{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0001","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-normalize；資料={\"n\":4,\"addition\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"multiplication\":[[0,0,0,0],[0,1,2,3],[0,2,0,2],[0,3,2,1]],\"coefficients\":[1,3,0,0]}。請依 [raw,zero,normalized,degree] 欄序回答。","answer":{"kind":"expression","value":"([1, 3, 0, 0], 0, [1, 3], 1)","explanation_zh":"Wave4第三層核對=([1, 3, 0, 0], 0, [1, 3], 1)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-normalize","seed":202610292001,"parameters":{"n":4,"addition":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"multiplication":[[0,0,0,0],[0,1,2,3],[0,2,0,2],[0,3,2,1]],"coefficients":[1,3,0,0],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-normalize","semantic_template":"qf-mod29-poly-normalize/{\"addition\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"coefficients\":[1,3,0,0],\"multiplication\":[[0,0,0,0],[0,1,2,3],[0,2,0,2],[0,3,2,1]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0002","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-normalize；資料={\"n\":4,\"addition\":[[0,1,2,3],[1,3,0,2],[2,0,3,1],[3,2,1,0]],\"multiplication\":[[0,0,0,0],[0,1,2,3],[0,2,1,3],[0,3,3,0]],\"coefficients\":[3,0,0,0]}。請依 [raw,zero,normalized,degree] 欄序回答。","answer":{"kind":"expression","value":"([3, 0, 0, 0], 0, [3], 0)","explanation_zh":"Wave4第三層核對=([3, 0, 0, 0], 0, [3], 0)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-normalize","seed":202610292002,"parameters":{"n":4,"addition":[[0,1,2,3],[1,3,0,2],[2,0,3,1],[3,2,1,0]],"multiplication":[[0,0,0,0],[0,1,2,3],[0,2,1,3],[0,3,3,0]],"coefficients":[3,0,0,0],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-normalize","semantic_template":"qf-mod29-poly-normalize/{\"addition\":[[0,1,2,3],[1,3,0,2],[2,0,3,1],[3,2,1,0]],\"coefficients\":[3,0,0,0],\"multiplication\":[[0,0,0,0],[0,1,2,3],[0,2,1,3],[0,3,3,0]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0003","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-normalize；資料={\"n\":4,\"addition\":[[0,1,2,3],[1,0,3,2],[2,3,1,0],[3,2,0,1]],\"multiplication\":[[0,0,0,0],[0,0,1,1],[0,1,2,3],[0,1,3,2]],\"coefficients\":[3,2,0,0]}。請依 [raw,zero,normalized,degree] 欄序回答。","answer":{"kind":"expression","value":"([3, 2, 0, 0], 0, [3, 2], 1)","explanation_zh":"Wave4第三層核對=([3, 2, 0, 0], 0, [3, 2], 1)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-normalize","seed":202610292003,"parameters":{"n":4,"addition":[[0,1,2,3],[1,0,3,2],[2,3,1,0],[3,2,0,1]],"multiplication":[[0,0,0,0],[0,0,1,1],[0,1,2,3],[0,1,3,2]],"coefficients":[3,2,0,0],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-normalize","semantic_template":"qf-mod29-poly-normalize/{\"addition\":[[0,1,2,3],[1,0,3,2],[2,3,1,0],[3,2,0,1]],\"coefficients\":[3,2,0,0],\"multiplication\":[[0,0,0,0],[0,0,1,1],[0,1,2,3],[0,1,3,2]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0004","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-normalize；資料={\"n\":4,\"addition\":[[0,1,2,3],[1,3,0,2],[2,0,3,1],[3,2,1,0]],\"multiplication\":[[0,0,0,0],[0,2,1,3],[0,1,2,3],[0,3,3,0]],\"coefficients\":[0,3,0,0]}。請依 [raw,zero,normalized,degree] 欄序回答。","answer":{"kind":"expression","value":"([0, 3, 0, 0], 0, [0, 3], 1)","explanation_zh":"Wave4第三層核對=([0, 3, 0, 0], 0, [0, 3], 1)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-normalize","seed":202610292004,"parameters":{"n":4,"addition":[[0,1,2,3],[1,3,0,2],[2,0,3,1],[3,2,1,0]],"multiplication":[[0,0,0,0],[0,2,1,3],[0,1,2,3],[0,3,3,0]],"coefficients":[0,3,0,0],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-normalize","semantic_template":"qf-mod29-poly-normalize/{\"addition\":[[0,1,2,3],[1,3,0,2],[2,0,3,1],[3,2,1,0]],\"coefficients\":[0,3,0,0],\"multiplication\":[[0,0,0,0],[0,2,1,3],[0,1,2,3],[0,3,3,0]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0005","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-normalize；資料={\"n\":4,\"addition\":[[0,1,2,3],[1,0,3,2],[2,3,1,0],[3,2,0,1]],\"multiplication\":[[0,0,0,0],[0,0,1,1],[0,1,3,2],[0,1,2,3]],\"coefficients\":[3,2,0,0]}。請依 [raw,zero,normalized,degree] 欄序回答。","answer":{"kind":"expression","value":"([3, 2, 0, 0], 0, [3, 2], 1)","explanation_zh":"Wave4第三層核對=([3, 2, 0, 0], 0, [3, 2], 1)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-normalize","seed":202610292005,"parameters":{"n":4,"addition":[[0,1,2,3],[1,0,3,2],[2,3,1,0],[3,2,0,1]],"multiplication":[[0,0,0,0],[0,0,1,1],[0,1,3,2],[0,1,2,3]],"coefficients":[3,2,0,0],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-normalize","semantic_template":"qf-mod29-poly-normalize/{\"addition\":[[0,1,2,3],[1,0,3,2],[2,3,1,0],[3,2,0,1]],\"coefficients\":[3,2,0,0],\"multiplication\":[[0,0,0,0],[0,0,1,1],[0,1,3,2],[0,1,2,3]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0006","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-normalize；資料={\"n\":4,\"addition\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"multiplication\":[[0,0,0,0],[0,3,2,1],[0,2,0,2],[0,1,2,3]],\"coefficients\":[2,0,0,0]}。請依 [raw,zero,normalized,degree] 欄序回答。","answer":{"kind":"expression","value":"([2, 0, 0, 0], 0, [2], 0)","explanation_zh":"Wave4第三層核對=([2, 0, 0, 0], 0, [2], 0)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-normalize","seed":202610292006,"parameters":{"n":4,"addition":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"multiplication":[[0,0,0,0],[0,3,2,1],[0,2,0,2],[0,1,2,3]],"coefficients":[2,0,0,0],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-normalize","semantic_template":"qf-mod29-poly-normalize/{\"addition\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"coefficients\":[2,0,0,0],\"multiplication\":[[0,0,0,0],[0,3,2,1],[0,2,0,2],[0,1,2,3]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0007","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-normalize；資料={\"n\":4,\"addition\":[[2,0,3,1],[0,1,2,3],[3,2,1,0],[1,3,0,2]],\"multiplication\":[[0,1,2,3],[1,1,1,1],[2,1,1,2],[3,1,2,0]],\"coefficients\":[3,0,1,1]}。請依 [raw,zero,normalized,degree] 欄序回答。","answer":{"kind":"expression","value":"([3, 0, 1, 1], 1, [3, 0], 1)","explanation_zh":"Wave4第三層核對=([3, 0, 1, 1], 1, [3, 0], 1)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-normalize","seed":202610292007,"parameters":{"n":4,"addition":[[2,0,3,1],[0,1,2,3],[3,2,1,0],[1,3,0,2]],"multiplication":[[0,1,2,3],[1,1,1,1],[2,1,1,2],[3,1,2,0]],"coefficients":[3,0,1,1],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-normalize","semantic_template":"qf-mod29-poly-normalize/{\"addition\":[[2,0,3,1],[0,1,2,3],[3,2,1,0],[1,3,0,2]],\"coefficients\":[3,0,1,1],\"multiplication\":[[0,1,2,3],[1,1,1,1],[2,1,1,2],[3,1,2,0]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0008","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-normalize；資料={\"n\":4,\"addition\":[[3,0,1,2],[0,1,2,3],[1,2,3,0],[2,3,0,1]],\"multiplication\":[[0,1,2,3],[1,1,1,1],[2,1,0,3],[3,1,3,1]],\"coefficients\":[1,3,1,1]}。請依 [raw,zero,normalized,degree] 欄序回答。","answer":{"kind":"expression","value":"([1, 3, 1, 1], 1, [1, 3], 1)","explanation_zh":"Wave4第三層核對=([1, 3, 1, 1], 1, [1, 3], 1)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-normalize","seed":202610292008,"parameters":{"n":4,"addition":[[3,0,1,2],[0,1,2,3],[1,2,3,0],[2,3,0,1]],"multiplication":[[0,1,2,3],[1,1,1,1],[2,1,0,3],[3,1,3,1]],"coefficients":[1,3,1,1],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-normalize","semantic_template":"qf-mod29-poly-normalize/{\"addition\":[[3,0,1,2],[0,1,2,3],[1,2,3,0],[2,3,0,1]],\"coefficients\":[1,3,1,1],\"multiplication\":[[0,1,2,3],[1,1,1,1],[2,1,0,3],[3,1,3,1]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0009","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-normalize；資料={\"n\":4,\"addition\":[[1,0,3,2],[0,1,2,3],[3,2,0,1],[2,3,1,0]],\"multiplication\":[[1,1,0,0],[1,1,1,1],[0,1,2,3],[0,1,3,2]],\"coefficients\":[2,3,1,1]}。請依 [raw,zero,normalized,degree] 欄序回答。","answer":{"kind":"expression","value":"([2, 3, 1, 1], 1, [2, 3], 1)","explanation_zh":"Wave4第三層核對=([2, 3, 1, 1], 1, [2, 3], 1)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-normalize","seed":202610292009,"parameters":{"n":4,"addition":[[1,0,3,2],[0,1,2,3],[3,2,0,1],[2,3,1,0]],"multiplication":[[1,1,0,0],[1,1,1,1],[0,1,2,3],[0,1,3,2]],"coefficients":[2,3,1,1],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-normalize","semantic_template":"qf-mod29-poly-normalize/{\"addition\":[[1,0,3,2],[0,1,2,3],[3,2,0,1],[2,3,1,0]],\"coefficients\":[2,3,1,1],\"multiplication\":[[1,1,0,0],[1,1,1,1],[0,1,2,3],[0,1,3,2]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0010","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-normalize；資料={\"n\":4,\"addition\":[[3,0,1,2],[0,1,2,3],[1,2,3,0],[2,3,0,1]],\"multiplication\":[[2,1,0,3],[1,1,1,1],[0,1,2,3],[3,1,3,1]],\"coefficients\":[3,1,1,1]}。請依 [raw,zero,normalized,degree] 欄序回答。","answer":{"kind":"expression","value":"([3, 1, 1, 1], 1, [3], 0)","explanation_zh":"Wave4第三層核對=([3, 1, 1, 1], 1, [3], 0)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-normalize","seed":202610292010,"parameters":{"n":4,"addition":[[3,0,1,2],[0,1,2,3],[1,2,3,0],[2,3,0,1]],"multiplication":[[2,1,0,3],[1,1,1,1],[0,1,2,3],[3,1,3,1]],"coefficients":[3,1,1,1],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-normalize","semantic_template":"qf-mod29-poly-normalize/{\"addition\":[[3,0,1,2],[0,1,2,3],[1,2,3,0],[2,3,0,1]],\"coefficients\":[3,1,1,1],\"multiplication\":[[2,1,0,3],[1,1,1,1],[0,1,2,3],[3,1,3,1]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0011","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-normalize；資料={\"n\":4,\"addition\":[[1,0,3,2],[0,1,2,3],[3,2,0,1],[2,3,1,0]],\"multiplication\":[[1,1,0,0],[1,1,1,1],[0,1,3,2],[0,1,2,3]],\"coefficients\":[2,3,1,1]}。請依 [raw,zero,normalized,degree] 欄序回答。","answer":{"kind":"expression","value":"([2, 3, 1, 1], 1, [2, 3], 1)","explanation_zh":"Wave4第三層核對=([2, 3, 1, 1], 1, [2, 3], 1)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-normalize","seed":202610292011,"parameters":{"n":4,"addition":[[1,0,3,2],[0,1,2,3],[3,2,0,1],[2,3,1,0]],"multiplication":[[1,1,0,0],[1,1,1,1],[0,1,3,2],[0,1,2,3]],"coefficients":[2,3,1,1],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-normalize","semantic_template":"qf-mod29-poly-normalize/{\"addition\":[[1,0,3,2],[0,1,2,3],[3,2,0,1],[2,3,1,0]],\"coefficients\":[2,3,1,1],\"multiplication\":[[1,1,0,0],[1,1,1,1],[0,1,3,2],[0,1,2,3]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0012","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-normalize；資料={\"n\":4,\"addition\":[[2,0,3,1],[0,1,2,3],[3,2,1,0],[1,3,0,2]],\"multiplication\":[[3,1,2,0],[1,1,1,1],[2,1,1,2],[0,1,2,3]],\"coefficients\":[1,2,1,1]}。請依 [raw,zero,normalized,degree] 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 1, 1], 1, [1, 2], 1)","explanation_zh":"Wave4第三層核對=([1, 2, 1, 1], 1, [1, 2], 1)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-normalize","seed":202610292012,"parameters":{"n":4,"addition":[[2,0,3,1],[0,1,2,3],[3,2,1,0],[1,3,0,2]],"multiplication":[[3,1,2,0],[1,1,1,1],[2,1,1,2],[0,1,2,3]],"coefficients":[1,2,1,1],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-normalize","semantic_template":"qf-mod29-poly-normalize/{\"addition\":[[2,0,3,1],[0,1,2,3],[3,2,1,0],[1,3,0,2]],\"coefficients\":[1,2,1,1],\"multiplication\":[[3,1,2,0],[1,1,1,1],[2,1,1,2],[0,1,2,3]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0013","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-normalize；資料={\"n\":4,\"addition\":[[1,3,0,2],[3,2,1,0],[0,1,2,3],[2,0,3,1]],\"multiplication\":[[0,1,2,3],[1,2,2,1],[2,2,2,2],[3,1,2,0]],\"coefficients\":[0,3,2,2]}。請依 [raw,zero,normalized,degree] 欄序回答。","answer":{"kind":"expression","value":"([0, 3, 2, 2], 2, [0, 3], 1)","explanation_zh":"Wave4第三層核對=([0, 3, 2, 2], 2, [0, 3], 1)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-normalize","seed":202610292013,"parameters":{"n":4,"addition":[[1,3,0,2],[3,2,1,0],[0,1,2,3],[2,0,3,1]],"multiplication":[[0,1,2,3],[1,2,2,1],[2,2,2,2],[3,1,2,0]],"coefficients":[0,3,2,2],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-normalize","semantic_template":"qf-mod29-poly-normalize/{\"addition\":[[1,3,0,2],[3,2,1,0],[0,1,2,3],[2,0,3,1]],\"coefficients\":[0,3,2,2],\"multiplication\":[[0,1,2,3],[1,2,2,1],[2,2,2,2],[3,1,2,0]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0014","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-normalize；資料={\"n\":4,\"addition\":[[3,2,0,1],[2,3,1,0],[0,1,2,3],[1,0,3,2]],\"multiplication\":[[0,1,2,3],[1,0,2,3],[2,2,2,2],[3,3,2,2]],\"coefficients\":[3,2,2,2]}。