{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r01-finite-integral-domain-0001","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-qualification；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]]}。請依 [n,zero,one,zero_divisor_pairs,domain,classification] 欄序回答。","answer":{"kind":"expression","value":"(4, 0, 1, [[2, 2]], 否, commutative_ring_with_zero_divisors)","explanation_zh":"Wave4第三層核對=(4, 0, 1, [[2, 2]], 否, commutative_ring_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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-qualification","seed":202610291001,"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]],"domain":"Complete_finite_integral_domain","operation":"integraldomain-qualification","semantic_template":"qf-mod29-integraldomain-qualification/{\"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]],\"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-r01-finite-integral-domain-0002","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-qualification；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,2,4,0,3],[2,4,3,1,0],[3,0,1,4,2],[4,3,0,2,1]],\"multiplication\":[[0,0,0,0,0],[0,1,2,3,4],[0,2,3,4,1],[0,3,4,1,2],[0,4,1,2,3]]}。請依 [n,zero,one,zero_divisor_pairs,domain,classification] 欄序回答。","answer":{"kind":"expression","value":"(5, 0, 1, [], 是, finite_integral_domain)","explanation_zh":"Wave4第三層核對=(5, 0, 1, [], 是, finite_integral_domain)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-qualification","seed":202610291002,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,2,4,0,3],[2,4,3,1,0],[3,0,1,4,2],[4,3,0,2,1]],"multiplication":[[0,0,0,0,0],[0,1,2,3,4],[0,2,3,4,1],[0,3,4,1,2],[0,4,1,2,3]],"domain":"Complete_finite_integral_domain","operation":"integraldomain-qualification","semantic_template":"qf-mod29-integraldomain-qualification/{\"addition\":[[0,1,2,3,4],[1,2,4,0,3],[2,4,3,1,0],[3,0,1,4,2],[4,3,0,2,1]],\"multiplication\":[[0,0,0,0,0],[0,1,2,3,4],[0,2,3,4,1],[0,3,4,1,2],[0,4,1,2,3]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0003","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-qualification；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],\"multiplication\":[[0,0,0,0,0],[0,1,2,3,4],[0,2,4,1,3],[0,3,1,4,2],[0,4,3,2,1]]}。請依 [n,zero,one,zero_divisor_pairs,domain,classification] 欄序回答。","answer":{"kind":"expression","value":"(5, 0, 1, [], 是, finite_integral_domain)","explanation_zh":"Wave4第三層核對=(5, 0, 1, [], 是, finite_integral_domain)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-qualification","seed":202610291003,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],"multiplication":[[0,0,0,0,0],[0,1,2,3,4],[0,2,4,1,3],[0,3,1,4,2],[0,4,3,2,1]],"domain":"Complete_finite_integral_domain","operation":"integraldomain-qualification","semantic_template":"qf-mod29-integraldomain-qualification/{\"addition\":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],\"multiplication\":[[0,0,0,0,0],[0,1,2,3,4],[0,2,4,1,3],[0,3,1,4,2],[0,4,3,2,1]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0004","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-qualification；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,3,0,4,2],[2,0,4,1,3],[3,4,1,2,0],[4,2,3,0,1]],\"multiplication\":[[0,0,0,0,0],[0,1,2,3,4],[0,2,1,4,3],[0,3,4,2,1],[0,4,3,1,2]]}。請依 [n,zero,one,zero_divisor_pairs,domain,classification] 欄序回答。","answer":{"kind":"expression","value":"(5, 0, 1, [], 是, finite_integral_domain)","explanation_zh":"Wave4第三層核對=(5, 0, 1, [], 是, finite_integral_domain)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-qualification","seed":202610291004,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,3,0,4,2],[2,0,4,1,3],[3,4,1,2,0],[4,2,3,0,1]],"multiplication":[[0,0,0,0,0],[0,1,2,3,4],[0,2,1,4,3],[0,3,4,2,1],[0,4,3,1,2]],"domain":"Complete_finite_integral_domain","operation":"integraldomain-qualification","semantic_template":"qf-mod29-integraldomain-qualification/{\"addition\":[[0,1,2,3,4],[1,3,0,4,2],[2,0,4,1,3],[3,4,1,2,0],[4,2,3,0,1]],\"multiplication\":[[0,0,0,0,0],[0,1,2,3,4],[0,2,1,4,3],[0,3,4,2,1],[0,4,3,1,2]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0005","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-qualification；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,4,3,0,2],[2,3,1,4,0],[3,0,4,2,1],[4,2,0,1,3]],\"multiplication\":[[0,0,0,0,0],[0,1,2,3,4],[0,2,3,4,1],[0,3,4,1,2],[0,4,1,2,3]]}。請依 [n,zero,one,zero_divisor_pairs,domain,classification] 欄序回答。","answer":{"kind":"expression","value":"(5, 0, 1, [], 是, finite_integral_domain)","explanation_zh":"Wave4第三層核對=(5, 0, 1, [], 是, finite_integral_domain)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-qualification","seed":202610291005,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,4,3,0,2],[2,3,1,4,0],[3,0,4,2,1],[4,2,0,1,3]],"multiplication":[[0,0,0,0,0],[0,1,2,3,4],[0,2,3,4,1],[0,3,4,1,2],[0,4,1,2,3]],"domain":"Complete_finite_integral_domain","operation":"integraldomain-qualification","semantic_template":"qf-mod29-integraldomain-qualification/{\"addition\":[[0,1,2,3,4],[1,4,3,0,2],[2,3,1,4,0],[3,0,4,2,1],[4,2,0,1,3]],\"multiplication\":[[0,0,0,0,0],[0,1,2,3,4],[0,2,3,4,1],[0,3,4,1,2],[0,4,1,2,3]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0006","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-qualification；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]]}。請依 [n,zero,one,zero_divisor_pairs,domain,classification] 欄序回答。","answer":{"kind":"expression","value":"(4, 0, 3, [[2, 2]], 否, commutative_ring_with_zero_divisors)","explanation_zh":"Wave4第三層核對=(4, 0, 3, [[2, 2]], 否, commutative_ring_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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-qualification","seed":202610291006,"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]],"domain":"Complete_finite_integral_domain","operation":"integraldomain-qualification","semantic_template":"qf-mod29-integraldomain-qualification/{\"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]],\"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-r01-finite-integral-domain-0007","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-qualification；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,4,3,0,2],[2,3,1,4,0],[3,0,4,2,1],[4,2,0,1,3]],\"multiplication\":[[0,0,0,0,0],[0,4,1,2,3],[0,1,2,3,4],[0,2,3,4,1],[0,3,4,1,2]]}。請依 [n,zero,one,zero_divisor_pairs,domain,classification] 欄序回答。","answer":{"kind":"expression","value":"(5, 0, 2, [], 是, finite_integral_domain)","explanation_zh":"Wave4第三層核對=(5, 0, 2, [], 是, finite_integral_domain)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-qualification","seed":202610291007,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,4,3,0,2],[2,3,1,4,0],[3,0,4,2,1],[4,2,0,1,3]],"multiplication":[[0,0,0,0,0],[0,4,1,2,3],[0,1,2,3,4],[0,2,3,4,1],[0,3,4,1,2]],"domain":"Complete_finite_integral_domain","operation":"integraldomain-qualification","semantic_template":"qf-mod29-integraldomain-qualification/{\"addition\":[[0,1,2,3,4],[1,4,3,0,2],[2,3,1,4,0],[3,0,4,2,1],[4,2,0,1,3]],\"multiplication\":[[0,0,0,0,0],[0,4,1,2,3],[0,1,2,3,4],[0,2,3,4,1],[0,3,4,1,2]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0008","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-qualification；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],\"multiplication\":[[0,0,0,0,0],[0,3,1,4,2],[0,1,2,3,4],[0,4,3,2,1],[0,2,4,1,3]]}。請依 [n,zero,one,zero_divisor_pairs,domain,classification] 欄序回答。","answer":{"kind":"expression","value":"(5, 0, 2, [], 是, finite_integral_domain)","explanation_zh":"Wave4第三層核對=(5, 0, 2, [], 是, finite_integral_domain)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-qualification","seed":202610291008,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],"multiplication":[[0,0,0,0,0],[0,3,1,4,2],[0,1,2,3,4],[0,4,3,2,1],[0,2,4,1,3]],"domain":"Complete_finite_integral_domain","operation":"integraldomain-qualification","semantic_template":"qf-mod29-integraldomain-qualification/{\"addition\":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],\"multiplication\":[[0,0,0,0,0],[0,3,1,4,2],[0,1,2,3,4],[0,4,3,2,1],[0,2,4,1,3]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0009","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-qualification；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,2,4,0,3],[2,4,3,1,0],[3,0,1,4,2],[4,3,0,2,1]],\"multiplication\":[[0,0,0,0,0],[0,4,1,2,3],[0,1,2,3,4],[0,2,3,4,1],[0,3,4,1,2]]}。請依 [n,zero,one,zero_divisor_pairs,domain,classification] 欄序回答。","answer":{"kind":"expression","value":"(5, 0, 2, [], 是, finite_integral_domain)","explanation_zh":"Wave4第三層核對=(5, 0, 2, [], 是, finite_integral_domain)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-qualification","seed":202610291009,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,2,4,0,3],[2,4,3,1,0],[3,0,1,4,2],[4,3,0,2,1]],"multiplication":[[0,0,0,0,0],[0,4,1,2,3],[0,1,2,3,4],[0,2,3,4,1],[0,3,4,1,2]],"domain":"Complete_finite_integral_domain","operation":"integraldomain-qualification","semantic_template":"qf-mod29-integraldomain-qualification/{\"addition\":[[0,1,2,3,4],[1,2,4,0,3],[2,4,3,1,0],[3,0,1,4,2],[4,3,0,2,1]],\"multiplication\":[[0,0,0,0,0],[0,4,1,2,3],[0,1,2,3,4],[0,2,3,4,1],[0,3,4,1,2]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0010","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-qualification；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,4,0,2,3],[2,0,3,4,1],[3,2,4,1,0],[4,3,1,0,2]],\"multiplication\":[[0,0,0,0,0],[0,2,1,4,3],[0,1,2,3,4],[0,4,3,1,2],[0,3,4,2,1]]}。請依 [n,zero,one,zero_divisor_pairs,domain,classification] 欄序回答。","answer":{"kind":"expression","value":"(5, 0, 2, [], 是, finite_integral_domain)","explanation_zh":"Wave4第三層核對=(5, 0, 2, [], 是, finite_integral_domain)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-qualification","seed":202610291010,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,4,0,2,3],[2,0,3,4,1],[3,2,4,1,0],[4,3,1,0,2]],"multiplication":[[0,0,0,0,0],[0,2,1,4,3],[0,1,2,3,4],[0,4,3,1,2],[0,3,4,2,1]],"domain":"Complete_finite_integral_domain","operation":"integraldomain-qualification","semantic_template":"qf-mod29-integraldomain-qualification/{\"addition\":[[0,1,2,3,4],[1,4,0,2,3],[2,0,3,4,1],[3,2,4,1,0],[4,3,1,0,2]],\"multiplication\":[[0,0,0,0,0],[0,2,1,4,3],[0,1,2,3,4],[0,4,3,1,2],[0,3,4,2,1]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0011","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-qualification；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]]}。