4,0,3,2],[2,4,0,5,1,3],[3,0,5,4,2,1],[4,3,1,2,5,0],[5,2,3,1,0,4]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,2,0,0],[0,3,2,1,5,4],[0,4,0,5,4,5],[0,5,0,4,5,4]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0096","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-identities；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,5,4,2,0,3],[2,4,5,1,3,0],[3,2,1,0,5,4],[4,0,3,5,2,1],[5,3,0,4,1,2]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,5,5],[0,3,0,3,3,0],[0,4,5,3,1,2],[0,5,5,0,2,2]]}。請依 [zero,one,distinct] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 是)","explanation_zh":"Wave4根節點核對=(0, 1, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-identities","seed":202610271096,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,5,4,2,0,3],[2,4,5,1,3,0],[3,2,1,0,5,4],[4,0,3,5,2,1],[5,3,0,4,1,2]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,5,5],[0,3,0,3,3,0],[0,4,5,3,1,2],[0,5,5,0,2,2]],"domain":"Complete_finite_unital_ring_tables","operation":"ring-identities","semantic_template":"qf-mod27-ring-identities/{\"addition\":[[0,1,2,3,4,5],[1,5,4,2,0,3],[2,4,5,1,3,0],[3,2,1,0,5,4],[4,0,3,5,2,1],[5,3,0,4,1,2]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,5,5],[0,3,0,3,3,0],[0,4,5,3,1,2],[0,5,5,0,2,2]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0097","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-identities；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,5,0,4,2,3],[2,0,4,5,3,1],[3,4,5,0,1,2],[4,2,3,1,5,0],[5,3,1,2,0,4]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,3,5,4],[0,3,3,3,0,0],[0,4,5,0,4,5],[0,5,4,0,5,4]]}。請依 [zero,one,distinct] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 是)","explanation_zh":"Wave4根節點核對=(0, 1, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-identities","seed":202610271097,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,5,0,4,2,3],[2,0,4,5,3,1],[3,4,5,0,1,2],[4,2,3,1,5,0],[5,3,1,2,0,4]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,3,5,4],[0,3,3,3,0,0],[0,4,5,0,4,5],[0,5,4,0,5,4]],"domain":"Complete_finite_unital_ring_tables","operation":"ring-identities","semantic_template":"qf-mod27-ring-identities/{\"addition\":[[0,1,2,3,4,5],[1,5,0,4,2,3],[2,0,4,5,3,1],[3,4,5,0,1,2],[4,2,3,1,5,0],[5,3,1,2,0,4]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,3,5,4],[0,3,3,3,0,0],[0,4,5,0,4,5],[0,5,4,0,5,4]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0098","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-identities；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,5,3,0,2,4],[2,3,5,4,1,0],[3,0,4,2,5,1],[4,2,1,5,0,3],[5,4,0,1,3,2]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,5,0,5],[0,3,5,1,4,2],[0,4,0,4,4,0],[0,5,5,2,0,2]]}。請依 [zero,one,distinct] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 是)","explanation_zh":"Wave4根節點核對=(0, 1, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-identities","seed":202610271098,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,5,3,0,2,4],[2,3,5,4,1,0],[3,0,4,2,5,1],[4,2,1,5,0,3],[5,4,0,1,3,2]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,5,0,5],[0,3,5,1,4,2],[0,4,0,4,4,0],[0,5,5,2,0,2]],"domain":"Complete_finite_unital_ring_tables","operation":"ring-identities","semantic_template":"qf-mod27-ring-identities/{\"addition\":[[0,1,2,3,4,5],[1,5,3,0,2,4],[2,3,5,4,1,0],[3,0,4,2,5,1],[4,2,1,5,0,3],[5,4,0,1,3,2]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,5,0,5],[0,3,5,1,4,2],[0,4,0,4,4,0],[0,5,5,2,0,2]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0099","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-identities；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,5,0,2,3,4],[2,0,3,4,5,1],[3,2,4,5,1,0],[4,3,5,1,0,2],[5,4,1,0,2,3]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,5,4,3],[0,3,5,3,0,5],[0,4,4,0,4,0],[0,5,3,5,0,3]]}。請依 [zero,one,distinct] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 是)","explanation_zh":"Wave4根節點核對=(0, 1, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-identities","seed":202610271099,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,5,0,2,3,4],[2,0,3,4,5,1],[3,2,4,5,1,0],[4,3,5,1,0,2],[5,4,1,0,2,3]],"multiplication":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,5,4,3],[0,3,5,3,0,5],[0,4,4,0,4,0],[0,5,3,5,0,3]],"domain":"Complete_finite_unital_ring_tables","operation":"ring-identities","semantic_template":"qf-mod27-ring-identities/{\"addition\":[[0,1,2,3,4,5],[1,5,0,2,3,4],[2,0,3,4,5,1],[3,2,4,5,1,0],[4,3,5,1,0,2],[5,4,1,0,2,3]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,5,4,3],[0,3,5,3,0,5],[0,4,4,0,4,0],[0,5,3,5,0,3]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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-20261027-r01-finite-unital-ring-0100","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-unital-ring"],"title_zh":"有限含一環","stem_zh":"同一完整有限載體上的加法表與乘法表。操作=ring-identities；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,4,3,5,0,2],[2,3,1,4,5,0],[3,5,4,0,2,1],[4,0,5,2,1,3],[5,2,0,1,3,4]],\"multiplication\":[[0,0,0,0,0,0],[0,4,1,0,1,4],[0,1,2,3,4,5],[0,0,3,3,0,3],[0,1,4,0,4,1],[0,4,5,3,1,2]]}。請依 [zero,one,distinct] 欄序回答。","answer":{"kind":"expression","value":"(0, 2, 是)","explanation_zh":"Wave4根節點核對=(0, 2, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限含一環"]},"variant":{"family_id":"qf-mod27-ring-identities","seed":202610271100,"parameters":{"n":6,"addition":[[0,1,2,3,4,5],[1,4,3,5,0,2],[2,3,1,4,5,0],[3,5,4,0,2,1],[4,0,5,2,1,3],[5,2,0,1,3,4]],"multiplication":[[0,0,0,0,0,0],[0,4,1,0,1,4],[0,1,2,3,4,5],[0,0,3,3,0,3],[0,1,4,0,4,1],[0,4,5,3,1,2]],"domain":"Complete_finite_unital_ring_tables","operation":"ring-identities","semantic_template":"qf-mod27-ring-identities/{\"addition\":[[0,1,2,3,4,5],[1,4,3,5,0,2],[2,3,1,4,5,0],[3,5,4,0,2,1],[4,0,5,2,1,3],[5,2,0,1,3,4]],\"multiplication\":[[0,0,0,0,0,0],[0,4,1,0,1,4],[0,1,2,3,4,5],[0,0,3,3,0,3],[0,1,4,0,4,1],[0,4,5,3,1,2]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、一般完備性、內積／有界性、mutual independence、directed graph、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"}}