請依 [raw,zero,normalized,degree] 欄序回答。","answer":{"kind":"expression","value":"([3, 2, 2, 2], 2, [3], 0)","explanation_zh":"Wave4第三層核對=([3, 2, 2, 2], 2, [3], 0)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-normalize","seed":202610292014,"parameters":{"n":4,"addition":[[3,2,0,1],[2,3,1,0],[0,1,2,3],[1,0,3,2]],"multiplication":[[0,1,2,3],[1,0,2,3],[2,2,2,2],[3,3,2,2]],"coefficients":[3,2,2,2],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-normalize","semantic_template":"qf-mod29-poly-normalize/{\"addition\":[[3,2,0,1],[2,3,1,0],[0,1,2,3],[1,0,3,2]],\"coefficients\":[3,2,2,2],\"multiplication\":[[0,1,2,3],[1,0,2,3],[2,2,2,2],[3,3,2,2]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0015","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-normalize；資料={\"n\":4,\"addition\":[[2,3,0,1],[3,0,1,2],[0,1,2,3],[1,2,3,0]],\"multiplication\":[[2,0,2,0],[0,1,2,3],[2,2,2,2],[0,3,2,1]],\"coefficients\":[3,1,2,2]}。請依 [raw,zero,normalized,degree] 欄序回答。","answer":{"kind":"expression","value":"([3, 1, 2, 2], 2, [3, 1], 1)","explanation_zh":"Wave4第三層核對=([3, 1, 2, 2], 2, [3, 1], 1)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-normalize","seed":202610292015,"parameters":{"n":4,"addition":[[2,3,0,1],[3,0,1,2],[0,1,2,3],[1,2,3,0]],"multiplication":[[2,0,2,0],[0,1,2,3],[2,2,2,2],[0,3,2,1]],"coefficients":[3,1,2,2],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-normalize","semantic_template":"qf-mod29-poly-normalize/{\"addition\":[[2,3,0,1],[3,0,1,2],[0,1,2,3],[1,2,3,0]],\"coefficients\":[3,1,2,2],\"multiplication\":[[2,0,2,0],[0,1,2,3],[2,2,2,2],[0,3,2,1]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0016","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-normalize；資料={\"n\":4,\"addition\":[[3,2,0,1],[2,3,1,0],[0,1,2,3],[1,0,3,2]],\"multiplication\":[[1,0,2,3],[0,1,2,3],[2,2,2,2],[3,3,2,2]],\"coefficients\":[2,3,2,2]}。請依 [raw,zero,normalized,degree] 欄序回答。","answer":{"kind":"expression","value":"([2, 3, 2, 2], 2, [2, 3], 1)","explanation_zh":"Wave4第三層核對=([2, 3, 2, 2], 2, [2, 3], 1)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-normalize","seed":202610292016,"parameters":{"n":4,"addition":[[3,2,0,1],[2,3,1,0],[0,1,2,3],[1,0,3,2]],"multiplication":[[1,0,2,3],[0,1,2,3],[2,2,2,2],[3,3,2,2]],"coefficients":[2,3,2,2],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-normalize","semantic_template":"qf-mod29-poly-normalize/{\"addition\":[[3,2,0,1],[2,3,1,0],[0,1,2,3],[1,0,3,2]],\"coefficients\":[2,3,2,2],\"multiplication\":[[1,0,2,3],[0,1,2,3],[2,2,2,2],[3,3,2,2]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0017","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-normalize；資料={\"n\":4,\"addition\":[[2,3,0,1],[3,0,1,2],[0,1,2,3],[1,2,3,0]],\"multiplication\":[[2,0,2,0],[0,3,2,1],[2,2,2,2],[0,1,2,3]],\"coefficients\":[3,1,2,2]}。請依 [raw,zero,normalized,degree] 欄序回答。","answer":{"kind":"expression","value":"([3, 1, 2, 2], 2, [3, 1], 1)","explanation_zh":"Wave4第三層核對=([3, 1, 2, 2], 2, [3, 1], 1)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-normalize","seed":202610292017,"parameters":{"n":4,"addition":[[2,3,0,1],[3,0,1,2],[0,1,2,3],[1,2,3,0]],"multiplication":[[2,0,2,0],[0,3,2,1],[2,2,2,2],[0,1,2,3]],"coefficients":[3,1,2,2],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-normalize","semantic_template":"qf-mod29-poly-normalize/{\"addition\":[[2,3,0,1],[3,0,1,2],[0,1,2,3],[1,2,3,0]],\"coefficients\":[3,1,2,2],\"multiplication\":[[2,0,2,0],[0,3,2,1],[2,2,2,2],[0,1,2,3]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0018","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-normalize；資料={\"n\":4,\"addition\":[[1,3,0,2],[3,2,1,0],[0,1,2,3],[2,0,3,1]],\"multiplication\":[[3,1,2,0],[1,2,2,1],[2,2,2,2],[0,1,2,3]],\"coefficients\":[1,2,2,2]}。請依 [raw,zero,normalized,degree] 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 2, 2], 2, [1], 0)","explanation_zh":"Wave4第三層核對=([1, 2, 2, 2], 2, [1], 0)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-normalize","seed":202610292018,"parameters":{"n":4,"addition":[[1,3,0,2],[3,2,1,0],[0,1,2,3],[2,0,3,1]],"multiplication":[[3,1,2,0],[1,2,2,1],[2,2,2,2],[0,1,2,3]],"coefficients":[1,2,2,2],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-normalize","semantic_template":"qf-mod29-poly-normalize/{\"addition\":[[1,3,0,2],[3,2,1,0],[0,1,2,3],[2,0,3,1]],\"coefficients\":[1,2,2,2],\"multiplication\":[[3,1,2,0],[1,2,2,1],[2,2,2,2],[0,1,2,3]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0019","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-normalize；資料={\"n\":4,\"addition\":[[1,2,3,0],[2,3,0,1],[3,0,1,2],[0,1,2,3]],\"multiplication\":[[0,1,2,3],[1,3,1,3],[2,1,0,3],[3,3,3,3]],\"coefficients\":[2,0,3,3]}。請依 [raw,zero,normalized,degree] 欄序回答。","answer":{"kind":"expression","value":"([2, 0, 3, 3], 3, [2, 0], 1)","explanation_zh":"Wave4第三層核對=([2, 0, 3, 3], 3, [2, 0], 1)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-normalize","seed":202610292019,"parameters":{"n":4,"addition":[[1,2,3,0],[2,3,0,1],[3,0,1,2],[0,1,2,3]],"multiplication":[[0,1,2,3],[1,3,1,3],[2,1,0,3],[3,3,3,3]],"coefficients":[2,0,3,3],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-normalize","semantic_template":"qf-mod29-poly-normalize/{\"addition\":[[1,2,3,0],[2,3,0,1],[3,0,1,2],[0,1,2,3]],\"coefficients\":[2,0,3,3],\"multiplication\":[[0,1,2,3],[1,3,1,3],[2,1,0,3],[3,3,3,3]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0020","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-normalize；資料={\"n\":4,\"addition\":[[2,3,1,0],[3,2,0,1],[1,0,3,2],[0,1,2,3]],\"multiplication\":[[0,1,2,3],[1,0,2,3],[2,2,3,3],[3,3,3,3]],\"coefficients\":[3,2,3,3]}。請依 [raw,zero,normalized,degree] 欄序回答。","answer":{"kind":"expression","value":"([3, 2, 3, 3], 3, [3, 2], 1)","explanation_zh":"Wave4第三層核對=([3, 2, 3, 3], 3, [3, 2], 1)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-normalize","seed":202610292020,"parameters":{"n":4,"addition":[[2,3,1,0],[3,2,0,1],[1,0,3,2],[0,1,2,3]],"multiplication":[[0,1,2,3],[1,0,2,3],[2,2,3,3],[3,3,3,3]],"coefficients":[3,2,3,3],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-normalize","semantic_template":"qf-mod29-poly-normalize/{\"addition\":[[2,3,1,0],[3,2,0,1],[1,0,3,2],[0,1,2,3]],\"coefficients\":[3,2,3,3],\"multiplication\":[[0,1,2,3],[1,0,2,3],[2,2,3,3],[3,3,3,3]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0021","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-normalize；資料={\"n\":4,\"addition\":[[3,2,1,0],[2,0,3,1],[1,3,0,2],[0,1,2,3]],\"multiplication\":[[3,0,0,3],[0,1,2,3],[0,2,1,3],[3,3,3,3]],\"coefficients\":[1,2,3,3]}。請依 [raw,zero,normalized,degree] 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 3, 3], 3, [1, 2], 1)","explanation_zh":"Wave4第三層核對=([1, 2, 3, 3], 3, [1, 2], 1)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-normalize","seed":202610292021,"parameters":{"n":4,"addition":[[3,2,1,0],[2,0,3,1],[1,3,0,2],[0,1,2,3]],"multiplication":[[3,0,0,3],[0,1,2,3],[0,2,1,3],[3,3,3,3]],"coefficients":[1,2,3,3],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-normalize","semantic_template":"qf-mod29-poly-normalize/{\"addition\":[[3,2,1,0],[2,0,3,1],[1,3,0,2],[0,1,2,3]],\"coefficients\":[1,2,3,3],\"multiplication\":[[3,0,0,3],[0,1,2,3],[0,2,1,3],[3,3,3,3]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0022","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-normalize；資料={\"n\":4,\"addition\":[[2,3,1,0],[3,2,0,1],[1,0,3,2],[0,1,2,3]],\"multiplication\":[[1,0,2,3],[0,1,2,3],[2,2,3,3],[3,3,3,3]],\"coefficients\":[2,3,3,3]}。請依 [raw,zero,normalized,degree] 欄序回答。","answer":{"kind":"expression","value":"([2, 3, 3, 3], 3, [2], 0)","explanation_zh":"Wave4第三層核對=([2, 3, 3, 3], 3, [2], 0)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-normalize","seed":202610292022,"parameters":{"n":4,"addition":[[2,3,1,0],[3,2,0,1],[1,0,3,2],[0,1,2,3]],"multiplication":[[1,0,2,3],[0,1,2,3],[2,2,3,3],[3,3,3,3]],"coefficients":[2,3,3,3],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-normalize","semantic_template":"qf-mod29-poly-normalize/{\"addition\":[[2,3,1,0],[3,2,0,1],[1,0,3,2],[0,1,2,3]],\"coefficients\":[2,3,3,3],\"multiplication\":[[1,0,2,3],[0,1,2,3],[2,2,3,3],[3,3,3,3]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0023","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-normalize；資料={\"n\":4,\"addition\":[[3,2,1,0],[2,0,3,1],[1,3,0,2],[0,1,2,3]],\"multiplication\":[[3,0,0,3],[0,2,1,3],[0,1,2,3],[3,3,3,3]],\"coefficients\":[1,2,3,3]}。請依 [raw,zero,normalized,degree] 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 3, 3], 3, [1, 2], 1)","explanation_zh":"Wave4第三層核對=([1, 2, 3, 3], 3, [1, 2], 1)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-normalize","seed":202610292023,"parameters":{"n":4,"addition":[[3,2,1,0],[2,0,3,1],[1,3,0,2],[0,1,2,3]],"multiplication":[[3,0,0,3],[0,2,1,3],[0,1,2,3],[3,3,3,3]],"coefficients":[1,2,3,3],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-normalize","semantic_template":"qf-mod29-poly-normalize/{\"addition\":[[3,2,1,0],[2,0,3,1],[1,3,0,2],[0,1,2,3]],\"coefficients\":[1,2,3,3],\"multiplication\":[[3,0,0,3],[0,2,1,3],[0,1,2,3],[3,3,3,3]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0024","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-normalize；資料={\"n\":4,\"addition\":[[1,2,3,0],[2,3,0,1],[3,0,1,2],[0,1,2,3]],\"multiplication\":[[2,1,0,3],[1,3,1,3],[0,1,2,3],[3,3,3,3]],\"coefficients\":[3,1,3,3]}。請依 [raw,zero,normalized,degree] 欄序回答。","answer":{"kind":"expression","value":"([3, 1, 3, 3], 3, [3, 1], 1)","explanation_zh":"Wave4第三層核對=([3, 1, 3, 3], 3, [3, 1], 1)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-normalize","seed":202610292024,"parameters":{"n":4,"addition":[[1,2,3,0],[2,3,0,1],[3,0,1,2],[0,1,2,3]],"multiplication":[[2,1,0,3],[1,3,1,3],[0,1,2,3],[3,3,3,3]],"coefficients":[3,1,3,3],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-normalize","semantic_template":"qf-mod29-poly-normalize/{\"addition\":[[1,2,3,0],[2,3,0,1],[3,0,1,2],[0,1,2,3]],\"coefficients\":[3,1,3,3],\"multiplication\":[[2,1,0,3],[1,3,1,3],[0,1,2,3],[3,3,3,3]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0025","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-normalize；資料={\"n\":4,\"addition\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"multiplication\":[[0,0,0,0],[0,1,2,3],[0,2,0,2],[0,3,2,1]],\"coefficients\":[1,3,0,0,0]}。請依 [raw,zero,normalized,degree] 欄序回答。","answer":{"kind":"expression","value":"([1, 3, 0, 0, 0], 0, [1, 3], 1)","explanation_zh":"Wave4第三層核對=([1, 3, 0, 0, 0], 0, [1, 3], 1)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-normalize","seed":202610292025,"parameters":{"n":4,"addition":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"multiplication":[[0,0,0,0],[0,1,2,3],[0,2,0,2],[0,3,2,1]],"coefficients":[1,3,0,0,0],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-normalize","semantic_template":"qf-mod29-poly-normalize/{\"addition\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"coefficients\":[1,3,0,0,0],\"multiplication\":[[0,0,0,0],[0,1,2,3],[0,2,0,2],[0,3,2,1]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0026","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-add；資料={\"n\":4,\"addition\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"multiplication\":[[0,0,0,0],[0,1,2,3],[0,2,0,2],[0,3,2,1]],\"left\":[1,3,0,0],\"right\":[2,1,0,0]}。請依 [left,right,sum,degree] 欄序回答。","answer":{"kind":"expression","value":"([1, 3], [2, 1], [3], 0)","explanation_zh":"Wave4第三層核對=([1, 3], [2, 1], [3], 0)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-add","seed":202610292026,"parameters":{"n":4,"addition":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"multiplication":[[0,0,0,0],[0,1,2,3],[0,2,0,2],[0,3,2,1]],"left":[1,3,0,0],"right":[2,1,0,0],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-add","semantic_template":"qf-mod29-poly-add/{\"addition\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"left\":[1,3,0,0],\"multiplication\":[[0,0,0,0],[0,1,2,3],[0,2,0,2],[0,3,2,1]],\"n\":4,\"right\":[2,1,0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0027","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-add；資料={\"n\":4,\"addition\":[[0,1,2,3],[1,3,0,2],[2,0,3,1],[3,2,1,0]],\"multiplication\":[[0,0,0,0],[0,1,2,3],[0,2,1,3],[0,3,3,0]],\"left\":[3,0,0,0],\"right\":[2,3,0,0]}。請依 [left,right,sum,degree] 欄序回答。","answer":{"kind":"expression","value":"([3], [2, 3], [1, 3], 1)","explanation_zh":"Wave4第三層核對=([3], [2, 3], [1, 3], 1)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-add","seed":202610292027,"parameters":{"n":4,"addition":[[0,1,2,3],[1,3,0,2],[2,0,3,1],[3,2,1,0]],"multiplication":[[0,0,0,0],[0,1,2,3],[0,2,1,3],[0,3,3,0]],"left":[3,0,0,0],"right":[2,3,0,0],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-add","semantic_template":"qf-mod29-poly-add/{\"addition\":[[0,1,2,3],[1,3,0,2],[2,0,3,1],[3,2,1,0]],\"left\":[3,0,0,0],\"multiplication\":[[0,0,0,0],[0,1,2,3],[0,2,1,3],[0,3,3,0]],\"n\":4,\"right\":[2,3,0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0028","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-add；資料={\"n\":4,\"addition\":[[0,1,2,3],[1,0,3,2],[2,3,1,0],[3,2,0,1]],\"multiplication\":[[0,0,0,0],[0,0,1,1],[0,1,2,3],[0,1,3,2]],\"left\":[3,2,0,0],\"right\":[0,3,0,0]}。