請依 [n,zero,one,zero_divisor_pairs,domain,classification] 欄序回答。","answer":{"kind":"expression","value":"(4, 1, 3, [[0, 0]], 否, commutative_ring_with_zero_divisors)","explanation_zh":"Wave4第三層核對=(4, 1, 3, [[0, 0]], 否, commutative_ring_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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-qualification","seed":202610291011,"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]],"domain":"Complete_finite_integral_domain","operation":"integraldomain-qualification","semantic_template":"qf-mod29-integraldomain-qualification/{\"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]],\"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-r01-finite-integral-domain-0012","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-qualification；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,3,0,4,2],[2,0,4,1,3],[3,4,1,2,0],[4,2,3,0,1]],\"multiplication\":[[0,0,0,0,0],[0,2,1,4,3],[0,1,2,3,4],[0,4,3,1,2],[0,3,4,2,1]]}。請依 [n,zero,one,zero_divisor_pairs,domain,classification] 欄序回答。","answer":{"kind":"expression","value":"(5, 0, 2, [], 是, finite_integral_domain)","explanation_zh":"Wave4第三層核對=(5, 0, 2, [], 是, finite_integral_domain)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-qualification","seed":202610291012,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,3,0,4,2],[2,0,4,1,3],[3,4,1,2,0],[4,2,3,0,1]],"multiplication":[[0,0,0,0,0],[0,2,1,4,3],[0,1,2,3,4],[0,4,3,1,2],[0,3,4,2,1]],"domain":"Complete_finite_integral_domain","operation":"integraldomain-qualification","semantic_template":"qf-mod29-integraldomain-qualification/{\"addition\":[[0,1,2,3,4],[1,3,0,4,2],[2,0,4,1,3],[3,4,1,2,0],[4,2,3,0,1]],\"multiplication\":[[0,0,0,0,0],[0,2,1,4,3],[0,1,2,3,4],[0,4,3,1,2],[0,3,4,2,1]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0013","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-qualification；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,4,0,2,3],[2,0,3,4,1],[3,2,4,1,0],[4,3,1,0,2]],\"multiplication\":[[0,0,0,0,0],[0,4,3,1,2],[0,3,4,2,1],[0,1,2,3,4],[0,2,1,4,3]]}。請依 [n,zero,one,zero_divisor_pairs,domain,classification] 欄序回答。","answer":{"kind":"expression","value":"(5, 0, 3, [], 是, finite_integral_domain)","explanation_zh":"Wave4第三層核對=(5, 0, 3, [], 是, finite_integral_domain)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-qualification","seed":202610291013,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,4,0,2,3],[2,0,3,4,1],[3,2,4,1,0],[4,3,1,0,2]],"multiplication":[[0,0,0,0,0],[0,4,3,1,2],[0,3,4,2,1],[0,1,2,3,4],[0,2,1,4,3]],"domain":"Complete_finite_integral_domain","operation":"integraldomain-qualification","semantic_template":"qf-mod29-integraldomain-qualification/{\"addition\":[[0,1,2,3,4],[1,4,0,2,3],[2,0,3,4,1],[3,2,4,1,0],[4,3,1,0,2]],\"multiplication\":[[0,0,0,0,0],[0,4,3,1,2],[0,3,4,2,1],[0,1,2,3,4],[0,2,1,4,3]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0014","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-qualification；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"multiplication\":[[0,0,0,0,0],[0,2,4,1,3],[0,4,3,2,1],[0,1,2,3,4],[0,3,1,4,2]]}。請依 [n,zero,one,zero_divisor_pairs,domain,classification] 欄序回答。","answer":{"kind":"expression","value":"(5, 0, 3, [], 是, finite_integral_domain)","explanation_zh":"Wave4第三層核對=(5, 0, 3, [], 是, finite_integral_domain)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-qualification","seed":202610291014,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"multiplication":[[0,0,0,0,0],[0,2,4,1,3],[0,4,3,2,1],[0,1,2,3,4],[0,3,1,4,2]],"domain":"Complete_finite_integral_domain","operation":"integraldomain-qualification","semantic_template":"qf-mod29-integraldomain-qualification/{\"addition\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"multiplication\":[[0,0,0,0,0],[0,2,4,1,3],[0,4,3,2,1],[0,1,2,3,4],[0,3,1,4,2]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0015","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-qualification；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,3,0,4,2],[2,0,4,1,3],[3,4,1,2,0],[4,2,3,0,1]],\"multiplication\":[[0,0,0,0,0],[0,4,3,1,2],[0,3,4,2,1],[0,1,2,3,4],[0,2,1,4,3]]}。請依 [n,zero,one,zero_divisor_pairs,domain,classification] 欄序回答。","answer":{"kind":"expression","value":"(5, 0, 3, [], 是, finite_integral_domain)","explanation_zh":"Wave4第三層核對=(5, 0, 3, [], 是, finite_integral_domain)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-qualification","seed":202610291015,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,3,0,4,2],[2,0,4,1,3],[3,4,1,2,0],[4,2,3,0,1]],"multiplication":[[0,0,0,0,0],[0,4,3,1,2],[0,3,4,2,1],[0,1,2,3,4],[0,2,1,4,3]],"domain":"Complete_finite_integral_domain","operation":"integraldomain-qualification","semantic_template":"qf-mod29-integraldomain-qualification/{\"addition\":[[0,1,2,3,4],[1,3,0,4,2],[2,0,4,1,3],[3,4,1,2,0],[4,2,3,0,1]],\"multiplication\":[[0,0,0,0,0],[0,4,3,1,2],[0,3,4,2,1],[0,1,2,3,4],[0,2,1,4,3]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0016","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-qualification；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]]}。請依 [n,zero,one,zero_divisor_pairs,domain,classification] 欄序回答。","answer":{"kind":"expression","value":"(4, 2, 1, [[3, 3]], 否, commutative_ring_with_zero_divisors)","explanation_zh":"Wave4第三層核對=(4, 2, 1, [[3, 3]], 否, commutative_ring_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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-qualification","seed":202610291016,"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]],"domain":"Complete_finite_integral_domain","operation":"integraldomain-qualification","semantic_template":"qf-mod29-integraldomain-qualification/{\"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]],\"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-r01-finite-integral-domain-0017","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-qualification；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],\"multiplication\":[[0,0,0,0,0],[0,2,4,1,3],[0,4,3,2,1],[0,1,2,3,4],[0,3,1,4,2]]}。請依 [n,zero,one,zero_divisor_pairs,domain,classification] 欄序回答。","answer":{"kind":"expression","value":"(5, 0, 3, [], 是, finite_integral_domain)","explanation_zh":"Wave4第三層核對=(5, 0, 3, [], 是, finite_integral_domain)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-qualification","seed":202610291017,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],"multiplication":[[0,0,0,0,0],[0,2,4,1,3],[0,4,3,2,1],[0,1,2,3,4],[0,3,1,4,2]],"domain":"Complete_finite_integral_domain","operation":"integraldomain-qualification","semantic_template":"qf-mod29-integraldomain-qualification/{\"addition\":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],\"multiplication\":[[0,0,0,0,0],[0,2,4,1,3],[0,4,3,2,1],[0,1,2,3,4],[0,3,1,4,2]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0018","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-qualification；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,2,4,0,3],[2,4,3,1,0],[3,0,1,4,2],[4,3,0,2,1]],\"multiplication\":[[0,0,0,0,0],[0,3,4,1,2],[0,4,1,2,3],[0,1,2,3,4],[0,2,3,4,1]]}。請依 [n,zero,one,zero_divisor_pairs,domain,classification] 欄序回答。","answer":{"kind":"expression","value":"(5, 0, 3, [], 是, finite_integral_domain)","explanation_zh":"Wave4第三層核對=(5, 0, 3, [], 是, finite_integral_domain)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-qualification","seed":202610291018,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,2,4,0,3],[2,4,3,1,0],[3,0,1,4,2],[4,3,0,2,1]],"multiplication":[[0,0,0,0,0],[0,3,4,1,2],[0,4,1,2,3],[0,1,2,3,4],[0,2,3,4,1]],"domain":"Complete_finite_integral_domain","operation":"integraldomain-qualification","semantic_template":"qf-mod29-integraldomain-qualification/{\"addition\":[[0,1,2,3,4],[1,2,4,0,3],[2,4,3,1,0],[3,0,1,4,2],[4,3,0,2,1]],\"multiplication\":[[0,0,0,0,0],[0,3,4,1,2],[0,4,1,2,3],[0,1,2,3,4],[0,2,3,4,1]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0019","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-qualification；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,3,0,4,2],[2,0,4,1,3],[3,4,1,2,0],[4,2,3,0,1]],\"multiplication\":[[0,0,0,0,0],[0,3,4,2,1],[0,4,3,1,2],[0,2,1,4,3],[0,1,2,3,4]]}。請依 [n,zero,one,zero_divisor_pairs,domain,classification] 欄序回答。","answer":{"kind":"expression","value":"(5, 0, 4, [], 是, finite_integral_domain)","explanation_zh":"Wave4第三層核對=(5, 0, 4, [], 是, finite_integral_domain)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-qualification","seed":202610291019,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,3,0,4,2],[2,0,4,1,3],[3,4,1,2,0],[4,2,3,0,1]],"multiplication":[[0,0,0,0,0],[0,3,4,2,1],[0,4,3,1,2],[0,2,1,4,3],[0,1,2,3,4]],"domain":"Complete_finite_integral_domain","operation":"integraldomain-qualification","semantic_template":"qf-mod29-integraldomain-qualification/{\"addition\":[[0,1,2,3,4],[1,3,0,4,2],[2,0,4,1,3],[3,4,1,2,0],[4,2,3,0,1]],\"multiplication\":[[0,0,0,0,0],[0,3,4,2,1],[0,4,3,1,2],[0,2,1,4,3],[0,1,2,3,4]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0020","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-qualification；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,2,4,0,3],[2,4,3,1,0],[3,0,1,4,2],[4,3,0,2,1]],\"multiplication\":[[0,0,0,0,0],[0,2,3,4,1],[0,3,4,1,2],[0,4,1,2,3],[0,1,2,3,4]]}。請依 [n,zero,one,zero_divisor_pairs,domain,classification] 欄序回答。","answer":{"kind":"expression","value":"(5, 0, 4, [], 是, finite_integral_domain)","explanation_zh":"Wave4第三層核對=(5, 0, 4, [], 是, finite_integral_domain)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-qualification","seed":202610291020,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,2,4,0,3],[2,4,3,1,0],[3,0,1,4,2],[4,3,0,2,1]],"multiplication":[[0,0,0,0,0],[0,2,3,4,1],[0,3,4,1,2],[0,4,1,2,3],[0,1,2,3,4]],"domain":"Complete_finite_integral_domain","operation":"integraldomain-qualification","semantic_template":"qf-mod29-integraldomain-qualification/{\"addition\":[[0,1,2,3,4],[1,2,4,0,3],[2,4,3,1,0],[3,0,1,4,2],[4,3,0,2,1]],\"multiplication\":[[0,0,0,0,0],[0,2,3,4,1],[0,3,4,1,2],[0,4,1,2,3],[0,1,2,3,4]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0021","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-qualification；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]]}。