請依 [left,right,sum,degree] 欄序回答。","answer":{"kind":"expression","value":"([3, 2], [0, 3], [3], 0)","explanation_zh":"Wave4第三層核對=([3, 2], [0, 3], [3], 0)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-add","seed":202610292028,"parameters":{"n":4,"addition":[[0,1,2,3],[1,0,3,2],[2,3,1,0],[3,2,0,1]],"multiplication":[[0,0,0,0],[0,0,1,1],[0,1,2,3],[0,1,3,2]],"left":[3,2,0,0],"right":[0,3,0,0],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-add","semantic_template":"qf-mod29-poly-add/{\"addition\":[[0,1,2,3],[1,0,3,2],[2,3,1,0],[3,2,0,1]],\"left\":[3,2,0,0],\"multiplication\":[[0,0,0,0],[0,0,1,1],[0,1,2,3],[0,1,3,2]],\"n\":4,\"right\":[0,3,0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0029","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-add；資料={\"n\":4,\"addition\":[[0,1,2,3],[1,3,0,2],[2,0,3,1],[3,2,1,0]],\"multiplication\":[[0,0,0,0],[0,2,1,3],[0,1,2,3],[0,3,3,0]],\"left\":[0,3,0,0],\"right\":[2,0,0,0]}。請依 [left,right,sum,degree] 欄序回答。","answer":{"kind":"expression","value":"([0, 3], [2], [2, 3], 1)","explanation_zh":"Wave4第三層核對=([0, 3], [2], [2, 3], 1)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-add","seed":202610292029,"parameters":{"n":4,"addition":[[0,1,2,3],[1,3,0,2],[2,0,3,1],[3,2,1,0]],"multiplication":[[0,0,0,0],[0,2,1,3],[0,1,2,3],[0,3,3,0]],"left":[0,3,0,0],"right":[2,0,0,0],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-add","semantic_template":"qf-mod29-poly-add/{\"addition\":[[0,1,2,3],[1,3,0,2],[2,0,3,1],[3,2,1,0]],\"left\":[0,3,0,0],\"multiplication\":[[0,0,0,0],[0,2,1,3],[0,1,2,3],[0,3,3,0]],\"n\":4,\"right\":[2,0,0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0030","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-add；資料={\"n\":4,\"addition\":[[0,1,2,3],[1,0,3,2],[2,3,1,0],[3,2,0,1]],\"multiplication\":[[0,0,0,0],[0,0,1,1],[0,1,3,2],[0,1,2,3]],\"left\":[3,2,0,0],\"right\":[1,3,0,0]}。請依 [left,right,sum,degree] 欄序回答。","answer":{"kind":"expression","value":"([3, 2], [1, 3], [2], 0)","explanation_zh":"Wave4第三層核對=([3, 2], [1, 3], [2], 0)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-add","seed":202610292030,"parameters":{"n":4,"addition":[[0,1,2,3],[1,0,3,2],[2,3,1,0],[3,2,0,1]],"multiplication":[[0,0,0,0],[0,0,1,1],[0,1,3,2],[0,1,2,3]],"left":[3,2,0,0],"right":[1,3,0,0],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-add","semantic_template":"qf-mod29-poly-add/{\"addition\":[[0,1,2,3],[1,0,3,2],[2,3,1,0],[3,2,0,1]],\"left\":[3,2,0,0],\"multiplication\":[[0,0,0,0],[0,0,1,1],[0,1,3,2],[0,1,2,3]],\"n\":4,\"right\":[1,3,0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0031","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-add；資料={\"n\":4,\"addition\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"multiplication\":[[0,0,0,0],[0,3,2,1],[0,2,0,2],[0,1,2,3]],\"left\":[2,0,0,0],\"right\":[1,2,0,0]}。請依 [left,right,sum,degree] 欄序回答。","answer":{"kind":"expression","value":"([2], [1, 2], [3, 2], 1)","explanation_zh":"Wave4第三層核對=([2], [1, 2], [3, 2], 1)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-add","seed":202610292031,"parameters":{"n":4,"addition":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"multiplication":[[0,0,0,0],[0,3,2,1],[0,2,0,2],[0,1,2,3]],"left":[2,0,0,0],"right":[1,2,0,0],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-add","semantic_template":"qf-mod29-poly-add/{\"addition\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"left\":[2,0,0,0],\"multiplication\":[[0,0,0,0],[0,3,2,1],[0,2,0,2],[0,1,2,3]],\"n\":4,\"right\":[1,2,0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0032","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-add；資料={\"n\":4,\"addition\":[[2,0,3,1],[0,1,2,3],[3,2,1,0],[1,3,0,2]],\"multiplication\":[[0,1,2,3],[1,1,1,1],[2,1,1,2],[3,1,2,0]],\"left\":[3,0,1,1],\"right\":[1,3,1,1]}。請依 [left,right,sum,degree] 欄序回答。","answer":{"kind":"expression","value":"([3, 0], [1, 3], [3], 0)","explanation_zh":"Wave4第三層核對=([3, 0], [1, 3], [3], 0)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-add","seed":202610292032,"parameters":{"n":4,"addition":[[2,0,3,1],[0,1,2,3],[3,2,1,0],[1,3,0,2]],"multiplication":[[0,1,2,3],[1,1,1,1],[2,1,1,2],[3,1,2,0]],"left":[3,0,1,1],"right":[1,3,1,1],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-add","semantic_template":"qf-mod29-poly-add/{\"addition\":[[2,0,3,1],[0,1,2,3],[3,2,1,0],[1,3,0,2]],\"left\":[3,0,1,1],\"multiplication\":[[0,1,2,3],[1,1,1,1],[2,1,1,2],[3,1,2,0]],\"n\":4,\"right\":[1,3,1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0033","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-add；資料={\"n\":4,\"addition\":[[3,0,1,2],[0,1,2,3],[1,2,3,0],[2,3,0,1]],\"multiplication\":[[0,1,2,3],[1,1,1,1],[2,1,0,3],[3,1,3,1]],\"left\":[1,3,1,1],\"right\":[0,1,1,1]}。請依 [left,right,sum,degree] 欄序回答。","answer":{"kind":"expression","value":"([1, 3], [0], [0, 3], 1)","explanation_zh":"Wave4第三層核對=([1, 3], [0], [0, 3], 1)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-add","seed":202610292033,"parameters":{"n":4,"addition":[[3,0,1,2],[0,1,2,3],[1,2,3,0],[2,3,0,1]],"multiplication":[[0,1,2,3],[1,1,1,1],[2,1,0,3],[3,1,3,1]],"left":[1,3,1,1],"right":[0,1,1,1],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-add","semantic_template":"qf-mod29-poly-add/{\"addition\":[[3,0,1,2],[0,1,2,3],[1,2,3,0],[2,3,0,1]],\"left\":[1,3,1,1],\"multiplication\":[[0,1,2,3],[1,1,1,1],[2,1,0,3],[3,1,3,1]],\"n\":4,\"right\":[0,1,1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0034","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-add；資料={\"n\":4,\"addition\":[[1,0,3,2],[0,1,2,3],[3,2,0,1],[2,3,1,0]],\"multiplication\":[[1,1,0,0],[1,1,1,1],[0,1,2,3],[0,1,3,2]],\"left\":[2,3,1,1],\"right\":[0,2,1,1]}。請依 [left,right,sum,degree] 欄序回答。","answer":{"kind":"expression","value":"([2, 3], [0, 2], [3], 0)","explanation_zh":"Wave4第三層核對=([2, 3], [0, 2], [3], 0)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-add","seed":202610292034,"parameters":{"n":4,"addition":[[1,0,3,2],[0,1,2,3],[3,2,0,1],[2,3,1,0]],"multiplication":[[1,1,0,0],[1,1,1,1],[0,1,2,3],[0,1,3,2]],"left":[2,3,1,1],"right":[0,2,1,1],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-add","semantic_template":"qf-mod29-poly-add/{\"addition\":[[1,0,3,2],[0,1,2,3],[3,2,0,1],[2,3,1,0]],\"left\":[2,3,1,1],\"multiplication\":[[1,1,0,0],[1,1,1,1],[0,1,2,3],[0,1,3,2]],\"n\":4,\"right\":[0,2,1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0035","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-add；資料={\"n\":4,\"addition\":[[3,0,1,2],[0,1,2,3],[1,2,3,0],[2,3,0,1]],\"multiplication\":[[2,1,0,3],[1,1,1,1],[0,1,2,3],[3,1,3,1]],\"left\":[3,1,1,1],\"right\":[0,3,1,1]}。請依 [left,right,sum,degree] 欄序回答。","answer":{"kind":"expression","value":"([3], [0, 3], [2, 3], 1)","explanation_zh":"Wave4第三層核對=([3], [0, 3], [2, 3], 1)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-add","seed":202610292035,"parameters":{"n":4,"addition":[[3,0,1,2],[0,1,2,3],[1,2,3,0],[2,3,0,1]],"multiplication":[[2,1,0,3],[1,1,1,1],[0,1,2,3],[3,1,3,1]],"left":[3,1,1,1],"right":[0,3,1,1],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-add","semantic_template":"qf-mod29-poly-add/{\"addition\":[[3,0,1,2],[0,1,2,3],[1,2,3,0],[2,3,0,1]],\"left\":[3,1,1,1],\"multiplication\":[[2,1,0,3],[1,1,1,1],[0,1,2,3],[3,1,3,1]],\"n\":4,\"right\":[0,3,1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0036","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-add；資料={\"n\":4,\"addition\":[[1,0,3,2],[0,1,2,3],[3,2,0,1],[2,3,1,0]],\"multiplication\":[[1,1,0,0],[1,1,1,1],[0,1,3,2],[0,1,2,3]],\"left\":[2,3,1,1],\"right\":[1,2,1,1]}。請依 [left,right,sum,degree] 欄序回答。","answer":{"kind":"expression","value":"([2, 3], [1, 2], [2], 0)","explanation_zh":"Wave4第三層核對=([2, 3], [1, 2], [2], 0)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-add","seed":202610292036,"parameters":{"n":4,"addition":[[1,0,3,2],[0,1,2,3],[3,2,0,1],[2,3,1,0]],"multiplication":[[1,1,0,0],[1,1,1,1],[0,1,3,2],[0,1,2,3]],"left":[2,3,1,1],"right":[1,2,1,1],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-add","semantic_template":"qf-mod29-poly-add/{\"addition\":[[1,0,3,2],[0,1,2,3],[3,2,0,1],[2,3,1,0]],\"left\":[2,3,1,1],\"multiplication\":[[1,1,0,0],[1,1,1,1],[0,1,3,2],[0,1,2,3]],\"n\":4,\"right\":[1,2,1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0037","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-add；資料={\"n\":4,\"addition\":[[2,0,3,1],[0,1,2,3],[3,2,1,0],[1,3,0,2]],\"multiplication\":[[3,1,2,0],[1,1,1,1],[2,1,1,2],[0,1,2,3]],\"left\":[1,2,1,1],\"right\":[3,1,1,1]}。請依 [left,right,sum,degree] 欄序回答。","answer":{"kind":"expression","value":"([1, 2], [3], [3, 2], 1)","explanation_zh":"Wave4第三層核對=([1, 2], [3], [3, 2], 1)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-add","seed":202610292037,"parameters":{"n":4,"addition":[[2,0,3,1],[0,1,2,3],[3,2,1,0],[1,3,0,2]],"multiplication":[[3,1,2,0],[1,1,1,1],[2,1,1,2],[0,1,2,3]],"left":[1,2,1,1],"right":[3,1,1,1],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-add","semantic_template":"qf-mod29-poly-add/{\"addition\":[[2,0,3,1],[0,1,2,3],[3,2,1,0],[1,3,0,2]],\"left\":[1,2,1,1],\"multiplication\":[[3,1,2,0],[1,1,1,1],[2,1,1,2],[0,1,2,3]],\"n\":4,\"right\":[3,1,1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0038","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-add；資料={\"n\":4,\"addition\":[[1,3,0,2],[3,2,1,0],[0,1,2,3],[2,0,3,1]],\"multiplication\":[[0,1,2,3],[1,2,2,1],[2,2,2,2],[3,1,2,0]],\"left\":[0,3,2,2],\"right\":[1,0,2,2]}。請依 [left,right,sum,degree] 欄序回答。","answer":{"kind":"expression","value":"([0, 3], [1, 0], [3], 0)","explanation_zh":"Wave4第三層核對=([0, 3], [1, 0], [3], 0)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-add","seed":202610292038,"parameters":{"n":4,"addition":[[1,3,0,2],[3,2,1,0],[0,1,2,3],[2,0,3,1]],"multiplication":[[0,1,2,3],[1,2,2,1],[2,2,2,2],[3,1,2,0]],"left":[0,3,2,2],"right":[1,0,2,2],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-add","semantic_template":"qf-mod29-poly-add/{\"addition\":[[1,3,0,2],[3,2,1,0],[0,1,2,3],[2,0,3,1]],\"left\":[0,3,2,2],\"multiplication\":[[0,1,2,3],[1,2,2,1],[2,2,2,2],[3,1,2,0]],\"n\":4,\"right\":[1,0,2,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0039","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-add；資料={\"n\":4,\"addition\":[[3,2,0,1],[2,3,1,0],[0,1,2,3],[1,0,3,2]],\"multiplication\":[[0,1,2,3],[1,0,2,3],[2,2,2,2],[3,3,2,2]],\"left\":[3,2,2,2],\"right\":[1,3,2,2]}。請依 [left,right,sum,degree] 欄序回答。","answer":{"kind":"expression","value":"([3], [1, 3], [0, 3], 1)","explanation_zh":"Wave4第三層核對=([3], [1, 3], [0, 3], 1)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-add","seed":202610292039,"parameters":{"n":4,"addition":[[3,2,0,1],[2,3,1,0],[0,1,2,3],[1,0,3,2]],"multiplication":[[0,1,2,3],[1,0,2,3],[2,2,2,2],[3,3,2,2]],"left":[3,2,2,2],"right":[1,3,2,2],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-add","semantic_template":"qf-mod29-poly-add/{\"addition\":[[3,2,0,1],[2,3,1,0],[0,1,2,3],[1,0,3,2]],\"left\":[3,2,2,2],\"multiplication\":[[0,1,2,3],[1,0,2,3],[2,2,2,2],[3,3,2,2]],\"n\":4,\"right\":[1,3,2,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0040","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-add；資料={\"n\":4,\"addition\":[[2,3,0,1],[3,0,1,2],[0,1,2,3],[1,2,3,0]],\"multiplication\":[[2,0,2,0],[0,1,2,3],[2,2,2,2],[0,3,2,1]],\"left\":[3,1,2,2],\"right\":[2,3,2,2]}。請依 [left,right,sum,degree] 欄序回答。","answer":{"kind":"expression","value":"([3, 1], [2, 3], [3], 0)","explanation_zh":"Wave4第三層核對=([3, 1], [2, 3], [3], 0)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-add","seed":202610292040,"parameters":{"n":4,"addition":[[2,3,0,1],[3,0,1,2],[0,1,2,3],[1,2,3,0]],"multiplication":[[2,0,2,0],[0,1,2,3],[2,2,2,2],[0,3,2,1]],"left":[3,1,2,2],"right":[2,3,2,2],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-add","semantic_template":"qf-mod29-poly-add/{\"addition\":[[2,3,0,1],[3,0,1,2],[0,1,2,3],[1,2,3,0]],\"left\":[3,1,2,2],\"multiplication\":[[2,0,2,0],[0,1,2,3],[2,2,2,2],[0,3,2,1]],\"n\":4,\"right\":[2,3,2,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0041","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-add；資料={\"n\":4,\"addition\":[[3,2,0,1],[2,3,1,0],[0,1,2,3],[1,0,3,2]],\"multiplication\":[[1,0,2,3],[0,1,2,3],[2,2,2,2],[3,3,2,2]],\"left\":[2,3,2,2],\"right\":[1,2,2,2]}。