請依 [n,zero,one,zero_divisor_pairs,domain,classification] 欄序回答。","answer":{"kind":"expression","value":"(4, 3, 1, [[0, 0]], 否, commutative_ring_with_zero_divisors)","explanation_zh":"Wave4第三層核對=(4, 3, 1, [[0, 0]], 否, commutative_ring_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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-qualification","seed":202610291021,"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]],"domain":"Complete_finite_integral_domain","operation":"integraldomain-qualification","semantic_template":"qf-mod29-integraldomain-qualification/{\"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]],\"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-r01-finite-integral-domain-0022","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-qualification；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],\"multiplication\":[[0,0,0,0,0],[0,4,3,2,1],[0,3,1,4,2],[0,2,4,1,3],[0,1,2,3,4]]}。請依 [n,zero,one,zero_divisor_pairs,domain,classification] 欄序回答。","answer":{"kind":"expression","value":"(5, 0, 4, [], 是, finite_integral_domain)","explanation_zh":"Wave4第三層核對=(5, 0, 4, [], 是, finite_integral_domain)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-qualification","seed":202610291022,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],"multiplication":[[0,0,0,0,0],[0,4,3,2,1],[0,3,1,4,2],[0,2,4,1,3],[0,1,2,3,4]],"domain":"Complete_finite_integral_domain","operation":"integraldomain-qualification","semantic_template":"qf-mod29-integraldomain-qualification/{\"addition\":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],\"multiplication\":[[0,0,0,0,0],[0,4,3,2,1],[0,3,1,4,2],[0,2,4,1,3],[0,1,2,3,4]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0023","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-qualification；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,4,3,0,2],[2,3,1,4,0],[3,0,4,2,1],[4,2,0,1,3]],\"multiplication\":[[0,0,0,0,0],[0,2,3,4,1],[0,3,4,1,2],[0,4,1,2,3],[0,1,2,3,4]]}。請依 [n,zero,one,zero_divisor_pairs,domain,classification] 欄序回答。","answer":{"kind":"expression","value":"(5, 0, 4, [], 是, finite_integral_domain)","explanation_zh":"Wave4第三層核對=(5, 0, 4, [], 是, finite_integral_domain)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-qualification","seed":202610291023,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,4,3,0,2],[2,3,1,4,0],[3,0,4,2,1],[4,2,0,1,3]],"multiplication":[[0,0,0,0,0],[0,2,3,4,1],[0,3,4,1,2],[0,4,1,2,3],[0,1,2,3,4]],"domain":"Complete_finite_integral_domain","operation":"integraldomain-qualification","semantic_template":"qf-mod29-integraldomain-qualification/{\"addition\":[[0,1,2,3,4],[1,4,3,0,2],[2,3,1,4,0],[3,0,4,2,1],[4,2,0,1,3]],\"multiplication\":[[0,0,0,0,0],[0,2,3,4,1],[0,3,4,1,2],[0,4,1,2,3],[0,1,2,3,4]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0024","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-qualification；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"multiplication\":[[0,0,0,0,0],[0,4,3,2,1],[0,3,1,4,2],[0,2,4,1,3],[0,1,2,3,4]]}。請依 [n,zero,one,zero_divisor_pairs,domain,classification] 欄序回答。","answer":{"kind":"expression","value":"(5, 0, 4, [], 是, finite_integral_domain)","explanation_zh":"Wave4第三層核對=(5, 0, 4, [], 是, finite_integral_domain)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-qualification","seed":202610291024,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"multiplication":[[0,0,0,0,0],[0,4,3,2,1],[0,3,1,4,2],[0,2,4,1,3],[0,1,2,3,4]],"domain":"Complete_finite_integral_domain","operation":"integraldomain-qualification","semantic_template":"qf-mod29-integraldomain-qualification/{\"addition\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"multiplication\":[[0,0,0,0,0],[0,4,3,2,1],[0,3,1,4,2],[0,2,4,1,3],[0,1,2,3,4]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0025","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-qualification；資料={\"n\":5,\"addition\":[[2,0,3,4,1],[0,1,2,3,4],[3,2,4,1,0],[4,3,1,0,2],[1,4,0,2,3]],\"multiplication\":[[0,1,2,3,4],[1,1,1,1,1],[2,1,4,0,3],[3,1,0,4,2],[4,1,3,2,0]]}。請依 [n,zero,one,zero_divisor_pairs,domain,classification] 欄序回答。","answer":{"kind":"expression","value":"(5, 1, 0, [], 是, finite_integral_domain)","explanation_zh":"Wave4第三層核對=(5, 1, 0, [], 是, finite_integral_domain)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-qualification","seed":202610291025,"parameters":{"n":5,"addition":[[2,0,3,4,1],[0,1,2,3,4],[3,2,4,1,0],[4,3,1,0,2],[1,4,0,2,3]],"multiplication":[[0,1,2,3,4],[1,1,1,1,1],[2,1,4,0,3],[3,1,0,4,2],[4,1,3,2,0]],"domain":"Complete_finite_integral_domain","operation":"integraldomain-qualification","semantic_template":"qf-mod29-integraldomain-qualification/{\"addition\":[[2,0,3,4,1],[0,1,2,3,4],[3,2,4,1,0],[4,3,1,0,2],[1,4,0,2,3]],\"multiplication\":[[0,1,2,3,4],[1,1,1,1,1],[2,1,4,0,3],[3,1,0,4,2],[4,1,3,2,0]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0026","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-zero-divisor-audit；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"multiplication\":[[0,0,0,0,0],[0,1,2,3,4],[0,2,4,1,3],[0,3,1,4,2],[0,4,3,2,1]]}。請依 [zero_divisor_pairs,no_nonzero_zero_divisors] 欄序回答。","answer":{"kind":"expression","value":"([], 是)","explanation_zh":"Wave4第三層核對=([], 是)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-zero-divisor-audit","seed":202610291026,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"multiplication":[[0,0,0,0,0],[0,1,2,3,4],[0,2,4,1,3],[0,3,1,4,2],[0,4,3,2,1]],"domain":"Complete_finite_integral_domain","operation":"integraldomain-zero-divisor-audit","semantic_template":"qf-mod29-integraldomain-zero-divisor-audit/{\"addition\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"multiplication\":[[0,0,0,0,0],[0,1,2,3,4],[0,2,4,1,3],[0,3,1,4,2],[0,4,3,2,1]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0027","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-zero-divisor-audit；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,2,4,0,3],[2,4,3,1,0],[3,0,1,4,2],[4,3,0,2,1]],\"multiplication\":[[0,0,0,0,0],[0,1,2,3,4],[0,2,3,4,1],[0,3,4,1,2],[0,4,1,2,3]]}。請依 [zero_divisor_pairs,no_nonzero_zero_divisors] 欄序回答。","answer":{"kind":"expression","value":"([], 是)","explanation_zh":"Wave4第三層核對=([], 是)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-zero-divisor-audit","seed":202610291027,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,2,4,0,3],[2,4,3,1,0],[3,0,1,4,2],[4,3,0,2,1]],"multiplication":[[0,0,0,0,0],[0,1,2,3,4],[0,2,3,4,1],[0,3,4,1,2],[0,4,1,2,3]],"domain":"Complete_finite_integral_domain","operation":"integraldomain-zero-divisor-audit","semantic_template":"qf-mod29-integraldomain-zero-divisor-audit/{\"addition\":[[0,1,2,3,4],[1,2,4,0,3],[2,4,3,1,0],[3,0,1,4,2],[4,3,0,2,1]],\"multiplication\":[[0,0,0,0,0],[0,1,2,3,4],[0,2,3,4,1],[0,3,4,1,2],[0,4,1,2,3]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0028","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-zero-divisor-audit；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],\"multiplication\":[[0,0,0,0,0],[0,1,2,3,4],[0,2,4,1,3],[0,3,1,4,2],[0,4,3,2,1]]}。請依 [zero_divisor_pairs,no_nonzero_zero_divisors] 欄序回答。","answer":{"kind":"expression","value":"([], 是)","explanation_zh":"Wave4第三層核對=([], 是)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-zero-divisor-audit","seed":202610291028,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],"multiplication":[[0,0,0,0,0],[0,1,2,3,4],[0,2,4,1,3],[0,3,1,4,2],[0,4,3,2,1]],"domain":"Complete_finite_integral_domain","operation":"integraldomain-zero-divisor-audit","semantic_template":"qf-mod29-integraldomain-zero-divisor-audit/{\"addition\":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],\"multiplication\":[[0,0,0,0,0],[0,1,2,3,4],[0,2,4,1,3],[0,3,1,4,2],[0,4,3,2,1]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0029","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-zero-divisor-audit；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,3,0,4,2],[2,0,4,1,3],[3,4,1,2,0],[4,2,3,0,1]],\"multiplication\":[[0,0,0,0,0],[0,1,2,3,4],[0,2,1,4,3],[0,3,4,2,1],[0,4,3,1,2]]}。請依 [zero_divisor_pairs,no_nonzero_zero_divisors] 欄序回答。","answer":{"kind":"expression","value":"([], 是)","explanation_zh":"Wave4第三層核對=([], 是)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-zero-divisor-audit","seed":202610291029,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,3,0,4,2],[2,0,4,1,3],[3,4,1,2,0],[4,2,3,0,1]],"multiplication":[[0,0,0,0,0],[0,1,2,3,4],[0,2,1,4,3],[0,3,4,2,1],[0,4,3,1,2]],"domain":"Complete_finite_integral_domain","operation":"integraldomain-zero-divisor-audit","semantic_template":"qf-mod29-integraldomain-zero-divisor-audit/{\"addition\":[[0,1,2,3,4],[1,3,0,4,2],[2,0,4,1,3],[3,4,1,2,0],[4,2,3,0,1]],\"multiplication\":[[0,0,0,0,0],[0,1,2,3,4],[0,2,1,4,3],[0,3,4,2,1],[0,4,3,1,2]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0030","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-zero-divisor-audit；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,4,3,0,2],[2,3,1,4,0],[3,0,4,2,1],[4,2,0,1,3]],\"multiplication\":[[0,0,0,0,0],[0,1,2,3,4],[0,2,3,4,1],[0,3,4,1,2],[0,4,1,2,3]]}。請依 [zero_divisor_pairs,no_nonzero_zero_divisors] 欄序回答。","answer":{"kind":"expression","value":"([], 是)","explanation_zh":"Wave4第三層核對=([], 是)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-zero-divisor-audit","seed":202610291030,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,4,3,0,2],[2,3,1,4,0],[3,0,4,2,1],[4,2,0,1,3]],"multiplication":[[0,0,0,0,0],[0,1,2,3,4],[0,2,3,4,1],[0,3,4,1,2],[0,4,1,2,3]],"domain":"Complete_finite_integral_domain","operation":"integraldomain-zero-divisor-audit","semantic_template":"qf-mod29-integraldomain-zero-divisor-audit/{\"addition\":[[0,1,2,3,4],[1,4,3,0,2],[2,3,1,4,0],[3,0,4,2,1],[4,2,0,1,3]],\"multiplication\":[[0,0,0,0,0],[0,1,2,3,4],[0,2,3,4,1],[0,3,4,1,2],[0,4,1,2,3]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0031","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-zero-divisor-audit；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,4,0,2,3],[2,0,3,4,1],[3,2,4,1,0],[4,3,1,0,2]],\"multiplication\":[[0,0,0,0,0],[0,1,2,3,4],[0,2,1,4,3],[0,3,4,2,1],[0,4,3,1,2]]}。請依 [zero_divisor_pairs,no_nonzero_zero_divisors] 欄序回答。","answer":{"kind":"expression","value":"([], 是)","explanation_zh":"Wave4第三層核對=([], 是)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-zero-divisor-audit","seed":202610291031,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,4,0,2,3],[2,0,3,4,1],[3,2,4,1,0],[4,3,1,0,2]],"multiplication":[[0,0,0,0,0],[0,1,2,3,4],[0,2,1,4,3],[0,3,4,2,1],[0,4,3,1,2]],"domain":"Complete_finite_integral_domain","operation":"integraldomain-zero-divisor-audit","semantic_template":"qf-mod29-integraldomain-zero-divisor-audit/{\"addition\":[[0,1,2,3,4],[1,4,0,2,3],[2,0,3,4,1],[3,2,4,1,0],[4,3,1,0,2]],\"multiplication\":[[0,0,0,0,0],[0,1,2,3,4],[0,2,1,4,3],[0,3,4,2,1],[0,4,3,1,2]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0032","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-zero-divisor-audit；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,4,3,0,2],[2,3,1,4,0],[3,0,4,2,1],[4,2,0,1,3]],\"multiplication\":[[0,0,0,0,0],[0,4,1,2,3],[0,1,2,3,4],[0,2,3,4,1],[0,3,4,1,2]]}。請依 [zero_divisor_pairs,no_nonzero_zero_divisors] 欄序回答。","answer":{"kind":"expression","value":"([], 是)","explanation_zh":"Wave4第三層核對=([], 是)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-zero-divisor-audit","seed":202610291032,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,4,3,0,2],[2,3,1,4,0],[3,0,4,2,1],[4,2,0,1,3]],"multiplication":[[0,0,0,0,0],[0,4,1,2,3],[0,1,2,3,4],[0,2,3,4,1],[0,3,4,1,2]],"domain":"Complete_finite_integral_domain","operation":"integraldomain-zero-divisor-audit","semantic_template":"qf-mod29-integraldomain-zero-divisor-audit/{\"addition\":[[0,1,2,3,4],[1,4,3,0,2],[2,3,1,4,0],[3,0,4,2,1],[4,2,0,1,3]],\"multiplication\":[[0,0,0,0,0],[0,4,1,2,3],[0,1,2,3,4],[0,2,3,4,1],[0,3,4,1,2]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0033","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-zero-divisor-audit；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],\"multiplication\":[[0,0,0,0,0],[0,3,1,4,2],[0,1,2,3,4],[0,4,3,2,1],[0,2,4,1,3]]}。請依 [zero_divisor_pairs,no_nonzero_zero_divisors] 欄序回答。","answer":{"kind":"expression","value":"([], 是)","explanation_zh":"Wave4第三層核對=([], 是)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-zero-divisor-audit","seed":202610291033,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],"multiplication":[[0,0,0,0,0],[0,3,1,4,2],[0,1,2,3,4],[0,4,3,2,1],[0,2,4,1,3]],"domain":"Complete_finite_integral_domain","operation":"integraldomain-zero-divisor-audit","semantic_template":"qf-mod29-integraldomain-zero-divisor-audit/{\"addition\":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],\"multiplication\":[[0,0,0,0,0],[0,3,1,4,2],[0,1,2,3,4],[0,4,3,2,1],[0,2,4,1,3]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0034","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-zero-divisor-audit；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,2,4,0,3],[2,4,3,1,0],[3,0,1,4,2],[4,3,0,2,1]],\"multiplication\":[[0,0,0,0,0],[0,4,1,2,3],[0,1,2,3,4],[0,2,3,4,1],[0,3,4,1,2]]}。請依 [zero_divisor_pairs,no_nonzero_zero_divisors] 欄序回答。","answer":{"kind":"expression","value":"([], 是)","explanation_zh":"Wave4第三層核對=([], 是)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-zero-divisor-audit","seed":202610291034,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,2,4,0,3],[2,4,3,1,0],[3,0,1,4,2],[4,3,0,2,1]],"multiplication":[[0,0,0,0,0],[0,4,1,2,3],[0,1,2,3,4],[0,2,3,4,1],[0,3,4,1,2]],"domain":"Complete_finite_integral_domain","operation":"integraldomain-zero-divisor-audit","semantic_template":"qf-mod29-integraldomain-zero-divisor-audit/{\"addition\":[[0,1,2,3,4],[1,2,4,0,3],[2,4,3,1,0],[3,0,1,4,2],[4,3,0,2,1]],\"multiplication\":[[0,0,0,0,0],[0,4,1,2,3],[0,1,2,3,4],[0,2,3,4,1],[0,3,4,1,2]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0035","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-zero-divisor-audit；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,4,0,2,3],[2,0,3,4,1],[3,2,4,1,0],[4,3,1,0,2]],\"multiplication\":[[0,0,0,0,0],[0,2,1,4,3],[0,1,2,3,4],[0,4,3,1,2],[0,3,4,2,1]]}。請依 [zero_divisor_pairs,no_nonzero_zero_divisors] 欄序回答。","answer":{"kind":"expression","value":"([], 是)","explanation_zh":"Wave4第三層核對=([], 是)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-zero-divisor-audit","seed":202610291035,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,4,0,2,3],[2,0,3,4,1],[3,2,4,1,0],[4,3,1,0,2]],"multiplication":[[0,0,0,0,0],[0,2,1,4,3],[0,1,2,3,4],[0,4,3,1,2],[0,3,4,2,1]],"domain":"Complete_finite_integral_domain","operation":"integraldomain-zero-divisor-audit","semantic_template":"qf-mod29-integraldomain-zero-divisor-audit/{\"addition\":[[0,1,2,3,4],[1,4,0,2,3],[2,0,3,4,1],[3,2,4,1,0],[4,3,1,0,2]],\"multiplication\":[[0,0,0,0,0],[0,2,1,4,3],[0,1,2,3,4],[0,4,3,1,2],[0,3,4,2,1]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0036","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-zero-divisor-audit；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"multiplication\":[[0,0,0,0,0],[0,3,1,4,2],[0,1,2,3,4],[0,4,3,2,1],[0,2,4,1,3]]}。請依 [zero_divisor_pairs,no_nonzero_zero_divisors] 欄序回答。","answer":{"kind":"expression","value":"([], 是)","explanation_zh":"Wave4第三層核對=([], 是)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-zero-divisor-audit","seed":202610291036,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"multiplication":[[0,0,0,0,0],[0,3,1,4,2],[0,1,2,3,4],[0,4,3,2,1],[0,2,4,1,3]],"domain":"Complete_finite_integral_domain","operation":"integraldomain-zero-divisor-audit","semantic_template":"qf-mod29-integraldomain-zero-divisor-audit/{\"addition\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"multiplication\":[[0,0,0,0,0],[0,3,1,4,2],[0,1,2,3,4],[0,4,3,2,1],[0,2,4,1,3]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0037","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-zero-divisor-audit；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,3,0,4,2],[2,0,4,1,3],[3,4,1,2,0],[4,2,3,0,1]],\"multiplication\":[[0,0,0,0,0],[0,2,1,4,3],[0,1,2,3,4],[0,4,3,1,2],[0,3,4,2,1]]}。請依 [zero_divisor_pairs,no_nonzero_zero_divisors] 欄序回答。","answer":{"kind":"expression","value":"([], 是)","explanation_zh":"Wave4第三層核對=([], 是)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-zero-divisor-audit","seed":202610291037,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,3,0,4,2],[2,0,4,1,3],[3,4,1,2,0],[4,2,3,0,1]],"multiplication":[[0,0,0,0,0],[0,2,1,4,3],[0,1,2,3,4],[0,4,3,1,2],[0,3,4,2,1]],"domain":"Complete_finite_integral_domain","operation":"integraldomain-zero-divisor-audit","semantic_template":"qf-mod29-integraldomain-zero-divisor-audit/{\"addition\":[[0,1,2,3,4],[1,3,0,4,2],[2,0,4,1,3],[3,4,1,2,0],[4,2,3,0,1]],\"multiplication\":[[0,0,0,0,0],[0,2,1,4,3],[0,1,2,3,4],[0,4,3,1,2],[0,3,4,2,1]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0038","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-zero-divisor-audit；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,4,0,2,3],[2,0,3,4,1],[3,2,4,1,0],[4,3,1,0,2]],\"multiplication\":[[0,0,0,0,0],[0,4,3,1,2],[0,3,4,2,1],[0,1,2,3,4],[0,2,1,4,3]]}。請依 [zero_divisor_pairs,no_nonzero_zero_divisors] 欄序回答。","answer":{"kind":"expression","value":"([], 是)","explanation_zh":"Wave4第三層核對=([], 是)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-zero-divisor-audit","seed":202610291038,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,4,0,2,3],[2,0,3,4,1],[3,2,4,1,0],[4,3,1,0,2]],"multiplication":[[0,0,0,0,0],[0,4,3,1,2],[0,3,4,2,1],[0,1,2,3,4],[0,2,1,4,3]],"domain":"Complete_finite_integral_domain","operation":"integraldomain-zero-divisor-audit","semantic_template":"qf-mod29-integraldomain-zero-divisor-audit/{\"addition\":[[0,1,2,3,4],[1,4,0,2,3],[2,0,3,4,1],[3,2,4,1,0],[4,3,1,0,2]],\"multiplication\":[[0,0,0,0,0],[0,4,3,1,2],[0,3,4,2,1],[0,1,2,3,4],[0,2,1,4,3]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0039","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-zero-divisor-audit；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"multiplication\":[[0,0,0,0,0],[0,2,4,1,3],[0,4,3,2,1],[0,1,2,3,4],[0,3,1,4,2]]}。請依 [zero_divisor_pairs,no_nonzero_zero_divisors] 欄序回答。","answer":{"kind":"expression","value":"([], 是)","explanation_zh":"Wave4第三層核對=([], 是)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-zero-divisor-audit","seed":202610291039,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"multiplication":[[0,0,0,0,0],[0,2,4,1,3],[0,4,3,2,1],[0,1,2,3,4],[0,3,1,4,2]],"domain":"Complete_finite_integral_domain","operation":"integraldomain-zero-divisor-audit","semantic_template":"qf-mod29-integraldomain-zero-divisor-audit/{\"addition\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"multiplication\":[[0,0,0,0,0],[0,2,4,1,3],[0,4,3,2,1],[0,1,2,3,4],[0,3,1,4,2]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0040","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-zero-divisor-audit；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,3,0,4,2],[2,0,4,1,3],[3,4,1,2,0],[4,2,3,0,1]],\"multiplication\":[[0,0,0,0,0],[0,4,3,1,2],[0,3,4,2,1],[0,1,2,3,4],[0,2,1,4,3]]}。請依 [zero_divisor_pairs,no_nonzero_zero_divisors] 欄序回答。","answer":{"kind":"expression","value":"([], 是)","explanation_zh":"Wave4第三層核對=([], 是)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-zero-divisor-audit","seed":202610291040,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,3,0,4,2],[2,0,4,1,3],[3,4,1,2,0],[4,2,3,0,1]],"multiplication":[[0,0,0,0,0],[0,4,3,1,2],[0,3,4,2,1],[0,1,2,3,4],[0,2,1,4,3]],"domain":"Complete_finite_integral_domain","operation":"integraldomain-zero-divisor-audit","semantic_template":"qf-mod29-integraldomain-zero-divisor-audit/{\"addition\":[[0,1,2,3,4],[1,3,0,4,2],[2,0,4,1,3],[3,4,1,2,0],[4,2,3,0,1]],\"multiplication\":[[0,0,0,0,0],[0,4,3,1,2],[0,3,4,2,1],[0,1,2,3,4],[0,2,1,4,3]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0041","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-zero-divisor-audit；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,4,3,0,2],[2,3,1,4,0],[3,0,4,2,1],[4,2,0,1,3]],\"multiplication\":[[0,0,0,0,0],[0,3,4,1,2],[0,4,1,2,3],[0,1,2,3,4],[0,2,3,4,1]]}。