請依 [left,right,sum,degree] 欄序回答。","answer":{"kind":"expression","value":"([2, 3], [1], [1, 3], 1)","explanation_zh":"Wave4第三層核對=([2, 3], [1], [1, 3], 1)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-add","seed":202610292041,"parameters":{"n":4,"addition":[[3,2,0,1],[2,3,1,0],[0,1,2,3],[1,0,3,2]],"multiplication":[[1,0,2,3],[0,1,2,3],[2,2,2,2],[3,3,2,2]],"left":[2,3,2,2],"right":[1,2,2,2],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-add","semantic_template":"qf-mod29-poly-add/{\"addition\":[[3,2,0,1],[2,3,1,0],[0,1,2,3],[1,0,3,2]],\"left\":[2,3,2,2],\"multiplication\":[[1,0,2,3],[0,1,2,3],[2,2,2,2],[3,3,2,2]],\"n\":4,\"right\":[1,2,2,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0042","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-add；資料={\"n\":4,\"addition\":[[2,3,0,1],[3,0,1,2],[0,1,2,3],[1,2,3,0]],\"multiplication\":[[2,0,2,0],[0,3,2,1],[2,2,2,2],[0,1,2,3]],\"left\":[3,1,2,2],\"right\":[0,3,2,2]}。請依 [left,right,sum,degree] 欄序回答。","answer":{"kind":"expression","value":"([3, 1], [0, 3], [1], 0)","explanation_zh":"Wave4第三層核對=([3, 1], [0, 3], [1], 0)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-add","seed":202610292042,"parameters":{"n":4,"addition":[[2,3,0,1],[3,0,1,2],[0,1,2,3],[1,2,3,0]],"multiplication":[[2,0,2,0],[0,3,2,1],[2,2,2,2],[0,1,2,3]],"left":[3,1,2,2],"right":[0,3,2,2],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-add","semantic_template":"qf-mod29-poly-add/{\"addition\":[[2,3,0,1],[3,0,1,2],[0,1,2,3],[1,2,3,0]],\"left\":[3,1,2,2],\"multiplication\":[[2,0,2,0],[0,3,2,1],[2,2,2,2],[0,1,2,3]],\"n\":4,\"right\":[0,3,2,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0043","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-add；資料={\"n\":4,\"addition\":[[1,3,0,2],[3,2,1,0],[0,1,2,3],[2,0,3,1]],\"multiplication\":[[3,1,2,0],[1,2,2,1],[2,2,2,2],[0,1,2,3]],\"left\":[1,2,2,2],\"right\":[0,1,2,2]}。請依 [left,right,sum,degree] 欄序回答。","answer":{"kind":"expression","value":"([1], [0, 1], [3, 1], 1)","explanation_zh":"Wave4第三層核對=([1], [0, 1], [3, 1], 1)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-add","seed":202610292043,"parameters":{"n":4,"addition":[[1,3,0,2],[3,2,1,0],[0,1,2,3],[2,0,3,1]],"multiplication":[[3,1,2,0],[1,2,2,1],[2,2,2,2],[0,1,2,3]],"left":[1,2,2,2],"right":[0,1,2,2],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-add","semantic_template":"qf-mod29-poly-add/{\"addition\":[[1,3,0,2],[3,2,1,0],[0,1,2,3],[2,0,3,1]],\"left\":[1,2,2,2],\"multiplication\":[[3,1,2,0],[1,2,2,1],[2,2,2,2],[0,1,2,3]],\"n\":4,\"right\":[0,1,2,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0044","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-add；資料={\"n\":4,\"addition\":[[1,2,3,0],[2,3,0,1],[3,0,1,2],[0,1,2,3]],\"multiplication\":[[0,1,2,3],[1,3,1,3],[2,1,0,3],[3,3,3,3]],\"left\":[2,0,3,3],\"right\":[3,2,3,3]}。請依 [left,right,sum,degree] 欄序回答。","answer":{"kind":"expression","value":"([2, 0], [3, 2], [2], 0)","explanation_zh":"Wave4第三層核對=([2, 0], [3, 2], [2], 0)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-add","seed":202610292044,"parameters":{"n":4,"addition":[[1,2,3,0],[2,3,0,1],[3,0,1,2],[0,1,2,3]],"multiplication":[[0,1,2,3],[1,3,1,3],[2,1,0,3],[3,3,3,3]],"left":[2,0,3,3],"right":[3,2,3,3],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-add","semantic_template":"qf-mod29-poly-add/{\"addition\":[[1,2,3,0],[2,3,0,1],[3,0,1,2],[0,1,2,3]],\"left\":[2,0,3,3],\"multiplication\":[[0,1,2,3],[1,3,1,3],[2,1,0,3],[3,3,3,3]],\"n\":4,\"right\":[3,2,3,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0045","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-add；資料={\"n\":4,\"addition\":[[2,3,1,0],[3,2,0,1],[1,0,3,2],[0,1,2,3]],\"multiplication\":[[0,1,2,3],[1,0,2,3],[2,2,3,3],[3,3,3,3]],\"left\":[3,2,3,3],\"right\":[0,3,3,3]}。請依 [left,right,sum,degree] 欄序回答。","answer":{"kind":"expression","value":"([3, 2], [0], [0, 2], 1)","explanation_zh":"Wave4第三層核對=([3, 2], [0], [0, 2], 1)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-add","seed":202610292045,"parameters":{"n":4,"addition":[[2,3,1,0],[3,2,0,1],[1,0,3,2],[0,1,2,3]],"multiplication":[[0,1,2,3],[1,0,2,3],[2,2,3,3],[3,3,3,3]],"left":[3,2,3,3],"right":[0,3,3,3],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-add","semantic_template":"qf-mod29-poly-add/{\"addition\":[[2,3,1,0],[3,2,0,1],[1,0,3,2],[0,1,2,3]],\"left\":[3,2,3,3],\"multiplication\":[[0,1,2,3],[1,0,2,3],[2,2,3,3],[3,3,3,3]],\"n\":4,\"right\":[0,3,3,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0046","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-add；資料={\"n\":4,\"addition\":[[3,2,1,0],[2,0,3,1],[1,3,0,2],[0,1,2,3]],\"multiplication\":[[3,0,0,3],[0,1,2,3],[0,2,1,3],[3,3,3,3]],\"left\":[1,2,3,3],\"right\":[0,1,3,3]}。請依 [left,right,sum,degree] 欄序回答。","answer":{"kind":"expression","value":"([1, 2], [0, 1], [2], 0)","explanation_zh":"Wave4第三層核對=([1, 2], [0, 1], [2], 0)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-add","seed":202610292046,"parameters":{"n":4,"addition":[[3,2,1,0],[2,0,3,1],[1,3,0,2],[0,1,2,3]],"multiplication":[[3,0,0,3],[0,1,2,3],[0,2,1,3],[3,3,3,3]],"left":[1,2,3,3],"right":[0,1,3,3],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-add","semantic_template":"qf-mod29-poly-add/{\"addition\":[[3,2,1,0],[2,0,3,1],[1,3,0,2],[0,1,2,3]],\"left\":[1,2,3,3],\"multiplication\":[[3,0,0,3],[0,1,2,3],[0,2,1,3],[3,3,3,3]],\"n\":4,\"right\":[0,1,3,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0047","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-add；資料={\"n\":4,\"addition\":[[2,3,1,0],[3,2,0,1],[1,0,3,2],[0,1,2,3]],\"multiplication\":[[1,0,2,3],[0,1,2,3],[2,2,3,3],[3,3,3,3]],\"left\":[2,3,3,3],\"right\":[0,2,3,3]}。請依 [left,right,sum,degree] 欄序回答。","answer":{"kind":"expression","value":"([2], [0, 2], [1, 2], 1)","explanation_zh":"Wave4第三層核對=([2], [0, 2], [1, 2], 1)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-add","seed":202610292047,"parameters":{"n":4,"addition":[[2,3,1,0],[3,2,0,1],[1,0,3,2],[0,1,2,3]],"multiplication":[[1,0,2,3],[0,1,2,3],[2,2,3,3],[3,3,3,3]],"left":[2,3,3,3],"right":[0,2,3,3],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-add","semantic_template":"qf-mod29-poly-add/{\"addition\":[[2,3,1,0],[3,2,0,1],[1,0,3,2],[0,1,2,3]],\"left\":[2,3,3,3],\"multiplication\":[[1,0,2,3],[0,1,2,3],[2,2,3,3],[3,3,3,3]],\"n\":4,\"right\":[0,2,3,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0048","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-add；資料={\"n\":4,\"addition\":[[3,2,1,0],[2,0,3,1],[1,3,0,2],[0,1,2,3]],\"multiplication\":[[3,0,0,3],[0,2,1,3],[0,1,2,3],[3,3,3,3]],\"left\":[1,2,3,3],\"right\":[3,1,3,3]}。請依 [left,right,sum,degree] 欄序回答。","answer":{"kind":"expression","value":"([1, 2], [3, 1], [1], 0)","explanation_zh":"Wave4第三層核對=([1, 2], [3, 1], [1], 0)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-add","seed":202610292048,"parameters":{"n":4,"addition":[[3,2,1,0],[2,0,3,1],[1,3,0,2],[0,1,2,3]],"multiplication":[[3,0,0,3],[0,2,1,3],[0,1,2,3],[3,3,3,3]],"left":[1,2,3,3],"right":[3,1,3,3],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-add","semantic_template":"qf-mod29-poly-add/{\"addition\":[[3,2,1,0],[2,0,3,1],[1,3,0,2],[0,1,2,3]],\"left\":[1,2,3,3],\"multiplication\":[[3,0,0,3],[0,2,1,3],[0,1,2,3],[3,3,3,3]],\"n\":4,\"right\":[3,1,3,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0049","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-add；資料={\"n\":4,\"addition\":[[1,2,3,0],[2,3,0,1],[3,0,1,2],[0,1,2,3]],\"multiplication\":[[2,1,0,3],[1,3,1,3],[0,1,2,3],[3,3,3,3]],\"left\":[3,1,3,3],\"right\":[2,3,3,3]}。請依 [left,right,sum,degree] 欄序回答。","answer":{"kind":"expression","value":"([3, 1], [2], [2, 1], 1)","explanation_zh":"Wave4第三層核對=([3, 1], [2], [2, 1], 1)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-add","seed":202610292049,"parameters":{"n":4,"addition":[[1,2,3,0],[2,3,0,1],[3,0,1,2],[0,1,2,3]],"multiplication":[[2,1,0,3],[1,3,1,3],[0,1,2,3],[3,3,3,3]],"left":[3,1,3,3],"right":[2,3,3,3],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-add","semantic_template":"qf-mod29-poly-add/{\"addition\":[[1,2,3,0],[2,3,0,1],[3,0,1,2],[0,1,2,3]],\"left\":[3,1,3,3],\"multiplication\":[[2,1,0,3],[1,3,1,3],[0,1,2,3],[3,3,3,3]],\"n\":4,\"right\":[2,3,3,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0050","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-add；資料={\"n\":4,\"addition\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"multiplication\":[[0,0,0,0],[0,1,2,3],[0,2,0,2],[0,3,2,1]],\"left\":[1,3,0,0,0],\"right\":[2,1,0,0,0]}。請依 [left,right,sum,degree] 欄序回答。","answer":{"kind":"expression","value":"([1, 3], [2, 1], [3], 0)","explanation_zh":"Wave4第三層核對=([1, 3], [2, 1], [3], 0)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-add","seed":202610292050,"parameters":{"n":4,"addition":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"multiplication":[[0,0,0,0],[0,1,2,3],[0,2,0,2],[0,3,2,1]],"left":[1,3,0,0,0],"right":[2,1,0,0,0],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-add","semantic_template":"qf-mod29-poly-add/{\"addition\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"left\":[1,3,0,0,0],\"multiplication\":[[0,0,0,0],[0,1,2,3],[0,2,0,2],[0,3,2,1]],\"n\":4,\"right\":[2,1,0,0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0051","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-multiply；資料={\"n\":4,\"addition\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"multiplication\":[[0,0,0,0],[0,1,2,3],[0,2,0,2],[0,3,2,1]],\"left\":[1,3,0,0],\"right\":[2,1,0,0]}。請依 [left,right,product,degree,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1, 3], [2, 1], [2, 3, 3], 2, degree_additivity_not_claimed_with_zero_divisors)","explanation_zh":"Wave4第三層核對=([1, 3], [2, 1], [2, 3, 3], 2, degree_additivity_not_claimed_with_zero_divisors)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-multiply","seed":202610292051,"parameters":{"n":4,"addition":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"multiplication":[[0,0,0,0],[0,1,2,3],[0,2,0,2],[0,3,2,1]],"left":[1,3,0,0],"right":[2,1,0,0],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-multiply","semantic_template":"qf-mod29-poly-multiply/{\"addition\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"left\":[1,3,0,0],\"multiplication\":[[0,0,0,0],[0,1,2,3],[0,2,0,2],[0,3,2,1]],\"n\":4,\"right\":[2,1,0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0052","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-multiply；資料={\"n\":4,\"addition\":[[0,1,2,3],[1,3,0,2],[2,0,3,1],[3,2,1,0]],\"multiplication\":[[0,0,0,0],[0,1,2,3],[0,2,1,3],[0,3,3,0]],\"left\":[3,0,0,0],\"right\":[2,3,0,0]}。請依 [left,right,product,degree,boundary] 欄序回答。","answer":{"kind":"expression","value":"([3], [2, 3], [3], 0, degree_additivity_not_claimed_with_zero_divisors)","explanation_zh":"Wave4第三層核對=([3], [2, 3], [3], 0, degree_additivity_not_claimed_with_zero_divisors)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-multiply","seed":202610292052,"parameters":{"n":4,"addition":[[0,1,2,3],[1,3,0,2],[2,0,3,1],[3,2,1,0]],"multiplication":[[0,0,0,0],[0,1,2,3],[0,2,1,3],[0,3,3,0]],"left":[3,0,0,0],"right":[2,3,0,0],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-multiply","semantic_template":"qf-mod29-poly-multiply/{\"addition\":[[0,1,2,3],[1,3,0,2],[2,0,3,1],[3,2,1,0]],\"left\":[3,0,0,0],\"multiplication\":[[0,0,0,0],[0,1,2,3],[0,2,1,3],[0,3,3,0]],\"n\":4,\"right\":[2,3,0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0053","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-multiply；資料={\"n\":4,\"addition\":[[0,1,2,3],[1,0,3,2],[2,3,1,0],[3,2,0,1]],\"multiplication\":[[0,0,0,0],[0,0,1,1],[0,1,2,3],[0,1,3,2]],\"left\":[3,2,0,0],\"right\":[0,3,0,0]}。請依 [left,right,product,degree,boundary] 欄序回答。","answer":{"kind":"expression","value":"([3, 2], [0, 3], [0, 2, 3], 2, degree_additivity_not_claimed_with_zero_divisors)","explanation_zh":"Wave4第三層核對=([3, 2], [0, 3], [0, 2, 3], 2, degree_additivity_not_claimed_with_zero_divisors)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-multiply","seed":202610292053,"parameters":{"n":4,"addition":[[0,1,2,3],[1,0,3,2],[2,3,1,0],[3,2,0,1]],"multiplication":[[0,0,0,0],[0,0,1,1],[0,1,2,3],[0,1,3,2]],"left":[3,2,0,0],"right":[0,3,0,0],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-multiply","semantic_template":"qf-mod29-poly-multiply/{\"addition\":[[0,1,2,3],[1,0,3,2],[2,3,1,0],[3,2,0,1]],\"left\":[3,2,0,0],\"multiplication\":[[0,0,0,0],[0,0,1,1],[0,1,2,3],[0,1,3,2]],\"n\":4,\"right\":[0,3,0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0054","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-multiply；資料={\"n\":4,\"addition\":[[0,1,2,3],[1,3,0,2],[2,0,3,1],[3,2,1,0]],\"multiplication\":[[0,0,0,0],[0,2,1,3],[0,1,2,3],[0,3,3,0]],\"left\":[0,3,0,0],\"right\":[2,0,0,0]}。請依 [left,right,product,degree,boundary] 欄序回答。","answer":{"kind":"expression","value":"([0, 3], [2], [0, 3], 1, degree_additivity_not_claimed_with_zero_divisors)","explanation_zh":"Wave4第三層核對=([0, 3], [2], [0, 3], 1, degree_additivity_not_claimed_with_zero_divisors)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-multiply","seed":202610292054,"parameters":{"n":4,"addition":[[0,1,2,3],[1,3,0,2],[2,0,3,1],[3,2,1,0]],"multiplication":[[0,0,0,0],[0,2,1,3],[0,1,2,3],[0,3,3,0]],"left":[0,3,0,0],"right":[2,0,0,0],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-multiply","semantic_template":"qf-mod29-poly-multiply/{\"addition\":[[0,1,2,3],[1,3,0,2],[2,0,3,1],[3,2,1,0]],\"left\":[0,3,0,0],\"multiplication\":[[0,0,0,0],[0,2,1,3],[0,1,2,3],[0,3,3,0]],\"n\":4,\"right\":[2,0,0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0055","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-multiply；資料={\"n\":4,\"addition\":[[0,1,2,3],[1,0,3,2],[2,3,1,0],[3,2,0,1]],\"multiplication\":[[0,0,0,0],[0,0,1,1],[0,1,3,2],[0,1,2,3]],\"left\":[3,2,0,0],\"right\":[1,3,0,0]}。請依 [left,right,product,degree,boundary] 欄序回答。","answer":{"kind":"expression","value":"([3, 2], [1, 3], [1, 2, 2], 2, degree_additivity_not_claimed_with_zero_divisors)","explanation_zh":"Wave4第三層核對=([3, 2], [1, 3], [1, 2, 2], 2, degree_additivity_not_claimed_with_zero_divisors)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-multiply","seed":202610292055,"parameters":{"n":4,"addition":[[0,1,2,3],[1,0,3,2],[2,3,1,0],[3,2,0,1]],"multiplication":[[0,0,0,0],[0,0,1,1],[0,1,3,2],[0,1,2,3]],"left":[3,2,0,0],"right":[1,3,0,0],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-multiply","semantic_template":"qf-mod29-poly-multiply/{\"addition\":[[0,1,2,3],[1,0,3,2],[2,3,1,0],[3,2,0,1]],\"left\":[3,2,0,0],\"multiplication\":[[0,0,0,0],[0,0,1,1],[0,1,3,2],[0,1,2,3]],\"n\":4,\"right\":[1,3,0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0056","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-multiply；資料={\"n\":4,\"addition\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"multiplication\":[[0,0,0,0],[0,3,2,1],[0,2,0,2],[0,1,2,3]],\"left\":[2,0,0,0],\"right\":[1,2,0,0]}。請依 [left,right,product,degree,boundary] 欄序回答。","answer":{"kind":"expression","value":"([2], [1, 2], [2], 0, degree_additivity_not_claimed_with_zero_divisors)","explanation_zh":"Wave4第三層核對=([2], [1, 2], [2], 0, degree_additivity_not_claimed_with_zero_divisors)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-multiply","seed":202610292056,"parameters":{"n":4,"addition":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"multiplication":[[0,0,0,0],[0,3,2,1],[0,2,0,2],[0,1,2,3]],"left":[2,0,0,0],"right":[1,2,0,0],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-multiply","semantic_template":"qf-mod29-poly-multiply/{\"addition\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"left\":[2,0,0,0],\"multiplication\":[[0,0,0,0],[0,3,2,1],[0,2,0,2],[0,1,2,3]],\"n\":4,\"right\":[1,2,0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0057","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-multiply；資料={\"n\":4,\"addition\":[[2,0,3,1],[0,1,2,3],[3,2,1,0],[1,3,0,2]],\"multiplication\":[[0,1,2,3],[1,1,1,1],[2,1,1,2],[3,1,2,0]],\"left\":[3,0,1,1],\"right\":[1,3,1,1]}。請依 [left,right,product,degree,boundary] 欄序回答。","answer":{"kind":"expression","value":"([3, 0], [1, 3], [1, 0, 3], 2, degree_additivity_not_claimed_with_zero_divisors)","explanation_zh":"Wave4第三層核對=([3, 0], [1, 3], [1, 0, 3], 2, degree_additivity_not_claimed_with_zero_divisors)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-multiply","seed":202610292057,"parameters":{"n":4,"addition":[[2,0,3,1],[0,1,2,3],[3,2,1,0],[1,3,0,2]],"multiplication":[[0,1,2,3],[1,1,1,1],[2,1,1,2],[3,1,2,0]],"left":[3,0,1,1],"right":[1,3,1,1],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-multiply","semantic_template":"qf-mod29-poly-multiply/{\"addition\":[[2,0,3,1],[0,1,2,3],[3,2,1,0],[1,3,0,2]],\"left\":[3,0,1,1],\"multiplication\":[[0,1,2,3],[1,1,1,1],[2,1,1,2],[3,1,2,0]],\"n\":4,\"right\":[1,3,1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0058","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-multiply；資料={\"n\":4,\"addition\":[[3,0,1,2],[0,1,2,3],[1,2,3,0],[2,3,0,1]],\"multiplication\":[[0,1,2,3],[1,1,1,1],[2,1,0,3],[3,1,3,1]],\"left\":[1,3,1,1],\"right\":[0,1,1,1]}。請依 [left,right,product,degree,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1, 3], [0], [1, 3], 1, degree_additivity_not_claimed_with_zero_divisors)","explanation_zh":"Wave4第三層核對=([1, 3], [0], [1, 3], 1, degree_additivity_not_claimed_with_zero_divisors)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-multiply","seed":202610292058,"parameters":{"n":4,"addition":[[3,0,1,2],[0,1,2,3],[1,2,3,0],[2,3,0,1]],"multiplication":[[0,1,2,3],[1,1,1,1],[2,1,0,3],[3,1,3,1]],"left":[1,3,1,1],"right":[0,1,1,1],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-multiply","semantic_template":"qf-mod29-poly-multiply/{\"addition\":[[3,0,1,2],[0,1,2,3],[1,2,3,0],[2,3,0,1]],\"left\":[1,3,1,1],\"multiplication\":[[0,1,2,3],[1,1,1,1],[2,1,0,3],[3,1,3,1]],\"n\":4,\"right\":[0,1,1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0059","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-multiply；資料={\"n\":4,\"addition\":[[1,0,3,2],[0,1,2,3],[3,2,0,1],[2,3,1,0]],\"multiplication\":[[1,1,0,0],[1,1,1,1],[0,1,2,3],[0,1,3,2]],\"left\":[2,3,1,1],\"right\":[0,2,1,1]}。請依 [left,right,product,degree,boundary] 欄序回答。","answer":{"kind":"expression","value":"([2, 3], [0, 2], [0, 3, 3], 2, degree_additivity_not_claimed_with_zero_divisors)","explanation_zh":"Wave4第三層核對=([2, 3], [0, 2], [0, 3, 3], 2, degree_additivity_not_claimed_with_zero_divisors)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-multiply","seed":202610292059,"parameters":{"n":4,"addition":[[1,0,3,2],[0,1,2,3],[3,2,0,1],[2,3,1,0]],"multiplication":[[1,1,0,0],[1,1,1,1],[0,1,2,3],[0,1,3,2]],"left":[2,3,1,1],"right":[0,2,1,1],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-multiply","semantic_template":"qf-mod29-poly-multiply/{\"addition\":[[1,0,3,2],[0,1,2,3],[3,2,0,1],[2,3,1,0]],\"left\":[2,3,1,1],\"multiplication\":[[1,1,0,0],[1,1,1,1],[0,1,2,3],[0,1,3,2]],\"n\":4,\"right\":[0,2,1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0060","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-multiply；資料={\"n\":4,\"addition\":[[3,0,1,2],[0,1,2,3],[1,2,3,0],[2,3,0,1]],\"multiplication\":[[2,1,0,3],[1,1,1,1],[0,1,2,3],[3,1,3,1]],\"left\":[3,1,1,1],\"right\":[0,3,1,1]}。請依 [left,right,product,degree,boundary] 欄序回答。","answer":{"kind":"expression","value":"([3], [0, 3], [3], 0, degree_additivity_not_claimed_with_zero_divisors)","explanation_zh":"Wave4第三層核對=([3], [0, 3], [3], 0, degree_additivity_not_claimed_with_zero_divisors)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-multiply","seed":202610292060,"parameters":{"n":4,"addition":[[3,0,1,2],[0,1,2,3],[1,2,3,0],[2,3,0,1]],"multiplication":[[2,1,0,3],[1,1,1,1],[0,1,2,3],[3,1,3,1]],"left":[3,1,1,1],"right":[0,3,1,1],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-multiply","semantic_template":"qf-mod29-poly-multiply/{\"addition\":[[3,0,1,2],[0,1,2,3],[1,2,3,0],[2,3,0,1]],\"left\":[3,1,1,1],\"multiplication\":[[2,1,0,3],[1,1,1,1],[0,1,2,3],[3,1,3,1]],\"n\":4,\"right\":[0,3,1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0061","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-multiply；資料={\"n\":4,\"addition\":[[1,0,3,2],[0,1,2,3],[3,2,0,1],[2,3,1,0]],\"multiplication\":[[1,1,0,0],[1,1,1,1],[0,1,3,2],[0,1,2,3]],\"left\":[2,3,1,1],\"right\":[1,2,1,1]}。請依 [left,right,product,degree,boundary] 欄序回答。","answer":{"kind":"expression","value":"([2, 3], [1, 2], [1, 3, 2], 2, degree_additivity_not_claimed_with_zero_divisors)","explanation_zh":"Wave4第三層核對=([2, 3], [1, 2], [1, 3, 2], 2, degree_additivity_not_claimed_with_zero_divisors)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-multiply","seed":202610292061,"parameters":{"n":4,"addition":[[1,0,3,2],[0,1,2,3],[3,2,0,1],[2,3,1,0]],"multiplication":[[1,1,0,0],[1,1,1,1],[0,1,3,2],[0,1,2,3]],"left":[2,3,1,1],"right":[1,2,1,1],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-multiply","semantic_template":"qf-mod29-poly-multiply/{\"addition\":[[1,0,3,2],[0,1,2,3],[3,2,0,1],[2,3,1,0]],\"left\":[2,3,1,1],\"multiplication\":[[1,1,0,0],[1,1,1,1],[0,1,3,2],[0,1,2,3]],\"n\":4,\"right\":[1,2,1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0062","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-multiply；資料={\"n\":4,\"addition\":[[2,0,3,1],[0,1,2,3],[3,2,1,0],[1,3,0,2]],\"multiplication\":[[3,1,2,0],[1,1,1,1],[2,1,1,2],[0,1,2,3]],\"left\":[1,2,1,1],\"right\":[3,1,1,1]}。請依 [left,right,product,degree,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1, 2], [3], [1, 2], 1, degree_additivity_not_claimed_with_zero_divisors)","explanation_zh":"Wave4第三層核對=([1, 2], [3], [1, 2], 1, degree_additivity_not_claimed_with_zero_divisors)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-multiply","seed":202610292062,"parameters":{"n":4,"addition":[[2,0,3,1],[0,1,2,3],[3,2,1,0],[1,3,0,2]],"multiplication":[[3,1,2,0],[1,1,1,1],[2,1,1,2],[0,1,2,3]],"left":[1,2,1,1],"right":[3,1,1,1],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-multiply","semantic_template":"qf-mod29-poly-multiply/{\"addition\":[[2,0,3,1],[0,1,2,3],[3,2,1,0],[1,3,0,2]],\"left\":[1,2,1,1],\"multiplication\":[[3,1,2,0],[1,1,1,1],[2,1,1,2],[0,1,2,3]],\"n\":4,\"right\":[3,1,1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0063","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-multiply；資料={\"n\":4,\"addition\":[[1,3,0,2],[3,2,1,0],[0,1,2,3],[2,0,3,1]],\"multiplication\":[[0,1,2,3],[1,2,2,1],[2,2,2,2],[3,1,2,0]],\"left\":[0,3,2,2],\"right\":[1,0,2,2]}。請依 [left,right,product,degree,boundary] 欄序回答。","answer":{"kind":"expression","value":"([0, 3], [1, 0], [1, 3, 3], 2, degree_additivity_not_claimed_with_zero_divisors)","explanation_zh":"Wave4第三層核對=([0, 3], [1, 0], [1, 3, 3], 2, degree_additivity_not_claimed_with_zero_divisors)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-multiply","seed":202610292063,"parameters":{"n":4,"addition":[[1,3,0,2],[3,2,1,0],[0,1,2,3],[2,0,3,1]],"multiplication":[[0,1,2,3],[1,2,2,1],[2,2,2,2],[3,1,2,0]],"left":[0,3,2,2],"right":[1,0,2,2],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-multiply","semantic_template":"qf-mod29-poly-multiply/{\"addition\":[[1,3,0,2],[3,2,1,0],[0,1,2,3],[2,0,3,1]],\"left\":[0,3,2,2],\"multiplication\":[[0,1,2,3],[1,2,2,1],[2,2,2,2],[3,1,2,0]],\"n\":4,\"right\":[1,0,2,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0064","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-multiply；資料={\"n\":4,\"addition\":[[3,2,0,1],[2,3,1,0],[0,1,2,3],[1,0,3,2]],\"multiplication\":[[0,1,2,3],[1,0,2,3],[2,2,2,2],[3,3,2,2]],\"left\":[3,2,2,2],\"right\":[1,3,2,2]}。請依 [left,right,product,degree,boundary] 欄序回答。","answer":{"kind":"expression","value":"([3], [1, 3], [3], 0, degree_additivity_not_claimed_with_zero_divisors)","explanation_zh":"Wave4第三層核對=([3], [1, 3], [3], 0, degree_additivity_not_claimed_with_zero_divisors)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-multiply","seed":202610292064,"parameters":{"n":4,"addition":[[3,2,0,1],[2,3,1,0],[0,1,2,3],[1,0,3,2]],"multiplication":[[0,1,2,3],[1,0,2,3],[2,2,2,2],[3,3,2,2]],"left":[3,2,2,2],"right":[1,3,2,2],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-multiply","semantic_template":"qf-mod29-poly-multiply/{\"addition\":[[3,2,0,1],[2,3,1,0],[0,1,2,3],[1,0,3,2]],\"left\":[3,2,2,2],\"multiplication\":[[0,1,2,3],[1,0,2,3],[2,2,2,2],[3,3,2,2]],\"n\":4,\"right\":[1,3,2,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0065","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-multiply；資料={\"n\":4,\"addition\":[[2,3,0,1],[3,0,1,2],[0,1,2,3],[1,2,3,0]],\"multiplication\":[[2,0,2,0],[0,1,2,3],[2,2,2,2],[0,3,2,1]],\"left\":[3,1,2,2],\"right\":[2,3,2,2]}。請依 [left,right,product,degree,boundary] 欄序回答。","answer":{"kind":"expression","value":"([3, 1], [2, 3], [2, 1, 3], 2, degree_additivity_not_claimed_with_zero_divisors)","explanation_zh":"Wave4第三層核對=([3, 1], [2, 3], [2, 1, 3], 2, degree_additivity_not_claimed_with_zero_divisors)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-multiply","seed":202610292065,"parameters":{"n":4,"addition":[[2,3,0,1],[3,0,1,2],[0,1,2,3],[1,2,3,0]],"multiplication":[[2,0,2,0],[0,1,2,3],[2,2,2,2],[0,3,2,1]],"left":[3,1,2,2],"right":[2,3,2,2],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-multiply","semantic_template":"qf-mod29-poly-multiply/{\"addition\":[[2,3,0,1],[3,0,1,2],[0,1,2,3],[1,2,3,0]],\"left\":[3,1,2,2],\"multiplication\":[[2,0,2,0],[0,1,2,3],[2,2,2,2],[0,3,2,1]],\"n\":4,\"right\":[2,3,2,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0066","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-multiply；資料={\"n\":4,\"addition\":[[3,2,0,1],[2,3,1,0],[0,1,2,3],[1,0,3,2]],\"multiplication\":[[1,0,2,3],[0,1,2,3],[2,2,2,2],[3,3,2,2]],\"left\":[2,3,2,2],\"right\":[1,2,2,2]}。