請依 [zero_divisor_pairs,no_nonzero_zero_divisors] 欄序回答。","answer":{"kind":"expression","value":"([], 是)","explanation_zh":"Wave4第三層核對=([], 是)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-zero-divisor-audit","seed":202610291041,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,4,3,0,2],[2,3,1,4,0],[3,0,4,2,1],[4,2,0,1,3]],"multiplication":[[0,0,0,0,0],[0,3,4,1,2],[0,4,1,2,3],[0,1,2,3,4],[0,2,3,4,1]],"domain":"Complete_finite_integral_domain","operation":"integraldomain-zero-divisor-audit","semantic_template":"qf-mod29-integraldomain-zero-divisor-audit/{\"addition\":[[0,1,2,3,4],[1,4,3,0,2],[2,3,1,4,0],[3,0,4,2,1],[4,2,0,1,3]],\"multiplication\":[[0,0,0,0,0],[0,3,4,1,2],[0,4,1,2,3],[0,1,2,3,4],[0,2,3,4,1]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0042","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-zero-divisor-audit；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],\"multiplication\":[[0,0,0,0,0],[0,2,4,1,3],[0,4,3,2,1],[0,1,2,3,4],[0,3,1,4,2]]}。請依 [zero_divisor_pairs,no_nonzero_zero_divisors] 欄序回答。","answer":{"kind":"expression","value":"([], 是)","explanation_zh":"Wave4第三層核對=([], 是)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-zero-divisor-audit","seed":202610291042,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],"multiplication":[[0,0,0,0,0],[0,2,4,1,3],[0,4,3,2,1],[0,1,2,3,4],[0,3,1,4,2]],"domain":"Complete_finite_integral_domain","operation":"integraldomain-zero-divisor-audit","semantic_template":"qf-mod29-integraldomain-zero-divisor-audit/{\"addition\":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],\"multiplication\":[[0,0,0,0,0],[0,2,4,1,3],[0,4,3,2,1],[0,1,2,3,4],[0,3,1,4,2]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0043","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-zero-divisor-audit；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,2,4,0,3],[2,4,3,1,0],[3,0,1,4,2],[4,3,0,2,1]],\"multiplication\":[[0,0,0,0,0],[0,3,4,1,2],[0,4,1,2,3],[0,1,2,3,4],[0,2,3,4,1]]}。請依 [zero_divisor_pairs,no_nonzero_zero_divisors] 欄序回答。","answer":{"kind":"expression","value":"([], 是)","explanation_zh":"Wave4第三層核對=([], 是)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-zero-divisor-audit","seed":202610291043,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,2,4,0,3],[2,4,3,1,0],[3,0,1,4,2],[4,3,0,2,1]],"multiplication":[[0,0,0,0,0],[0,3,4,1,2],[0,4,1,2,3],[0,1,2,3,4],[0,2,3,4,1]],"domain":"Complete_finite_integral_domain","operation":"integraldomain-zero-divisor-audit","semantic_template":"qf-mod29-integraldomain-zero-divisor-audit/{\"addition\":[[0,1,2,3,4],[1,2,4,0,3],[2,4,3,1,0],[3,0,1,4,2],[4,3,0,2,1]],\"multiplication\":[[0,0,0,0,0],[0,3,4,1,2],[0,4,1,2,3],[0,1,2,3,4],[0,2,3,4,1]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0044","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-zero-divisor-audit；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,3,0,4,2],[2,0,4,1,3],[3,4,1,2,0],[4,2,3,0,1]],\"multiplication\":[[0,0,0,0,0],[0,3,4,2,1],[0,4,3,1,2],[0,2,1,4,3],[0,1,2,3,4]]}。請依 [zero_divisor_pairs,no_nonzero_zero_divisors] 欄序回答。","answer":{"kind":"expression","value":"([], 是)","explanation_zh":"Wave4第三層核對=([], 是)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-zero-divisor-audit","seed":202610291044,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,3,0,4,2],[2,0,4,1,3],[3,4,1,2,0],[4,2,3,0,1]],"multiplication":[[0,0,0,0,0],[0,3,4,2,1],[0,4,3,1,2],[0,2,1,4,3],[0,1,2,3,4]],"domain":"Complete_finite_integral_domain","operation":"integraldomain-zero-divisor-audit","semantic_template":"qf-mod29-integraldomain-zero-divisor-audit/{\"addition\":[[0,1,2,3,4],[1,3,0,4,2],[2,0,4,1,3],[3,4,1,2,0],[4,2,3,0,1]],\"multiplication\":[[0,0,0,0,0],[0,3,4,2,1],[0,4,3,1,2],[0,2,1,4,3],[0,1,2,3,4]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0045","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-zero-divisor-audit；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,2,4,0,3],[2,4,3,1,0],[3,0,1,4,2],[4,3,0,2,1]],\"multiplication\":[[0,0,0,0,0],[0,2,3,4,1],[0,3,4,1,2],[0,4,1,2,3],[0,1,2,3,4]]}。請依 [zero_divisor_pairs,no_nonzero_zero_divisors] 欄序回答。","answer":{"kind":"expression","value":"([], 是)","explanation_zh":"Wave4第三層核對=([], 是)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-zero-divisor-audit","seed":202610291045,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,2,4,0,3],[2,4,3,1,0],[3,0,1,4,2],[4,3,0,2,1]],"multiplication":[[0,0,0,0,0],[0,2,3,4,1],[0,3,4,1,2],[0,4,1,2,3],[0,1,2,3,4]],"domain":"Complete_finite_integral_domain","operation":"integraldomain-zero-divisor-audit","semantic_template":"qf-mod29-integraldomain-zero-divisor-audit/{\"addition\":[[0,1,2,3,4],[1,2,4,0,3],[2,4,3,1,0],[3,0,1,4,2],[4,3,0,2,1]],\"multiplication\":[[0,0,0,0,0],[0,2,3,4,1],[0,3,4,1,2],[0,4,1,2,3],[0,1,2,3,4]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0046","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-zero-divisor-audit；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,4,0,2,3],[2,0,3,4,1],[3,2,4,1,0],[4,3,1,0,2]],\"multiplication\":[[0,0,0,0,0],[0,3,4,2,1],[0,4,3,1,2],[0,2,1,4,3],[0,1,2,3,4]]}。請依 [zero_divisor_pairs,no_nonzero_zero_divisors] 欄序回答。","answer":{"kind":"expression","value":"([], 是)","explanation_zh":"Wave4第三層核對=([], 是)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-zero-divisor-audit","seed":202610291046,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,4,0,2,3],[2,0,3,4,1],[3,2,4,1,0],[4,3,1,0,2]],"multiplication":[[0,0,0,0,0],[0,3,4,2,1],[0,4,3,1,2],[0,2,1,4,3],[0,1,2,3,4]],"domain":"Complete_finite_integral_domain","operation":"integraldomain-zero-divisor-audit","semantic_template":"qf-mod29-integraldomain-zero-divisor-audit/{\"addition\":[[0,1,2,3,4],[1,4,0,2,3],[2,0,3,4,1],[3,2,4,1,0],[4,3,1,0,2]],\"multiplication\":[[0,0,0,0,0],[0,3,4,2,1],[0,4,3,1,2],[0,2,1,4,3],[0,1,2,3,4]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0047","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-zero-divisor-audit；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],\"multiplication\":[[0,0,0,0,0],[0,4,3,2,1],[0,3,1,4,2],[0,2,4,1,3],[0,1,2,3,4]]}。請依 [zero_divisor_pairs,no_nonzero_zero_divisors] 欄序回答。","answer":{"kind":"expression","value":"([], 是)","explanation_zh":"Wave4第三層核對=([], 是)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-zero-divisor-audit","seed":202610291047,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],"multiplication":[[0,0,0,0,0],[0,4,3,2,1],[0,3,1,4,2],[0,2,4,1,3],[0,1,2,3,4]],"domain":"Complete_finite_integral_domain","operation":"integraldomain-zero-divisor-audit","semantic_template":"qf-mod29-integraldomain-zero-divisor-audit/{\"addition\":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],\"multiplication\":[[0,0,0,0,0],[0,4,3,2,1],[0,3,1,4,2],[0,2,4,1,3],[0,1,2,3,4]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0048","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-zero-divisor-audit；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,4,3,0,2],[2,3,1,4,0],[3,0,4,2,1],[4,2,0,1,3]],\"multiplication\":[[0,0,0,0,0],[0,2,3,4,1],[0,3,4,1,2],[0,4,1,2,3],[0,1,2,3,4]]}。請依 [zero_divisor_pairs,no_nonzero_zero_divisors] 欄序回答。","answer":{"kind":"expression","value":"([], 是)","explanation_zh":"Wave4第三層核對=([], 是)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-zero-divisor-audit","seed":202610291048,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,4,3,0,2],[2,3,1,4,0],[3,0,4,2,1],[4,2,0,1,3]],"multiplication":[[0,0,0,0,0],[0,2,3,4,1],[0,3,4,1,2],[0,4,1,2,3],[0,1,2,3,4]],"domain":"Complete_finite_integral_domain","operation":"integraldomain-zero-divisor-audit","semantic_template":"qf-mod29-integraldomain-zero-divisor-audit/{\"addition\":[[0,1,2,3,4],[1,4,3,0,2],[2,3,1,4,0],[3,0,4,2,1],[4,2,0,1,3]],\"multiplication\":[[0,0,0,0,0],[0,2,3,4,1],[0,3,4,1,2],[0,4,1,2,3],[0,1,2,3,4]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0049","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-zero-divisor-audit；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"multiplication\":[[0,0,0,0,0],[0,4,3,2,1],[0,3,1,4,2],[0,2,4,1,3],[0,1,2,3,4]]}。請依 [zero_divisor_pairs,no_nonzero_zero_divisors] 欄序回答。","answer":{"kind":"expression","value":"([], 是)","explanation_zh":"Wave4第三層核對=([], 是)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-zero-divisor-audit","seed":202610291049,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"multiplication":[[0,0,0,0,0],[0,4,3,2,1],[0,3,1,4,2],[0,2,4,1,3],[0,1,2,3,4]],"domain":"Complete_finite_integral_domain","operation":"integraldomain-zero-divisor-audit","semantic_template":"qf-mod29-integraldomain-zero-divisor-audit/{\"addition\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"multiplication\":[[0,0,0,0,0],[0,4,3,2,1],[0,3,1,4,2],[0,2,4,1,3],[0,1,2,3,4]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0050","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-zero-divisor-audit；資料={\"n\":5,\"addition\":[[2,0,3,4,1],[0,1,2,3,4],[3,2,4,1,0],[4,3,1,0,2],[1,4,0,2,3]],\"multiplication\":[[0,1,2,3,4],[1,1,1,1,1],[2,1,4,0,3],[3,1,0,4,2],[4,1,3,2,0]]}。請依 [zero_divisor_pairs,no_nonzero_zero_divisors] 欄序回答。","answer":{"kind":"expression","value":"([], 是)","explanation_zh":"Wave4第三層核對=([], 是)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-zero-divisor-audit","seed":202610291050,"parameters":{"n":5,"addition":[[2,0,3,4,1],[0,1,2,3,4],[3,2,4,1,0],[4,3,1,0,2],[1,4,0,2,3]],"multiplication":[[0,1,2,3,4],[1,1,1,1,1],[2,1,4,0,3],[3,1,0,4,2],[4,1,3,2,0]],"domain":"Complete_finite_integral_domain","operation":"integraldomain-zero-divisor-audit","semantic_template":"qf-mod29-integraldomain-zero-divisor-audit/{\"addition\":[[2,0,3,4,1],[0,1,2,3,4],[3,2,4,1,0],[4,3,1,0,2],[1,4,0,2,3]],\"multiplication\":[[0,1,2,3,4],[1,1,1,1,1],[2,1,4,0,3],[3,1,0,4,2],[4,1,3,2,0]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0051","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-cancellation；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"multiplication\":[[0,0,0,0,0],[0,1,2,3,4],[0,2,4,1,3],[0,3,1,4,2],[0,4,3,2,1]],\"a\":1,\"b\":1,\"c\":2}。