請依 [left,right,product,degree,boundary] 欄序回答。","answer":{"kind":"expression","value":"([2, 3], [1], [2, 3], 1, degree_additivity_not_claimed_with_zero_divisors)","explanation_zh":"Wave4第三層核對=([2, 3], [1], [2, 3], 1, degree_additivity_not_claimed_with_zero_divisors)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-multiply","seed":202610292066,"parameters":{"n":4,"addition":[[3,2,0,1],[2,3,1,0],[0,1,2,3],[1,0,3,2]],"multiplication":[[1,0,2,3],[0,1,2,3],[2,2,2,2],[3,3,2,2]],"left":[2,3,2,2],"right":[1,2,2,2],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-multiply","semantic_template":"qf-mod29-poly-multiply/{\"addition\":[[3,2,0,1],[2,3,1,0],[0,1,2,3],[1,0,3,2]],\"left\":[2,3,2,2],\"multiplication\":[[1,0,2,3],[0,1,2,3],[2,2,2,2],[3,3,2,2]],\"n\":4,\"right\":[1,2,2,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0067","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-multiply；資料={\"n\":4,\"addition\":[[2,3,0,1],[3,0,1,2],[0,1,2,3],[1,2,3,0]],\"multiplication\":[[2,0,2,0],[0,3,2,1],[2,2,2,2],[0,1,2,3]],\"left\":[3,1,2,2],\"right\":[0,3,2,2]}。請依 [left,right,product,degree,boundary] 欄序回答。","answer":{"kind":"expression","value":"([3, 1], [0, 3], [0, 1, 1], 2, degree_additivity_not_claimed_with_zero_divisors)","explanation_zh":"Wave4第三層核對=([3, 1], [0, 3], [0, 1, 1], 2, degree_additivity_not_claimed_with_zero_divisors)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-multiply","seed":202610292067,"parameters":{"n":4,"addition":[[2,3,0,1],[3,0,1,2],[0,1,2,3],[1,2,3,0]],"multiplication":[[2,0,2,0],[0,3,2,1],[2,2,2,2],[0,1,2,3]],"left":[3,1,2,2],"right":[0,3,2,2],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-multiply","semantic_template":"qf-mod29-poly-multiply/{\"addition\":[[2,3,0,1],[3,0,1,2],[0,1,2,3],[1,2,3,0]],\"left\":[3,1,2,2],\"multiplication\":[[2,0,2,0],[0,3,2,1],[2,2,2,2],[0,1,2,3]],\"n\":4,\"right\":[0,3,2,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0068","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-multiply；資料={\"n\":4,\"addition\":[[1,3,0,2],[3,2,1,0],[0,1,2,3],[2,0,3,1]],\"multiplication\":[[3,1,2,0],[1,2,2,1],[2,2,2,2],[0,1,2,3]],\"left\":[1,2,2,2],\"right\":[0,1,2,2]}。請依 [left,right,product,degree,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1], [0, 1], [1], 0, degree_additivity_not_claimed_with_zero_divisors)","explanation_zh":"Wave4第三層核對=([1], [0, 1], [1], 0, degree_additivity_not_claimed_with_zero_divisors)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-multiply","seed":202610292068,"parameters":{"n":4,"addition":[[1,3,0,2],[3,2,1,0],[0,1,2,3],[2,0,3,1]],"multiplication":[[3,1,2,0],[1,2,2,1],[2,2,2,2],[0,1,2,3]],"left":[1,2,2,2],"right":[0,1,2,2],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-multiply","semantic_template":"qf-mod29-poly-multiply/{\"addition\":[[1,3,0,2],[3,2,1,0],[0,1,2,3],[2,0,3,1]],\"left\":[1,2,2,2],\"multiplication\":[[3,1,2,0],[1,2,2,1],[2,2,2,2],[0,1,2,3]],\"n\":4,\"right\":[0,1,2,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0069","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-multiply；資料={\"n\":4,\"addition\":[[1,2,3,0],[2,3,0,1],[3,0,1,2],[0,1,2,3]],\"multiplication\":[[0,1,2,3],[1,3,1,3],[2,1,0,3],[3,3,3,3]],\"left\":[2,0,3,3],\"right\":[3,2,3,3]}。請依 [left,right,product,degree,boundary] 欄序回答。","answer":{"kind":"expression","value":"([2, 0], [3, 2], [3, 0, 2], 2, degree_additivity_not_claimed_with_zero_divisors)","explanation_zh":"Wave4第三層核對=([2, 0], [3, 2], [3, 0, 2], 2, degree_additivity_not_claimed_with_zero_divisors)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-multiply","seed":202610292069,"parameters":{"n":4,"addition":[[1,2,3,0],[2,3,0,1],[3,0,1,2],[0,1,2,3]],"multiplication":[[0,1,2,3],[1,3,1,3],[2,1,0,3],[3,3,3,3]],"left":[2,0,3,3],"right":[3,2,3,3],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-multiply","semantic_template":"qf-mod29-poly-multiply/{\"addition\":[[1,2,3,0],[2,3,0,1],[3,0,1,2],[0,1,2,3]],\"left\":[2,0,3,3],\"multiplication\":[[0,1,2,3],[1,3,1,3],[2,1,0,3],[3,3,3,3]],\"n\":4,\"right\":[3,2,3,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0070","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-multiply；資料={\"n\":4,\"addition\":[[2,3,1,0],[3,2,0,1],[1,0,3,2],[0,1,2,3]],\"multiplication\":[[0,1,2,3],[1,0,2,3],[2,2,3,3],[3,3,3,3]],\"left\":[3,2,3,3],\"right\":[0,3,3,3]}。請依 [left,right,product,degree,boundary] 欄序回答。","answer":{"kind":"expression","value":"([3, 2], [0], [3, 2], 1, degree_additivity_not_claimed_with_zero_divisors)","explanation_zh":"Wave4第三層核對=([3, 2], [0], [3, 2], 1, degree_additivity_not_claimed_with_zero_divisors)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-multiply","seed":202610292070,"parameters":{"n":4,"addition":[[2,3,1,0],[3,2,0,1],[1,0,3,2],[0,1,2,3]],"multiplication":[[0,1,2,3],[1,0,2,3],[2,2,3,3],[3,3,3,3]],"left":[3,2,3,3],"right":[0,3,3,3],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-multiply","semantic_template":"qf-mod29-poly-multiply/{\"addition\":[[2,3,1,0],[3,2,0,1],[1,0,3,2],[0,1,2,3]],\"left\":[3,2,3,3],\"multiplication\":[[0,1,2,3],[1,0,2,3],[2,2,3,3],[3,3,3,3]],\"n\":4,\"right\":[0,3,3,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0071","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-multiply；資料={\"n\":4,\"addition\":[[3,2,1,0],[2,0,3,1],[1,3,0,2],[0,1,2,3]],\"multiplication\":[[3,0,0,3],[0,1,2,3],[0,2,1,3],[3,3,3,3]],\"left\":[1,2,3,3],\"right\":[0,1,3,3]}。請依 [left,right,product,degree,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1, 2], [0, 1], [0, 2, 2], 2, degree_additivity_not_claimed_with_zero_divisors)","explanation_zh":"Wave4第三層核對=([1, 2], [0, 1], [0, 2, 2], 2, degree_additivity_not_claimed_with_zero_divisors)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-multiply","seed":202610292071,"parameters":{"n":4,"addition":[[3,2,1,0],[2,0,3,1],[1,3,0,2],[0,1,2,3]],"multiplication":[[3,0,0,3],[0,1,2,3],[0,2,1,3],[3,3,3,3]],"left":[1,2,3,3],"right":[0,1,3,3],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-multiply","semantic_template":"qf-mod29-poly-multiply/{\"addition\":[[3,2,1,0],[2,0,3,1],[1,3,0,2],[0,1,2,3]],\"left\":[1,2,3,3],\"multiplication\":[[3,0,0,3],[0,1,2,3],[0,2,1,3],[3,3,3,3]],\"n\":4,\"right\":[0,1,3,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0072","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-multiply；資料={\"n\":4,\"addition\":[[2,3,1,0],[3,2,0,1],[1,0,3,2],[0,1,2,3]],\"multiplication\":[[1,0,2,3],[0,1,2,3],[2,2,3,3],[3,3,3,3]],\"left\":[2,3,3,3],\"right\":[0,2,3,3]}。請依 [left,right,product,degree,boundary] 欄序回答。","answer":{"kind":"expression","value":"([2], [0, 2], [2], 0, degree_additivity_not_claimed_with_zero_divisors)","explanation_zh":"Wave4第三層核對=([2], [0, 2], [2], 0, degree_additivity_not_claimed_with_zero_divisors)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-multiply","seed":202610292072,"parameters":{"n":4,"addition":[[2,3,1,0],[3,2,0,1],[1,0,3,2],[0,1,2,3]],"multiplication":[[1,0,2,3],[0,1,2,3],[2,2,3,3],[3,3,3,3]],"left":[2,3,3,3],"right":[0,2,3,3],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-multiply","semantic_template":"qf-mod29-poly-multiply/{\"addition\":[[2,3,1,0],[3,2,0,1],[1,0,3,2],[0,1,2,3]],\"left\":[2,3,3,3],\"multiplication\":[[1,0,2,3],[0,1,2,3],[2,2,3,3],[3,3,3,3]],\"n\":4,\"right\":[0,2,3,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0073","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-multiply；資料={\"n\":4,\"addition\":[[3,2,1,0],[2,0,3,1],[1,3,0,2],[0,1,2,3]],\"multiplication\":[[3,0,0,3],[0,2,1,3],[0,1,2,3],[3,3,3,3]],\"left\":[1,2,3,3],\"right\":[3,1,3,3]}。請依 [left,right,product,degree,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1, 2], [3, 1], [3, 2, 1], 2, degree_additivity_not_claimed_with_zero_divisors)","explanation_zh":"Wave4第三層核對=([1, 2], [3, 1], [3, 2, 1], 2, degree_additivity_not_claimed_with_zero_divisors)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-multiply","seed":202610292073,"parameters":{"n":4,"addition":[[3,2,1,0],[2,0,3,1],[1,3,0,2],[0,1,2,3]],"multiplication":[[3,0,0,3],[0,2,1,3],[0,1,2,3],[3,3,3,3]],"left":[1,2,3,3],"right":[3,1,3,3],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-multiply","semantic_template":"qf-mod29-poly-multiply/{\"addition\":[[3,2,1,0],[2,0,3,1],[1,3,0,2],[0,1,2,3]],\"left\":[1,2,3,3],\"multiplication\":[[3,0,0,3],[0,2,1,3],[0,1,2,3],[3,3,3,3]],\"n\":4,\"right\":[3,1,3,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0074","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-multiply；資料={\"n\":4,\"addition\":[[1,2,3,0],[2,3,0,1],[3,0,1,2],[0,1,2,3]],\"multiplication\":[[2,1,0,3],[1,3,1,3],[0,1,2,3],[3,3,3,3]],\"left\":[3,1,3,3],\"right\":[2,3,3,3]}。請依 [left,right,product,degree,boundary] 欄序回答。","answer":{"kind":"expression","value":"([3, 1], [2], [3, 1], 1, degree_additivity_not_claimed_with_zero_divisors)","explanation_zh":"Wave4第三層核對=([3, 1], [2], [3, 1], 1, degree_additivity_not_claimed_with_zero_divisors)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-multiply","seed":202610292074,"parameters":{"n":4,"addition":[[1,2,3,0],[2,3,0,1],[3,0,1,2],[0,1,2,3]],"multiplication":[[2,1,0,3],[1,3,1,3],[0,1,2,3],[3,3,3,3]],"left":[3,1,3,3],"right":[2,3,3,3],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-multiply","semantic_template":"qf-mod29-poly-multiply/{\"addition\":[[1,2,3,0],[2,3,0,1],[3,0,1,2],[0,1,2,3]],\"left\":[3,1,3,3],\"multiplication\":[[2,1,0,3],[1,3,1,3],[0,1,2,3],[3,3,3,3]],\"n\":4,\"right\":[2,3,3,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0075","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-multiply；資料={\"n\":4,\"addition\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"multiplication\":[[0,0,0,0],[0,1,2,3],[0,2,0,2],[0,3,2,1]],\"left\":[1,3,0,0,0],\"right\":[2,1,0,0,0]}。請依 [left,right,product,degree,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1, 3], [2, 1], [2, 3, 3], 2, degree_additivity_not_claimed_with_zero_divisors)","explanation_zh":"Wave4第三層核對=([1, 3], [2, 1], [2, 3, 3], 2, degree_additivity_not_claimed_with_zero_divisors)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-multiply","seed":202610292075,"parameters":{"n":4,"addition":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"multiplication":[[0,0,0,0],[0,1,2,3],[0,2,0,2],[0,3,2,1]],"left":[1,3,0,0,0],"right":[2,1,0,0,0],"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-multiply","semantic_template":"qf-mod29-poly-multiply/{\"addition\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"left\":[1,3,0,0,0],\"multiplication\":[[0,0,0,0],[0,1,2,3],[0,2,0,2],[0,3,2,1]],\"n\":4,\"right\":[2,1,0,0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0076","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-infinite-carrier-witness；資料={\"n\":4,\"addition\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"multiplication\":[[0,0,0,0],[0,1,2,3],[0,2,0,2],[0,3,2,1]],\"degree\":1}。請依 [degree,monomial,coefficient_count,distinct,boundary] 欄序回答。","answer":{"kind":"expression","value":"(1, [0, 1], 2, 是, R[x]_carrier_infinite_when_zero_ne_one)","explanation_zh":"Wave4第三層核對=(1, [0, 1], 2, 是, R[x]_carrier_infinite_when_zero_ne_one)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-infinite-carrier-witness","seed":202610292076,"parameters":{"n":4,"addition":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"multiplication":[[0,0,0,0],[0,1,2,3],[0,2,0,2],[0,3,2,1]],"degree":1,"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-infinite-carrier-witness","semantic_template":"qf-mod29-poly-infinite-carrier-witness/{\"addition\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"degree\":1,\"multiplication\":[[0,0,0,0],[0,1,2,3],[0,2,0,2],[0,3,2,1]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0077","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-infinite-carrier-witness；資料={\"n\":4,\"addition\":[[0,1,2,3],[1,3,0,2],[2,0,3,1],[3,2,1,0]],\"multiplication\":[[0,0,0,0],[0,1,2,3],[0,2,1,3],[0,3,3,0]],\"degree\":2}。請依 [degree,monomial,coefficient_count,distinct,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, [0, 0, 1], 3, 是, R[x]_carrier_infinite_when_zero_ne_one)","explanation_zh":"Wave4第三層核對=(2, [0, 0, 1], 3, 是, R[x]_carrier_infinite_when_zero_ne_one)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-infinite-carrier-witness","seed":202610292077,"parameters":{"n":4,"addition":[[0,1,2,3],[1,3,0,2],[2,0,3,1],[3,2,1,0]],"multiplication":[[0,0,0,0],[0,1,2,3],[0,2,1,3],[0,3,3,0]],"degree":2,"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-infinite-carrier-witness","semantic_template":"qf-mod29-poly-infinite-carrier-witness/{\"addition\":[[0,1,2,3],[1,3,0,2],[2,0,3,1],[3,2,1,0]],\"degree\":2,\"multiplication\":[[0,0,0,0],[0,1,2,3],[0,2,1,3],[0,3,3,0]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0078","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-infinite-carrier-witness；資料={\"n\":4,\"addition\":[[0,1,2,3],[1,0,3,2],[2,3,1,0],[3,2,0,1]],\"multiplication\":[[0,0,0,0],[0,0,1,1],[0,1,2,3],[0,1,3,2]],\"degree\":3}。請依 [degree,monomial,coefficient_count,distinct,boundary] 欄序回答。","answer":{"kind":"expression","value":"(3, [0, 0, 0, 2], 4, 是, R[x]_carrier_infinite_when_zero_ne_one)","explanation_zh":"Wave4第三層核對=(3, [0, 0, 0, 2], 4, 是, R[x]_carrier_infinite_when_zero_ne_one)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-infinite-carrier-witness","seed":202610292078,"parameters":{"n":4,"addition":[[0,1,2,3],[1,0,3,2],[2,3,1,0],[3,2,0,1]],"multiplication":[[0,0,0,0],[0,0,1,1],[0,1,2,3],[0,1,3,2]],"degree":3,"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-infinite-carrier-witness","semantic_template":"qf-mod29-poly-infinite-carrier-witness/{\"addition\":[[0,1,2,3],[1,0,3,2],[2,3,1,0],[3,2,0,1]],\"degree\":3,\"multiplication\":[[0,0,0,0],[0,0,1,1],[0,1,2,3],[0,1,3,2]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0079","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-infinite-carrier-witness；資料={\"n\":4,\"addition\":[[0,1,2,3],[1,3,0,2],[2,0,3,1],[3,2,1,0]],\"multiplication\":[[0,0,0,0],[0,2,1,3],[0,1,2,3],[0,3,3,0]],\"degree\":4}。請依 [degree,monomial,coefficient_count,distinct,boundary] 欄序回答。","answer":{"kind":"expression","value":"(4, [0, 0, 0, 0, 2], 5, 是, R[x]_carrier_infinite_when_zero_ne_one)","explanation_zh":"Wave4第三層核對=(4, [0, 0, 0, 0, 2], 5, 是, R[x]_carrier_infinite_when_zero_ne_one)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-infinite-carrier-witness","seed":202610292079,"parameters":{"n":4,"addition":[[0,1,2,3],[1,3,0,2],[2,0,3,1],[3,2,1,0]],"multiplication":[[0,0,0,0],[0,2,1,3],[0,1,2,3],[0,3,3,0]],"degree":4,"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-infinite-carrier-witness","semantic_template":"qf-mod29-poly-infinite-carrier-witness/{\"addition\":[[0,1,2,3],[1,3,0,2],[2,0,3,1],[3,2,1,0]],\"degree\":4,\"multiplication\":[[0,0,0,0],[0,2,1,3],[0,1,2,3],[0,3,3,0]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0080","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-infinite-carrier-witness；資料={\"n\":4,\"addition\":[[0,1,2,3],[1,0,3,2],[2,3,1,0],[3,2,0,1]],\"multiplication\":[[0,0,0,0],[0,0,1,1],[0,1,3,2],[0,1,2,3]],\"degree\":5}。請依 [degree,monomial,coefficient_count,distinct,boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, [0, 0, 0, 0, 0, 3], 6, 是, R[x]_carrier_infinite_when_zero_ne_one)","explanation_zh":"Wave4第三層核對=(5, [0, 0, 0, 0, 0, 3], 6, 是, R[x]_carrier_infinite_when_zero_ne_one)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-infinite-carrier-witness","seed":202610292080,"parameters":{"n":4,"addition":[[0,1,2,3],[1,0,3,2],[2,3,1,0],[3,2,0,1]],"multiplication":[[0,0,0,0],[0,0,1,1],[0,1,3,2],[0,1,2,3]],"degree":5,"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-infinite-carrier-witness","semantic_template":"qf-mod29-poly-infinite-carrier-witness/{\"addition\":[[0,1,2,3],[1,0,3,2],[2,3,1,0],[3,2,0,1]],\"degree\":5,\"multiplication\":[[0,0,0,0],[0,0,1,1],[0,1,3,2],[0,1,2,3]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0081","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-infinite-carrier-witness；資料={\"n\":4,\"addition\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"multiplication\":[[0,0,0,0],[0,3,2,1],[0,2,0,2],[0,1,2,3]],\"degree\":6}。請依 [degree,monomial,coefficient_count,distinct,boundary] 欄序回答。","answer":{"kind":"expression","value":"(6, [0, 0, 0, 0, 0, 0, 3], 7, 是, R[x]_carrier_infinite_when_zero_ne_one)","explanation_zh":"Wave4第三層核對=(6, [0, 0, 0, 0, 0, 0, 3], 7, 是, R[x]_carrier_infinite_when_zero_ne_one)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-infinite-carrier-witness","seed":202610292081,"parameters":{"n":4,"addition":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"multiplication":[[0,0,0,0],[0,3,2,1],[0,2,0,2],[0,1,2,3]],"degree":6,"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-infinite-carrier-witness","semantic_template":"qf-mod29-poly-infinite-carrier-witness/{\"addition\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"degree\":6,\"multiplication\":[[0,0,0,0],[0,3,2,1],[0,2,0,2],[0,1,2,3]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0082","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-infinite-carrier-witness；資料={\"n\":4,\"addition\":[[2,0,3,1],[0,1,2,3],[3,2,1,0],[1,3,0,2]],\"multiplication\":[[0,1,2,3],[1,1,1,1],[2,1,1,2],[3,1,2,0]],\"degree\":7}。請依 [degree,monomial,coefficient_count,distinct,boundary] 欄序回答。","answer":{"kind":"expression","value":"(7, [1, 1, 1, 1, 1, 1, 1, 0], 8, 是, R[x]_carrier_infinite_when_zero_ne_one)","explanation_zh":"Wave4第三層核對=(7, [1, 1, 1, 1, 1, 1, 1, 0], 8, 是, R[x]_carrier_infinite_when_zero_ne_one)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-infinite-carrier-witness","seed":202610292082,"parameters":{"n":4,"addition":[[2,0,3,1],[0,1,2,3],[3,2,1,0],[1,3,0,2]],"multiplication":[[0,1,2,3],[1,1,1,1],[2,1,1,2],[3,1,2,0]],"degree":7,"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-infinite-carrier-witness","semantic_template":"qf-mod29-poly-infinite-carrier-witness/{\"addition\":[[2,0,3,1],[0,1,2,3],[3,2,1,0],[1,3,0,2]],\"degree\":7,\"multiplication\":[[0,1,2,3],[1,1,1,1],[2,1,1,2],[3,1,2,0]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0083","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-infinite-carrier-witness；資料={\"n\":4,\"addition\":[[3,0,1,2],[0,1,2,3],[1,2,3,0],[2,3,0,1]],\"multiplication\":[[0,1,2,3],[1,1,1,1],[2,1,0,3],[3,1,3,1]],\"degree\":8}。請依 [degree,monomial,coefficient_count,distinct,boundary] 欄序回答。","answer":{"kind":"expression","value":"(8, [1, 1, 1, 1, 1, 1, 1, 1, 0], 9, 是, R[x]_carrier_infinite_when_zero_ne_one)","explanation_zh":"Wave4第三層核對=(8, [1, 1, 1, 1, 1, 1, 1, 1, 0], 9, 是, R[x]_carrier_infinite_when_zero_ne_one)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-infinite-carrier-witness","seed":202610292083,"parameters":{"n":4,"addition":[[3,0,1,2],[0,1,2,3],[1,2,3,0],[2,3,0,1]],"multiplication":[[0,1,2,3],[1,1,1,1],[2,1,0,3],[3,1,3,1]],"degree":8,"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-infinite-carrier-witness","semantic_template":"qf-mod29-poly-infinite-carrier-witness/{\"addition\":[[3,0,1,2],[0,1,2,3],[1,2,3,0],[2,3,0,1]],\"degree\":8,\"multiplication\":[[0,1,2,3],[1,1,1,1],[2,1,0,3],[3,1,3,1]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0084","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-infinite-carrier-witness；資料={\"n\":4,\"addition\":[[1,0,3,2],[0,1,2,3],[3,2,0,1],[2,3,1,0]],\"multiplication\":[[1,1,0,0],[1,1,1,1],[0,1,2,3],[0,1,3,2]],\"degree\":9}。請依 [degree,monomial,coefficient_count,distinct,boundary] 欄序回答。","answer":{"kind":"expression","value":"(9, [1, 1, 1, 1, 1, 1, 1, 1, 1, 2], 10, 是, R[x]_carrier_infinite_when_zero_ne_one)","explanation_zh":"Wave4第三層核對=(9, [1, 1, 1, 1, 1, 1, 1, 1, 1, 2], 10, 是, R[x]_carrier_infinite_when_zero_ne_one)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-infinite-carrier-witness","seed":202610292084,"parameters":{"n":4,"addition":[[1,0,3,2],[0,1,2,3],[3,2,0,1],[2,3,1,0]],"multiplication":[[1,1,0,0],[1,1,1,1],[0,1,2,3],[0,1,3,2]],"degree":9,"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-infinite-carrier-witness","semantic_template":"qf-mod29-poly-infinite-carrier-witness/{\"addition\":[[1,0,3,2],[0,1,2,3],[3,2,0,1],[2,3,1,0]],\"degree\":9,\"multiplication\":[[1,1,0,0],[1,1,1,1],[0,1,2,3],[0,1,3,2]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0085","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-infinite-carrier-witness；資料={\"n\":4,\"addition\":[[3,0,1,2],[0,1,2,3],[1,2,3,0],[2,3,0,1]],\"multiplication\":[[2,1,0,3],[1,1,1,1],[0,1,2,3],[3,1,3,1]],\"degree\":10}。請依 [degree,monomial,coefficient_count,distinct,boundary] 欄序回答。","answer":{"kind":"expression","value":"(10, [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2], 11, 是, R[x]_carrier_infinite_when_zero_ne_one)","explanation_zh":"Wave4第三層核對=(10, [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2], 11, 是, R[x]_carrier_infinite_when_zero_ne_one)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-infinite-carrier-witness","seed":202610292085,"parameters":{"n":4,"addition":[[3,0,1,2],[0,1,2,3],[1,2,3,0],[2,3,0,1]],"multiplication":[[2,1,0,3],[1,1,1,1],[0,1,2,3],[3,1,3,1]],"degree":10,"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-infinite-carrier-witness","semantic_template":"qf-mod29-poly-infinite-carrier-witness/{\"addition\":[[3,0,1,2],[0,1,2,3],[1,2,3,0],[2,3,0,1]],\"degree\":10,\"multiplication\":[[2,1,0,3],[1,1,1,1],[0,1,2,3],[3,1,3,1]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0086","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-infinite-carrier-witness；資料={\"n\":4,\"addition\":[[1,0,3,2],[0,1,2,3],[3,2,0,1],[2,3,1,0]],\"multiplication\":[[1,1,0,0],[1,1,1,1],[0,1,3,2],[0,1,2,3]],\"degree\":11}。請依 [degree,monomial,coefficient_count,distinct,boundary] 欄序回答。","answer":{"kind":"expression","value":"(11, [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3], 12, 是, R[x]_carrier_infinite_when_zero_ne_one)","explanation_zh":"Wave4第三層核對=(11, [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3], 12, 是, R[x]_carrier_infinite_when_zero_ne_one)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-infinite-carrier-witness","seed":202610292086,"parameters":{"n":4,"addition":[[1,0,3,2],[0,1,2,3],[3,2,0,1],[2,3,1,0]],"multiplication":[[1,1,0,0],[1,1,1,1],[0,1,3,2],[0,1,2,3]],"degree":11,"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-infinite-carrier-witness","semantic_template":"qf-mod29-poly-infinite-carrier-witness/{\"addition\":[[1,0,3,2],[0,1,2,3],[3,2,0,1],[2,3,1,0]],\"degree\":11,\"multiplication\":[[1,1,0,0],[1,1,1,1],[0,1,3,2],[0,1,2,3]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0087","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-infinite-carrier-witness；資料={\"n\":4,\"addition\":[[2,0,3,1],[0,1,2,3],[3,2,1,0],[1,3,0,2]],\"multiplication\":[[3,1,2,0],[1,1,1,1],[2,1,1,2],[0,1,2,3]],\"degree\":12}。請依 [degree,monomial,coefficient_count,distinct,boundary] 欄序回答。","answer":{"kind":"expression","value":"(12, [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3], 13, 是, R[x]_carrier_infinite_when_zero_ne_one)","explanation_zh":"Wave4第三層核對=(12, [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3], 13, 是, R[x]_carrier_infinite_when_zero_ne_one)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-infinite-carrier-witness","seed":202610292087,"parameters":{"n":4,"addition":[[2,0,3,1],[0,1,2,3],[3,2,1,0],[1,3,0,2]],"multiplication":[[3,1,2,0],[1,1,1,1],[2,1,1,2],[0,1,2,3]],"degree":12,"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-infinite-carrier-witness","semantic_template":"qf-mod29-poly-infinite-carrier-witness/{\"addition\":[[2,0,3,1],[0,1,2,3],[3,2,1,0],[1,3,0,2]],\"degree\":12,\"multiplication\":[[3,1,2,0],[1,1,1,1],[2,1,1,2],[0,1,2,3]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0088","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-infinite-carrier-witness；資料={\"n\":4,\"addition\":[[1,3,0,2],[3,2,1,0],[0,1,2,3],[2,0,3,1]],\"multiplication\":[[0,1,2,3],[1,2,2,1],[2,2,2,2],[3,1,2,0]],\"degree\":13}。請依 [degree,monomial,coefficient_count,distinct,boundary] 欄序回答。","answer":{"kind":"expression","value":"(13, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0], 14, 是, R[x]_carrier_infinite_when_zero_ne_one)","explanation_zh":"Wave4第三層核對=(13, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0], 14, 是, R[x]_carrier_infinite_when_zero_ne_one)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-infinite-carrier-witness","seed":202610292088,"parameters":{"n":4,"addition":[[1,3,0,2],[3,2,1,0],[0,1,2,3],[2,0,3,1]],"multiplication":[[0,1,2,3],[1,2,2,1],[2,2,2,2],[3,1,2,0]],"degree":13,"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-infinite-carrier-witness","semantic_template":"qf-mod29-poly-infinite-carrier-witness/{\"addition\":[[1,3,0,2],[3,2,1,0],[0,1,2,3],[2,0,3,1]],\"degree\":13,\"multiplication\":[[0,1,2,3],[1,2,2,1],[2,2,2,2],[3,1,2,0]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0089","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-infinite-carrier-witness；資料={\"n\":4,\"addition\":[[3,2,0,1],[2,3,1,0],[0,1,2,3],[1,0,3,2]],\"multiplication\":[[0,1,2,3],[1,0,2,3],[2,2,2,2],[3,3,2,2]],\"degree\":14}。請依 [degree,monomial,coefficient_count,distinct,boundary] 欄序回答。","answer":{"kind":"expression","value":"(14, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0], 15, 是, R[x]_carrier_infinite_when_zero_ne_one)","explanation_zh":"Wave4第三層核對=(14, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0], 15, 是, R[x]_carrier_infinite_when_zero_ne_one)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-infinite-carrier-witness","seed":202610292089,"parameters":{"n":4,"addition":[[3,2,0,1],[2,3,1,0],[0,1,2,3],[1,0,3,2]],"multiplication":[[0,1,2,3],[1,0,2,3],[2,2,2,2],[3,3,2,2]],"degree":14,"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-infinite-carrier-witness","semantic_template":"qf-mod29-poly-infinite-carrier-witness/{\"addition\":[[3,2,0,1],[2,3,1,0],[0,1,2,3],[1,0,3,2]],\"degree\":14,\"multiplication\":[[0,1,2,3],[1,0,2,3],[2,2,2,2],[3,3,2,2]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0090","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-infinite-carrier-witness；資料={\"n\":4,\"addition\":[[2,3,0,1],[3,0,1,2],[0,1,2,3],[1,2,3,0]],\"multiplication\":[[2,0,2,0],[0,1,2,3],[2,2,2,2],[0,3,2,1]],\"degree\":15}。