請依 [a,b,c,ab,ac,premise_equal,cancellation_valid,certificate] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, 2, 1, 2, 否, 是, nonzero_cancellation_in_domain)","explanation_zh":"Wave4第三層核對=(1, 1, 2, 1, 2, 否, 是, nonzero_cancellation_in_domain)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-cancellation","seed":202610291051,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"multiplication":[[0,0,0,0,0],[0,1,2,3,4],[0,2,4,1,3],[0,3,1,4,2],[0,4,3,2,1]],"a":1,"b":1,"c":2,"domain":"Complete_finite_integral_domain","operation":"integraldomain-cancellation","semantic_template":"qf-mod29-integraldomain-cancellation/{\"a\":1,\"addition\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"b\":1,\"c\":2,\"multiplication\":[[0,0,0,0,0],[0,1,2,3,4],[0,2,4,1,3],[0,3,1,4,2],[0,4,3,2,1]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0052","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-cancellation；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,2,4,0,3],[2,4,3,1,0],[3,0,1,4,2],[4,3,0,2,1]],\"multiplication\":[[0,0,0,0,0],[0,1,2,3,4],[0,2,3,4,1],[0,3,4,1,2],[0,4,1,2,3]],\"a\":2,\"b\":2,\"c\":2}。請依 [a,b,c,ab,ac,premise_equal,cancellation_valid,certificate] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 2, 3, 3, 是, 是, nonzero_cancellation_in_domain)","explanation_zh":"Wave4第三層核對=(2, 2, 2, 3, 3, 是, 是, nonzero_cancellation_in_domain)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-cancellation","seed":202610291052,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,2,4,0,3],[2,4,3,1,0],[3,0,1,4,2],[4,3,0,2,1]],"multiplication":[[0,0,0,0,0],[0,1,2,3,4],[0,2,3,4,1],[0,3,4,1,2],[0,4,1,2,3]],"a":2,"b":2,"c":2,"domain":"Complete_finite_integral_domain","operation":"integraldomain-cancellation","semantic_template":"qf-mod29-integraldomain-cancellation/{\"a\":2,\"addition\":[[0,1,2,3,4],[1,2,4,0,3],[2,4,3,1,0],[3,0,1,4,2],[4,3,0,2,1]],\"b\":2,\"c\":2,\"multiplication\":[[0,0,0,0,0],[0,1,2,3,4],[0,2,3,4,1],[0,3,4,1,2],[0,4,1,2,3]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0053","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-cancellation；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],\"multiplication\":[[0,0,0,0,0],[0,1,2,3,4],[0,2,4,1,3],[0,3,1,4,2],[0,4,3,2,1]],\"a\":2,\"b\":2,\"c\":2}。請依 [a,b,c,ab,ac,premise_equal,cancellation_valid,certificate] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 2, 4, 4, 是, 是, nonzero_cancellation_in_domain)","explanation_zh":"Wave4第三層核對=(2, 2, 2, 4, 4, 是, 是, nonzero_cancellation_in_domain)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-cancellation","seed":202610291053,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],"multiplication":[[0,0,0,0,0],[0,1,2,3,4],[0,2,4,1,3],[0,3,1,4,2],[0,4,3,2,1]],"a":2,"b":2,"c":2,"domain":"Complete_finite_integral_domain","operation":"integraldomain-cancellation","semantic_template":"qf-mod29-integraldomain-cancellation/{\"a\":2,\"addition\":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],\"b\":2,\"c\":2,\"multiplication\":[[0,0,0,0,0],[0,1,2,3,4],[0,2,4,1,3],[0,3,1,4,2],[0,4,3,2,1]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0054","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-cancellation；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,3,0,4,2],[2,0,4,1,3],[3,4,1,2,0],[4,2,3,0,1]],\"multiplication\":[[0,0,0,0,0],[0,1,2,3,4],[0,2,1,4,3],[0,3,4,2,1],[0,4,3,1,2]],\"a\":2,\"b\":2,\"c\":2}。請依 [a,b,c,ab,ac,premise_equal,cancellation_valid,certificate] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 2, 1, 1, 是, 是, nonzero_cancellation_in_domain)","explanation_zh":"Wave4第三層核對=(2, 2, 2, 1, 1, 是, 是, nonzero_cancellation_in_domain)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-cancellation","seed":202610291054,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,3,0,4,2],[2,0,4,1,3],[3,4,1,2,0],[4,2,3,0,1]],"multiplication":[[0,0,0,0,0],[0,1,2,3,4],[0,2,1,4,3],[0,3,4,2,1],[0,4,3,1,2]],"a":2,"b":2,"c":2,"domain":"Complete_finite_integral_domain","operation":"integraldomain-cancellation","semantic_template":"qf-mod29-integraldomain-cancellation/{\"a\":2,\"addition\":[[0,1,2,3,4],[1,3,0,4,2],[2,0,4,1,3],[3,4,1,2,0],[4,2,3,0,1]],\"b\":2,\"c\":2,\"multiplication\":[[0,0,0,0,0],[0,1,2,3,4],[0,2,1,4,3],[0,3,4,2,1],[0,4,3,1,2]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0055","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-cancellation；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,4,3,0,2],[2,3,1,4,0],[3,0,4,2,1],[4,2,0,1,3]],\"multiplication\":[[0,0,0,0,0],[0,1,2,3,4],[0,2,3,4,1],[0,3,4,1,2],[0,4,1,2,3]],\"a\":1,\"b\":0,\"c\":1}。請依 [a,b,c,ab,ac,premise_equal,cancellation_valid,certificate] 欄序回答。","answer":{"kind":"expression","value":"(1, 0, 1, 0, 1, 否, 是, nonzero_cancellation_in_domain)","explanation_zh":"Wave4第三層核對=(1, 0, 1, 0, 1, 否, 是, nonzero_cancellation_in_domain)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-cancellation","seed":202610291055,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,4,3,0,2],[2,3,1,4,0],[3,0,4,2,1],[4,2,0,1,3]],"multiplication":[[0,0,0,0,0],[0,1,2,3,4],[0,2,3,4,1],[0,3,4,1,2],[0,4,1,2,3]],"a":1,"b":0,"c":1,"domain":"Complete_finite_integral_domain","operation":"integraldomain-cancellation","semantic_template":"qf-mod29-integraldomain-cancellation/{\"a\":1,\"addition\":[[0,1,2,3,4],[1,4,3,0,2],[2,3,1,4,0],[3,0,4,2,1],[4,2,0,1,3]],\"b\":0,\"c\":1,\"multiplication\":[[0,0,0,0,0],[0,1,2,3,4],[0,2,3,4,1],[0,3,4,1,2],[0,4,1,2,3]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0056","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-cancellation；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,4,0,2,3],[2,0,3,4,1],[3,2,4,1,0],[4,3,1,0,2]],\"multiplication\":[[0,0,0,0,0],[0,1,2,3,4],[0,2,1,4,3],[0,3,4,2,1],[0,4,3,1,2]],\"a\":4,\"b\":1,\"c\":1}。請依 [a,b,c,ab,ac,premise_equal,cancellation_valid,certificate] 欄序回答。","answer":{"kind":"expression","value":"(4, 1, 1, 4, 4, 是, 是, nonzero_cancellation_in_domain)","explanation_zh":"Wave4第三層核對=(4, 1, 1, 4, 4, 是, 是, nonzero_cancellation_in_domain)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-cancellation","seed":202610291056,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,4,0,2,3],[2,0,3,4,1],[3,2,4,1,0],[4,3,1,0,2]],"multiplication":[[0,0,0,0,0],[0,1,2,3,4],[0,2,1,4,3],[0,3,4,2,1],[0,4,3,1,2]],"a":4,"b":1,"c":1,"domain":"Complete_finite_integral_domain","operation":"integraldomain-cancellation","semantic_template":"qf-mod29-integraldomain-cancellation/{\"a\":4,\"addition\":[[0,1,2,3,4],[1,4,0,2,3],[2,0,3,4,1],[3,2,4,1,0],[4,3,1,0,2]],\"b\":1,\"c\":1,\"multiplication\":[[0,0,0,0,0],[0,1,2,3,4],[0,2,1,4,3],[0,3,4,2,1],[0,4,3,1,2]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0057","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-cancellation；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,4,3,0,2],[2,3,1,4,0],[3,0,4,2,1],[4,2,0,1,3]],\"multiplication\":[[0,0,0,0,0],[0,4,1,2,3],[0,1,2,3,4],[0,2,3,4,1],[0,3,4,1,2]],\"a\":3,\"b\":1,\"c\":1}。請依 [a,b,c,ab,ac,premise_equal,cancellation_valid,certificate] 欄序回答。","answer":{"kind":"expression","value":"(3, 1, 1, 2, 2, 是, 是, nonzero_cancellation_in_domain)","explanation_zh":"Wave4第三層核對=(3, 1, 1, 2, 2, 是, 是, nonzero_cancellation_in_domain)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-cancellation","seed":202610291057,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,4,3,0,2],[2,3,1,4,0],[3,0,4,2,1],[4,2,0,1,3]],"multiplication":[[0,0,0,0,0],[0,4,1,2,3],[0,1,2,3,4],[0,2,3,4,1],[0,3,4,1,2]],"a":3,"b":1,"c":1,"domain":"Complete_finite_integral_domain","operation":"integraldomain-cancellation","semantic_template":"qf-mod29-integraldomain-cancellation/{\"a\":3,\"addition\":[[0,1,2,3,4],[1,4,3,0,2],[2,3,1,4,0],[3,0,4,2,1],[4,2,0,1,3]],\"b\":1,\"c\":1,\"multiplication\":[[0,0,0,0,0],[0,4,1,2,3],[0,1,2,3,4],[0,2,3,4,1],[0,3,4,1,2]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0058","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-cancellation；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],\"multiplication\":[[0,0,0,0,0],[0,3,1,4,2],[0,1,2,3,4],[0,4,3,2,1],[0,2,4,1,3]],\"a\":3,\"b\":4,\"c\":4}。請依 [a,b,c,ab,ac,premise_equal,cancellation_valid,certificate] 欄序回答。","answer":{"kind":"expression","value":"(3, 4, 4, 1, 1, 是, 是, nonzero_cancellation_in_domain)","explanation_zh":"Wave4第三層核對=(3, 4, 4, 1, 1, 是, 是, nonzero_cancellation_in_domain)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-cancellation","seed":202610291058,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],"multiplication":[[0,0,0,0,0],[0,3,1,4,2],[0,1,2,3,4],[0,4,3,2,1],[0,2,4,1,3]],"a":3,"b":4,"c":4,"domain":"Complete_finite_integral_domain","operation":"integraldomain-cancellation","semantic_template":"qf-mod29-integraldomain-cancellation/{\"a\":3,\"addition\":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],\"b\":4,\"c\":4,\"multiplication\":[[0,0,0,0,0],[0,3,1,4,2],[0,1,2,3,4],[0,4,3,2,1],[0,2,4,1,3]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0059","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-cancellation；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,2,4,0,3],[2,4,3,1,0],[3,0,1,4,2],[4,3,0,2,1]],\"multiplication\":[[0,0,0,0,0],[0,4,1,2,3],[0,1,2,3,4],[0,2,3,4,1],[0,3,4,1,2]],\"a\":2,\"b\":4,\"c\":0}。請依 [a,b,c,ab,ac,premise_equal,cancellation_valid,certificate] 欄序回答。","answer":{"kind":"expression","value":"(2, 4, 0, 4, 0, 否, 是, nonzero_cancellation_in_domain)","explanation_zh":"Wave4第三層核對=(2, 4, 0, 4, 0, 否, 是, nonzero_cancellation_in_domain)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-cancellation","seed":202610291059,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,2,4,0,3],[2,4,3,1,0],[3,0,1,4,2],[4,3,0,2,1]],"multiplication":[[0,0,0,0,0],[0,4,1,2,3],[0,1,2,3,4],[0,2,3,4,1],[0,3,4,1,2]],"a":2,"b":4,"c":0,"domain":"Complete_finite_integral_domain","operation":"integraldomain-cancellation","semantic_template":"qf-mod29-integraldomain-cancellation/{\"a\":2,\"addition\":[[0,1,2,3,4],[1,2,4,0,3],[2,4,3,1,0],[3,0,1,4,2],[4,3,0,2,1]],\"b\":4,\"c\":0,\"multiplication\":[[0,0,0,0,0],[0,4,1,2,3],[0,1,2,3,4],[0,2,3,4,1],[0,3,4,1,2]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0060","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-cancellation；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,4,0,2,3],[2,0,3,4,1],[3,2,4,1,0],[4,3,1,0,2]],\"multiplication\":[[0,0,0,0,0],[0,2,1,4,3],[0,1,2,3,4],[0,4,3,1,2],[0,3,4,2,1]],\"a\":3,\"b\":0,\"c\":0}。