請依 [degree,monomial,coefficient_count,distinct,boundary] 欄序回答。","answer":{"kind":"expression","value":"(15, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], 16, 是, R[x]_carrier_infinite_when_zero_ne_one)","explanation_zh":"Wave4第三層核對=(15, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], 16, 是, R[x]_carrier_infinite_when_zero_ne_one)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-infinite-carrier-witness","seed":202610292090,"parameters":{"n":4,"addition":[[2,3,0,1],[3,0,1,2],[0,1,2,3],[1,2,3,0]],"multiplication":[[2,0,2,0],[0,1,2,3],[2,2,2,2],[0,3,2,1]],"degree":15,"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-infinite-carrier-witness","semantic_template":"qf-mod29-poly-infinite-carrier-witness/{\"addition\":[[2,3,0,1],[3,0,1,2],[0,1,2,3],[1,2,3,0]],\"degree\":15,\"multiplication\":[[2,0,2,0],[0,1,2,3],[2,2,2,2],[0,3,2,1]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0091","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-infinite-carrier-witness；資料={\"n\":4,\"addition\":[[3,2,0,1],[2,3,1,0],[0,1,2,3],[1,0,3,2]],\"multiplication\":[[1,0,2,3],[0,1,2,3],[2,2,2,2],[3,3,2,2]],\"degree\":16}。請依 [degree,monomial,coefficient_count,distinct,boundary] 欄序回答。","answer":{"kind":"expression","value":"(16, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], 17, 是, R[x]_carrier_infinite_when_zero_ne_one)","explanation_zh":"Wave4第三層核對=(16, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], 17, 是, R[x]_carrier_infinite_when_zero_ne_one)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-infinite-carrier-witness","seed":202610292091,"parameters":{"n":4,"addition":[[3,2,0,1],[2,3,1,0],[0,1,2,3],[1,0,3,2]],"multiplication":[[1,0,2,3],[0,1,2,3],[2,2,2,2],[3,3,2,2]],"degree":16,"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-infinite-carrier-witness","semantic_template":"qf-mod29-poly-infinite-carrier-witness/{\"addition\":[[3,2,0,1],[2,3,1,0],[0,1,2,3],[1,0,3,2]],\"degree\":16,\"multiplication\":[[1,0,2,3],[0,1,2,3],[2,2,2,2],[3,3,2,2]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0092","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-infinite-carrier-witness；資料={\"n\":4,\"addition\":[[2,3,0,1],[3,0,1,2],[0,1,2,3],[1,2,3,0]],\"multiplication\":[[2,0,2,0],[0,3,2,1],[2,2,2,2],[0,1,2,3]],\"degree\":17}。請依 [degree,monomial,coefficient_count,distinct,boundary] 欄序回答。","answer":{"kind":"expression","value":"(17, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3], 18, 是, R[x]_carrier_infinite_when_zero_ne_one)","explanation_zh":"Wave4第三層核對=(17, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3], 18, 是, R[x]_carrier_infinite_when_zero_ne_one)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-infinite-carrier-witness","seed":202610292092,"parameters":{"n":4,"addition":[[2,3,0,1],[3,0,1,2],[0,1,2,3],[1,2,3,0]],"multiplication":[[2,0,2,0],[0,3,2,1],[2,2,2,2],[0,1,2,3]],"degree":17,"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-infinite-carrier-witness","semantic_template":"qf-mod29-poly-infinite-carrier-witness/{\"addition\":[[2,3,0,1],[3,0,1,2],[0,1,2,3],[1,2,3,0]],\"degree\":17,\"multiplication\":[[2,0,2,0],[0,3,2,1],[2,2,2,2],[0,1,2,3]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0093","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-infinite-carrier-witness；資料={\"n\":4,\"addition\":[[1,3,0,2],[3,2,1,0],[0,1,2,3],[2,0,3,1]],\"multiplication\":[[3,1,2,0],[1,2,2,1],[2,2,2,2],[0,1,2,3]],\"degree\":18}。請依 [degree,monomial,coefficient_count,distinct,boundary] 欄序回答。","answer":{"kind":"expression","value":"(18, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3], 19, 是, R[x]_carrier_infinite_when_zero_ne_one)","explanation_zh":"Wave4第三層核對=(18, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3], 19, 是, R[x]_carrier_infinite_when_zero_ne_one)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-infinite-carrier-witness","seed":202610292093,"parameters":{"n":4,"addition":[[1,3,0,2],[3,2,1,0],[0,1,2,3],[2,0,3,1]],"multiplication":[[3,1,2,0],[1,2,2,1],[2,2,2,2],[0,1,2,3]],"degree":18,"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-infinite-carrier-witness","semantic_template":"qf-mod29-poly-infinite-carrier-witness/{\"addition\":[[1,3,0,2],[3,2,1,0],[0,1,2,3],[2,0,3,1]],\"degree\":18,\"multiplication\":[[3,1,2,0],[1,2,2,1],[2,2,2,2],[0,1,2,3]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0094","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-infinite-carrier-witness；資料={\"n\":4,\"addition\":[[1,2,3,0],[2,3,0,1],[3,0,1,2],[0,1,2,3]],\"multiplication\":[[0,1,2,3],[1,3,1,3],[2,1,0,3],[3,3,3,3]],\"degree\":19}。請依 [degree,monomial,coefficient_count,distinct,boundary] 欄序回答。","answer":{"kind":"expression","value":"(19, [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 0], 20, 是, R[x]_carrier_infinite_when_zero_ne_one)","explanation_zh":"Wave4第三層核對=(19, [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 0], 20, 是, R[x]_carrier_infinite_when_zero_ne_one)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-infinite-carrier-witness","seed":202610292094,"parameters":{"n":4,"addition":[[1,2,3,0],[2,3,0,1],[3,0,1,2],[0,1,2,3]],"multiplication":[[0,1,2,3],[1,3,1,3],[2,1,0,3],[3,3,3,3]],"degree":19,"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-infinite-carrier-witness","semantic_template":"qf-mod29-poly-infinite-carrier-witness/{\"addition\":[[1,2,3,0],[2,3,0,1],[3,0,1,2],[0,1,2,3]],\"degree\":19,\"multiplication\":[[0,1,2,3],[1,3,1,3],[2,1,0,3],[3,3,3,3]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0095","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-infinite-carrier-witness；資料={\"n\":4,\"addition\":[[2,3,1,0],[3,2,0,1],[1,0,3,2],[0,1,2,3]],\"multiplication\":[[0,1,2,3],[1,0,2,3],[2,2,3,3],[3,3,3,3]],\"degree\":20}。請依 [degree,monomial,coefficient_count,distinct,boundary] 欄序回答。","answer":{"kind":"expression","value":"(20, [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 0], 21, 是, R[x]_carrier_infinite_when_zero_ne_one)","explanation_zh":"Wave4第三層核對=(20, [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 0], 21, 是, R[x]_carrier_infinite_when_zero_ne_one)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-infinite-carrier-witness","seed":202610292095,"parameters":{"n":4,"addition":[[2,3,1,0],[3,2,0,1],[1,0,3,2],[0,1,2,3]],"multiplication":[[0,1,2,3],[1,0,2,3],[2,2,3,3],[3,3,3,3]],"degree":20,"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-infinite-carrier-witness","semantic_template":"qf-mod29-poly-infinite-carrier-witness/{\"addition\":[[2,3,1,0],[3,2,0,1],[1,0,3,2],[0,1,2,3]],\"degree\":20,\"multiplication\":[[0,1,2,3],[1,0,2,3],[2,2,3,3],[3,3,3,3]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0096","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-infinite-carrier-witness；資料={\"n\":4,\"addition\":[[3,2,1,0],[2,0,3,1],[1,3,0,2],[0,1,2,3]],\"multiplication\":[[3,0,0,3],[0,1,2,3],[0,2,1,3],[3,3,3,3]],\"degree\":21}。請依 [degree,monomial,coefficient_count,distinct,boundary] 欄序回答。","answer":{"kind":"expression","value":"(21, [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1], 22, 是, R[x]_carrier_infinite_when_zero_ne_one)","explanation_zh":"Wave4第三層核對=(21, [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1], 22, 是, R[x]_carrier_infinite_when_zero_ne_one)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-infinite-carrier-witness","seed":202610292096,"parameters":{"n":4,"addition":[[3,2,1,0],[2,0,3,1],[1,3,0,2],[0,1,2,3]],"multiplication":[[3,0,0,3],[0,1,2,3],[0,2,1,3],[3,3,3,3]],"degree":21,"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-infinite-carrier-witness","semantic_template":"qf-mod29-poly-infinite-carrier-witness/{\"addition\":[[3,2,1,0],[2,0,3,1],[1,3,0,2],[0,1,2,3]],\"degree\":21,\"multiplication\":[[3,0,0,3],[0,1,2,3],[0,2,1,3],[3,3,3,3]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0097","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-infinite-carrier-witness；資料={\"n\":4,\"addition\":[[2,3,1,0],[3,2,0,1],[1,0,3,2],[0,1,2,3]],\"multiplication\":[[1,0,2,3],[0,1,2,3],[2,2,3,3],[3,3,3,3]],\"degree\":22}。請依 [degree,monomial,coefficient_count,distinct,boundary] 欄序回答。","answer":{"kind":"expression","value":"(22, [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1], 23, 是, R[x]_carrier_infinite_when_zero_ne_one)","explanation_zh":"Wave4第三層核對=(22, [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1], 23, 是, R[x]_carrier_infinite_when_zero_ne_one)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-infinite-carrier-witness","seed":202610292097,"parameters":{"n":4,"addition":[[2,3,1,0],[3,2,0,1],[1,0,3,2],[0,1,2,3]],"multiplication":[[1,0,2,3],[0,1,2,3],[2,2,3,3],[3,3,3,3]],"degree":22,"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-infinite-carrier-witness","semantic_template":"qf-mod29-poly-infinite-carrier-witness/{\"addition\":[[2,3,1,0],[3,2,0,1],[1,0,3,2],[0,1,2,3]],\"degree\":22,\"multiplication\":[[1,0,2,3],[0,1,2,3],[2,2,3,3],[3,3,3,3]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0098","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-infinite-carrier-witness；資料={\"n\":4,\"addition\":[[3,2,1,0],[2,0,3,1],[1,3,0,2],[0,1,2,3]],\"multiplication\":[[3,0,0,3],[0,2,1,3],[0,1,2,3],[3,3,3,3]],\"degree\":23}。請依 [degree,monomial,coefficient_count,distinct,boundary] 欄序回答。","answer":{"kind":"expression","value":"(23, [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2], 24, 是, R[x]_carrier_infinite_when_zero_ne_one)","explanation_zh":"Wave4第三層核對=(23, [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2], 24, 是, R[x]_carrier_infinite_when_zero_ne_one)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-infinite-carrier-witness","seed":202610292098,"parameters":{"n":4,"addition":[[3,2,1,0],[2,0,3,1],[1,3,0,2],[0,1,2,3]],"multiplication":[[3,0,0,3],[0,2,1,3],[0,1,2,3],[3,3,3,3]],"degree":23,"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-infinite-carrier-witness","semantic_template":"qf-mod29-poly-infinite-carrier-witness/{\"addition\":[[3,2,1,0],[2,0,3,1],[1,3,0,2],[0,1,2,3]],\"degree\":23,\"multiplication\":[[3,0,0,3],[0,2,1,3],[0,1,2,3],[3,3,3,3]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0099","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-infinite-carrier-witness；資料={\"n\":4,\"addition\":[[1,2,3,0],[2,3,0,1],[3,0,1,2],[0,1,2,3]],\"multiplication\":[[2,1,0,3],[1,3,1,3],[0,1,2,3],[3,3,3,3]],\"degree\":24}。請依 [degree,monomial,coefficient_count,distinct,boundary] 欄序回答。","answer":{"kind":"expression","value":"(24, [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2], 25, 是, R[x]_carrier_infinite_when_zero_ne_one)","explanation_zh":"Wave4第三層核對=(24, [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2], 25, 是, R[x]_carrier_infinite_when_zero_ne_one)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-infinite-carrier-witness","seed":202610292099,"parameters":{"n":4,"addition":[[1,2,3,0],[2,3,0,1],[3,0,1,2],[0,1,2,3]],"multiplication":[[2,1,0,3],[1,3,1,3],[0,1,2,3],[3,3,3,3]],"degree":24,"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-infinite-carrier-witness","semantic_template":"qf-mod29-poly-infinite-carrier-witness/{\"addition\":[[1,2,3,0],[2,3,0,1],[3,0,1,2],[0,1,2,3]],\"degree\":24,\"multiplication\":[[2,1,0,3],[1,3,1,3],[0,1,2,3],[3,3,3,3]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0100","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"],"title_zh":"有限交換環上的多項式環","stem_zh":"係數屬同一完整有限交換環且多項式有限支撐。操作=poly-infinite-carrier-witness；資料={\"n\":4,\"addition\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"multiplication\":[[0,0,0,0],[0,1,2,3],[0,2,0,2],[0,3,2,1]],\"degree\":25}。請依 [degree,monomial,coefficient_count,distinct,boundary] 欄序回答。","answer":{"kind":"expression","value":"(25, [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], 26, 是, R[x]_carrier_infinite_when_zero_ne_one)","explanation_zh":"Wave4第三層核對=(25, [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], 26, 是, R[x]_carrier_infinite_when_zero_ne_one)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限交換環上的多項式環"]},"variant":{"family_id":"qf-mod29-poly-infinite-carrier-witness","seed":202610292100,"parameters":{"n":4,"addition":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"multiplication":[[0,0,0,0],[0,1,2,3],[0,2,0,2],[0,3,2,1]],"degree":25,"domain":"Finite_support_polynomial_over_finite_commutative_ring","operation":"poly-infinite-carrier-witness","semantic_template":"qf-mod29-poly-infinite-carrier-witness/{\"addition\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"degree\":25,\"multiplication\":[[0,0,0,0],[0,1,2,3],[0,2,0,2],[0,3,2,1]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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"}}