請依 [a,b,c,ab,ac,premise_equal,cancellation_valid,certificate] 欄序回答。","answer":{"kind":"expression","value":"(3, 0, 0, 0, 0, 是, 是, nonzero_cancellation_in_domain)","explanation_zh":"Wave4第三層核對=(3, 0, 0, 0, 0, 是, 是, nonzero_cancellation_in_domain)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-cancellation","seed":202610291060,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,4,0,2,3],[2,0,3,4,1],[3,2,4,1,0],[4,3,1,0,2]],"multiplication":[[0,0,0,0,0],[0,2,1,4,3],[0,1,2,3,4],[0,4,3,1,2],[0,3,4,2,1]],"a":3,"b":0,"c":0,"domain":"Complete_finite_integral_domain","operation":"integraldomain-cancellation","semantic_template":"qf-mod29-integraldomain-cancellation/{\"a\":3,\"addition\":[[0,1,2,3,4],[1,4,0,2,3],[2,0,3,4,1],[3,2,4,1,0],[4,3,1,0,2]],\"b\":0,\"c\":0,\"multiplication\":[[0,0,0,0,0],[0,2,1,4,3],[0,1,2,3,4],[0,4,3,1,2],[0,3,4,2,1]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0061","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-cancellation；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"multiplication\":[[0,0,0,0,0],[0,3,1,4,2],[0,1,2,3,4],[0,4,3,2,1],[0,2,4,1,3]],\"a\":1,\"b\":2,\"c\":2}。請依 [a,b,c,ab,ac,premise_equal,cancellation_valid,certificate] 欄序回答。","answer":{"kind":"expression","value":"(1, 2, 2, 1, 1, 是, 是, nonzero_cancellation_in_domain)","explanation_zh":"Wave4第三層核對=(1, 2, 2, 1, 1, 是, 是, nonzero_cancellation_in_domain)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-cancellation","seed":202610291061,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"multiplication":[[0,0,0,0,0],[0,3,1,4,2],[0,1,2,3,4],[0,4,3,2,1],[0,2,4,1,3]],"a":1,"b":2,"c":2,"domain":"Complete_finite_integral_domain","operation":"integraldomain-cancellation","semantic_template":"qf-mod29-integraldomain-cancellation/{\"a\":1,\"addition\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"b\":2,\"c\":2,\"multiplication\":[[0,0,0,0,0],[0,3,1,4,2],[0,1,2,3,4],[0,4,3,2,1],[0,2,4,1,3]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0062","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-cancellation；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,3,0,4,2],[2,0,4,1,3],[3,4,1,2,0],[4,2,3,0,1]],\"multiplication\":[[0,0,0,0,0],[0,2,1,4,3],[0,1,2,3,4],[0,4,3,1,2],[0,3,4,2,1]],\"a\":1,\"b\":4,\"c\":4}。請依 [a,b,c,ab,ac,premise_equal,cancellation_valid,certificate] 欄序回答。","answer":{"kind":"expression","value":"(1, 4, 4, 3, 3, 是, 是, nonzero_cancellation_in_domain)","explanation_zh":"Wave4第三層核對=(1, 4, 4, 3, 3, 是, 是, nonzero_cancellation_in_domain)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-cancellation","seed":202610291062,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,3,0,4,2],[2,0,4,1,3],[3,4,1,2,0],[4,2,3,0,1]],"multiplication":[[0,0,0,0,0],[0,2,1,4,3],[0,1,2,3,4],[0,4,3,1,2],[0,3,4,2,1]],"a":1,"b":4,"c":4,"domain":"Complete_finite_integral_domain","operation":"integraldomain-cancellation","semantic_template":"qf-mod29-integraldomain-cancellation/{\"a\":1,\"addition\":[[0,1,2,3,4],[1,3,0,4,2],[2,0,4,1,3],[3,4,1,2,0],[4,2,3,0,1]],\"b\":4,\"c\":4,\"multiplication\":[[0,0,0,0,0],[0,2,1,4,3],[0,1,2,3,4],[0,4,3,1,2],[0,3,4,2,1]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0063","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-cancellation；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,4,0,2,3],[2,0,3,4,1],[3,2,4,1,0],[4,3,1,0,2]],\"multiplication\":[[0,0,0,0,0],[0,4,3,1,2],[0,3,4,2,1],[0,1,2,3,4],[0,2,1,4,3]],\"a\":3,\"b\":2,\"c\":4}。請依 [a,b,c,ab,ac,premise_equal,cancellation_valid,certificate] 欄序回答。","answer":{"kind":"expression","value":"(3, 2, 4, 2, 4, 否, 是, nonzero_cancellation_in_domain)","explanation_zh":"Wave4第三層核對=(3, 2, 4, 2, 4, 否, 是, nonzero_cancellation_in_domain)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-cancellation","seed":202610291063,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,4,0,2,3],[2,0,3,4,1],[3,2,4,1,0],[4,3,1,0,2]],"multiplication":[[0,0,0,0,0],[0,4,3,1,2],[0,3,4,2,1],[0,1,2,3,4],[0,2,1,4,3]],"a":3,"b":2,"c":4,"domain":"Complete_finite_integral_domain","operation":"integraldomain-cancellation","semantic_template":"qf-mod29-integraldomain-cancellation/{\"a\":3,\"addition\":[[0,1,2,3,4],[1,4,0,2,3],[2,0,3,4,1],[3,2,4,1,0],[4,3,1,0,2]],\"b\":2,\"c\":4,\"multiplication\":[[0,0,0,0,0],[0,4,3,1,2],[0,3,4,2,1],[0,1,2,3,4],[0,2,1,4,3]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0064","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-cancellation；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"multiplication\":[[0,0,0,0,0],[0,2,4,1,3],[0,4,3,2,1],[0,1,2,3,4],[0,3,1,4,2]],\"a\":1,\"b\":2,\"c\":2}。請依 [a,b,c,ab,ac,premise_equal,cancellation_valid,certificate] 欄序回答。","answer":{"kind":"expression","value":"(1, 2, 2, 4, 4, 是, 是, nonzero_cancellation_in_domain)","explanation_zh":"Wave4第三層核對=(1, 2, 2, 4, 4, 是, 是, nonzero_cancellation_in_domain)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-cancellation","seed":202610291064,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"multiplication":[[0,0,0,0,0],[0,2,4,1,3],[0,4,3,2,1],[0,1,2,3,4],[0,3,1,4,2]],"a":1,"b":2,"c":2,"domain":"Complete_finite_integral_domain","operation":"integraldomain-cancellation","semantic_template":"qf-mod29-integraldomain-cancellation/{\"a\":1,\"addition\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"b\":2,\"c\":2,\"multiplication\":[[0,0,0,0,0],[0,2,4,1,3],[0,4,3,2,1],[0,1,2,3,4],[0,3,1,4,2]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0065","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-cancellation；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,3,0,4,2],[2,0,4,1,3],[3,4,1,2,0],[4,2,3,0,1]],\"multiplication\":[[0,0,0,0,0],[0,4,3,1,2],[0,3,4,2,1],[0,1,2,3,4],[0,2,1,4,3]],\"a\":1,\"b\":0,\"c\":0}。請依 [a,b,c,ab,ac,premise_equal,cancellation_valid,certificate] 欄序回答。","answer":{"kind":"expression","value":"(1, 0, 0, 0, 0, 是, 是, nonzero_cancellation_in_domain)","explanation_zh":"Wave4第三層核對=(1, 0, 0, 0, 0, 是, 是, nonzero_cancellation_in_domain)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-cancellation","seed":202610291065,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,3,0,4,2],[2,0,4,1,3],[3,4,1,2,0],[4,2,3,0,1]],"multiplication":[[0,0,0,0,0],[0,4,3,1,2],[0,3,4,2,1],[0,1,2,3,4],[0,2,1,4,3]],"a":1,"b":0,"c":0,"domain":"Complete_finite_integral_domain","operation":"integraldomain-cancellation","semantic_template":"qf-mod29-integraldomain-cancellation/{\"a\":1,\"addition\":[[0,1,2,3,4],[1,3,0,4,2],[2,0,4,1,3],[3,4,1,2,0],[4,2,3,0,1]],\"b\":0,\"c\":0,\"multiplication\":[[0,0,0,0,0],[0,4,3,1,2],[0,3,4,2,1],[0,1,2,3,4],[0,2,1,4,3]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0066","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-cancellation；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,4,3,0,2],[2,3,1,4,0],[3,0,4,2,1],[4,2,0,1,3]],\"multiplication\":[[0,0,0,0,0],[0,3,4,1,2],[0,4,1,2,3],[0,1,2,3,4],[0,2,3,4,1]],\"a\":1,\"b\":3,\"c\":3}。請依 [a,b,c,ab,ac,premise_equal,cancellation_valid,certificate] 欄序回答。","answer":{"kind":"expression","value":"(1, 3, 3, 1, 1, 是, 是, nonzero_cancellation_in_domain)","explanation_zh":"Wave4第三層核對=(1, 3, 3, 1, 1, 是, 是, nonzero_cancellation_in_domain)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-cancellation","seed":202610291066,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,4,3,0,2],[2,3,1,4,0],[3,0,4,2,1],[4,2,0,1,3]],"multiplication":[[0,0,0,0,0],[0,3,4,1,2],[0,4,1,2,3],[0,1,2,3,4],[0,2,3,4,1]],"a":1,"b":3,"c":3,"domain":"Complete_finite_integral_domain","operation":"integraldomain-cancellation","semantic_template":"qf-mod29-integraldomain-cancellation/{\"a\":1,\"addition\":[[0,1,2,3,4],[1,4,3,0,2],[2,3,1,4,0],[3,0,4,2,1],[4,2,0,1,3]],\"b\":3,\"c\":3,\"multiplication\":[[0,0,0,0,0],[0,3,4,1,2],[0,4,1,2,3],[0,1,2,3,4],[0,2,3,4,1]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0067","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-cancellation；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],\"multiplication\":[[0,0,0,0,0],[0,2,4,1,3],[0,4,3,2,1],[0,1,2,3,4],[0,3,1,4,2]],\"a\":3,\"b\":4,\"c\":1}。請依 [a,b,c,ab,ac,premise_equal,cancellation_valid,certificate] 欄序回答。","answer":{"kind":"expression","value":"(3, 4, 1, 4, 1, 否, 是, nonzero_cancellation_in_domain)","explanation_zh":"Wave4第三層核對=(3, 4, 1, 4, 1, 否, 是, nonzero_cancellation_in_domain)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-cancellation","seed":202610291067,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],"multiplication":[[0,0,0,0,0],[0,2,4,1,3],[0,4,3,2,1],[0,1,2,3,4],[0,3,1,4,2]],"a":3,"b":4,"c":1,"domain":"Complete_finite_integral_domain","operation":"integraldomain-cancellation","semantic_template":"qf-mod29-integraldomain-cancellation/{\"a\":3,\"addition\":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],\"b\":4,\"c\":1,\"multiplication\":[[0,0,0,0,0],[0,2,4,1,3],[0,4,3,2,1],[0,1,2,3,4],[0,3,1,4,2]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0068","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-cancellation；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,2,4,0,3],[2,4,3,1,0],[3,0,1,4,2],[4,3,0,2,1]],\"multiplication\":[[0,0,0,0,0],[0,3,4,1,2],[0,4,1,2,3],[0,1,2,3,4],[0,2,3,4,1]],\"a\":4,\"b\":2,\"c\":2}。請依 [a,b,c,ab,ac,premise_equal,cancellation_valid,certificate] 欄序回答。","answer":{"kind":"expression","value":"(4, 2, 2, 3, 3, 是, 是, nonzero_cancellation_in_domain)","explanation_zh":"Wave4第三層核對=(4, 2, 2, 3, 3, 是, 是, nonzero_cancellation_in_domain)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-cancellation","seed":202610291068,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,2,4,0,3],[2,4,3,1,0],[3,0,1,4,2],[4,3,0,2,1]],"multiplication":[[0,0,0,0,0],[0,3,4,1,2],[0,4,1,2,3],[0,1,2,3,4],[0,2,3,4,1]],"a":4,"b":2,"c":2,"domain":"Complete_finite_integral_domain","operation":"integraldomain-cancellation","semantic_template":"qf-mod29-integraldomain-cancellation/{\"a\":4,\"addition\":[[0,1,2,3,4],[1,2,4,0,3],[2,4,3,1,0],[3,0,1,4,2],[4,3,0,2,1]],\"b\":2,\"c\":2,\"multiplication\":[[0,0,0,0,0],[0,3,4,1,2],[0,4,1,2,3],[0,1,2,3,4],[0,2,3,4,1]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0069","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-cancellation；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,3,0,4,2],[2,0,4,1,3],[3,4,1,2,0],[4,2,3,0,1]],\"multiplication\":[[0,0,0,0,0],[0,3,4,2,1],[0,4,3,1,2],[0,2,1,4,3],[0,1,2,3,4]],\"a\":2,\"b\":3,\"c\":3}。請依 [a,b,c,ab,ac,premise_equal,cancellation_valid,certificate] 欄序回答。","answer":{"kind":"expression","value":"(2, 3, 3, 1, 1, 是, 是, nonzero_cancellation_in_domain)","explanation_zh":"Wave4第三層核對=(2, 3, 3, 1, 1, 是, 是, nonzero_cancellation_in_domain)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-cancellation","seed":202610291069,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,3,0,4,2],[2,0,4,1,3],[3,4,1,2,0],[4,2,3,0,1]],"multiplication":[[0,0,0,0,0],[0,3,4,2,1],[0,4,3,1,2],[0,2,1,4,3],[0,1,2,3,4]],"a":2,"b":3,"c":3,"domain":"Complete_finite_integral_domain","operation":"integraldomain-cancellation","semantic_template":"qf-mod29-integraldomain-cancellation/{\"a\":2,\"addition\":[[0,1,2,3,4],[1,3,0,4,2],[2,0,4,1,3],[3,4,1,2,0],[4,2,3,0,1]],\"b\":3,\"c\":3,\"multiplication\":[[0,0,0,0,0],[0,3,4,2,1],[0,4,3,1,2],[0,2,1,4,3],[0,1,2,3,4]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0070","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-cancellation；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,2,4,0,3],[2,4,3,1,0],[3,0,1,4,2],[4,3,0,2,1]],\"multiplication\":[[0,0,0,0,0],[0,2,3,4,1],[0,3,4,1,2],[0,4,1,2,3],[0,1,2,3,4]],\"a\":2,\"b\":0,\"c\":0}。請依 [a,b,c,ab,ac,premise_equal,cancellation_valid,certificate] 欄序回答。","answer":{"kind":"expression","value":"(2, 0, 0, 0, 0, 是, 是, nonzero_cancellation_in_domain)","explanation_zh":"Wave4第三層核對=(2, 0, 0, 0, 0, 是, 是, nonzero_cancellation_in_domain)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-cancellation","seed":202610291070,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,2,4,0,3],[2,4,3,1,0],[3,0,1,4,2],[4,3,0,2,1]],"multiplication":[[0,0,0,0,0],[0,2,3,4,1],[0,3,4,1,2],[0,4,1,2,3],[0,1,2,3,4]],"a":2,"b":0,"c":0,"domain":"Complete_finite_integral_domain","operation":"integraldomain-cancellation","semantic_template":"qf-mod29-integraldomain-cancellation/{\"a\":2,\"addition\":[[0,1,2,3,4],[1,2,4,0,3],[2,4,3,1,0],[3,0,1,4,2],[4,3,0,2,1]],\"b\":0,\"c\":0,\"multiplication\":[[0,0,0,0,0],[0,2,3,4,1],[0,3,4,1,2],[0,4,1,2,3],[0,1,2,3,4]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0071","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-cancellation；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,4,0,2,3],[2,0,3,4,1],[3,2,4,1,0],[4,3,1,0,2]],\"multiplication\":[[0,0,0,0,0],[0,3,4,2,1],[0,4,3,1,2],[0,2,1,4,3],[0,1,2,3,4]],\"a\":4,\"b\":4,\"c\":2}。請依 [a,b,c,ab,ac,premise_equal,cancellation_valid,certificate] 欄序回答。","answer":{"kind":"expression","value":"(4, 4, 2, 4, 2, 否, 是, nonzero_cancellation_in_domain)","explanation_zh":"Wave4第三層核對=(4, 4, 2, 4, 2, 否, 是, nonzero_cancellation_in_domain)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-cancellation","seed":202610291071,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,4,0,2,3],[2,0,3,4,1],[3,2,4,1,0],[4,3,1,0,2]],"multiplication":[[0,0,0,0,0],[0,3,4,2,1],[0,4,3,1,2],[0,2,1,4,3],[0,1,2,3,4]],"a":4,"b":4,"c":2,"domain":"Complete_finite_integral_domain","operation":"integraldomain-cancellation","semantic_template":"qf-mod29-integraldomain-cancellation/{\"a\":4,\"addition\":[[0,1,2,3,4],[1,4,0,2,3],[2,0,3,4,1],[3,2,4,1,0],[4,3,1,0,2]],\"b\":4,\"c\":2,\"multiplication\":[[0,0,0,0,0],[0,3,4,2,1],[0,4,3,1,2],[0,2,1,4,3],[0,1,2,3,4]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0072","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-cancellation；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],\"multiplication\":[[0,0,0,0,0],[0,4,3,2,1],[0,3,1,4,2],[0,2,4,1,3],[0,1,2,3,4]],\"a\":2,\"b\":2,\"c\":2}。請依 [a,b,c,ab,ac,premise_equal,cancellation_valid,certificate] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 2, 1, 1, 是, 是, nonzero_cancellation_in_domain)","explanation_zh":"Wave4第三層核對=(2, 2, 2, 1, 1, 是, 是, nonzero_cancellation_in_domain)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-cancellation","seed":202610291072,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],"multiplication":[[0,0,0,0,0],[0,4,3,2,1],[0,3,1,4,2],[0,2,4,1,3],[0,1,2,3,4]],"a":2,"b":2,"c":2,"domain":"Complete_finite_integral_domain","operation":"integraldomain-cancellation","semantic_template":"qf-mod29-integraldomain-cancellation/{\"a\":2,\"addition\":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],\"b\":2,\"c\":2,\"multiplication\":[[0,0,0,0,0],[0,4,3,2,1],[0,3,1,4,2],[0,2,4,1,3],[0,1,2,3,4]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0073","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-cancellation；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,4,3,0,2],[2,3,1,4,0],[3,0,4,2,1],[4,2,0,1,3]],\"multiplication\":[[0,0,0,0,0],[0,2,3,4,1],[0,3,4,1,2],[0,4,1,2,3],[0,1,2,3,4]],\"a\":1,\"b\":1,\"c\":1}。請依 [a,b,c,ab,ac,premise_equal,cancellation_valid,certificate] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, 1, 2, 2, 是, 是, nonzero_cancellation_in_domain)","explanation_zh":"Wave4第三層核對=(1, 1, 1, 2, 2, 是, 是, nonzero_cancellation_in_domain)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-cancellation","seed":202610291073,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,4,3,0,2],[2,3,1,4,0],[3,0,4,2,1],[4,2,0,1,3]],"multiplication":[[0,0,0,0,0],[0,2,3,4,1],[0,3,4,1,2],[0,4,1,2,3],[0,1,2,3,4]],"a":1,"b":1,"c":1,"domain":"Complete_finite_integral_domain","operation":"integraldomain-cancellation","semantic_template":"qf-mod29-integraldomain-cancellation/{\"a\":1,\"addition\":[[0,1,2,3,4],[1,4,3,0,2],[2,3,1,4,0],[3,0,4,2,1],[4,2,0,1,3]],\"b\":1,\"c\":1,\"multiplication\":[[0,0,0,0,0],[0,2,3,4,1],[0,3,4,1,2],[0,4,1,2,3],[0,1,2,3,4]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0074","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-cancellation；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"multiplication\":[[0,0,0,0,0],[0,4,3,2,1],[0,3,1,4,2],[0,2,4,1,3],[0,1,2,3,4]],\"a\":1,\"b\":1,\"c\":1}。請依 [a,b,c,ab,ac,premise_equal,cancellation_valid,certificate] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, 1, 4, 4, 是, 是, nonzero_cancellation_in_domain)","explanation_zh":"Wave4第三層核對=(1, 1, 1, 4, 4, 是, 是, nonzero_cancellation_in_domain)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-cancellation","seed":202610291074,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"multiplication":[[0,0,0,0,0],[0,4,3,2,1],[0,3,1,4,2],[0,2,4,1,3],[0,1,2,3,4]],"a":1,"b":1,"c":1,"domain":"Complete_finite_integral_domain","operation":"integraldomain-cancellation","semantic_template":"qf-mod29-integraldomain-cancellation/{\"a\":1,\"addition\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"b\":1,\"c\":1,\"multiplication\":[[0,0,0,0,0],[0,4,3,2,1],[0,3,1,4,2],[0,2,4,1,3],[0,1,2,3,4]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0075","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-cancellation；資料={\"n\":5,\"addition\":[[2,0,3,4,1],[0,1,2,3,4],[3,2,4,1,0],[4,3,1,0,2],[1,4,0,2,3]],\"multiplication\":[[0,1,2,3,4],[1,1,1,1,1],[2,1,4,0,3],[3,1,0,4,2],[4,1,3,2,0]],\"a\":0,\"b\":1,\"c\":0}。請依 [a,b,c,ab,ac,premise_equal,cancellation_valid,certificate] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 0, 1, 0, 否, 是, nonzero_cancellation_in_domain)","explanation_zh":"Wave4第三層核對=(0, 1, 0, 1, 0, 否, 是, nonzero_cancellation_in_domain)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-cancellation","seed":202610291075,"parameters":{"n":5,"addition":[[2,0,3,4,1],[0,1,2,3,4],[3,2,4,1,0],[4,3,1,0,2],[1,4,0,2,3]],"multiplication":[[0,1,2,3,4],[1,1,1,1,1],[2,1,4,0,3],[3,1,0,4,2],[4,1,3,2,0]],"a":0,"b":1,"c":0,"domain":"Complete_finite_integral_domain","operation":"integraldomain-cancellation","semantic_template":"qf-mod29-integraldomain-cancellation/{\"a\":0,\"addition\":[[2,0,3,4,1],[0,1,2,3,4],[3,2,4,1,0],[4,3,1,0,2],[1,4,0,2,3]],\"b\":1,\"c\":0,\"multiplication\":[[0,1,2,3,4],[1,1,1,1,1],[2,1,4,0,3],[3,1,0,4,2],[4,1,3,2,0]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0076","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-finite-inverse；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"multiplication\":[[0,0,0,0,0],[0,1,2,3,4],[0,2,4,1,3],[0,3,1,4,2],[0,4,3,2,1]],\"element\":1}。請依 [element,one,inverse,found,finite_theorem_step] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, 1, 是, finite_injective_multiplication_implies_surjective)","explanation_zh":"Wave4第三層核對=(1, 1, 1, 是, finite_injective_multiplication_implies_surjective)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-finite-inverse","seed":202610291076,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"multiplication":[[0,0,0,0,0],[0,1,2,3,4],[0,2,4,1,3],[0,3,1,4,2],[0,4,3,2,1]],"element":1,"domain":"Complete_finite_integral_domain","operation":"integraldomain-finite-inverse","semantic_template":"qf-mod29-integraldomain-finite-inverse/{\"addition\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"element\":1,\"multiplication\":[[0,0,0,0,0],[0,1,2,3,4],[0,2,4,1,3],[0,3,1,4,2],[0,4,3,2,1]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0077","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-finite-inverse；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,2,4,0,3],[2,4,3,1,0],[3,0,1,4,2],[4,3,0,2,1]],\"multiplication\":[[0,0,0,0,0],[0,1,2,3,4],[0,2,3,4,1],[0,3,4,1,2],[0,4,1,2,3]],\"element\":2}。請依 [element,one,inverse,found,finite_theorem_step] 欄序回答。","answer":{"kind":"expression","value":"(2, 1, 4, 是, finite_injective_multiplication_implies_surjective)","explanation_zh":"Wave4第三層核對=(2, 1, 4, 是, finite_injective_multiplication_implies_surjective)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-finite-inverse","seed":202610291077,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,2,4,0,3],[2,4,3,1,0],[3,0,1,4,2],[4,3,0,2,1]],"multiplication":[[0,0,0,0,0],[0,1,2,3,4],[0,2,3,4,1],[0,3,4,1,2],[0,4,1,2,3]],"element":2,"domain":"Complete_finite_integral_domain","operation":"integraldomain-finite-inverse","semantic_template":"qf-mod29-integraldomain-finite-inverse/{\"addition\":[[0,1,2,3,4],[1,2,4,0,3],[2,4,3,1,0],[3,0,1,4,2],[4,3,0,2,1]],\"element\":2,\"multiplication\":[[0,0,0,0,0],[0,1,2,3,4],[0,2,3,4,1],[0,3,4,1,2],[0,4,1,2,3]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0078","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-finite-inverse；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],\"multiplication\":[[0,0,0,0,0],[0,1,2,3,4],[0,2,4,1,3],[0,3,1,4,2],[0,4,3,2,1]],\"element\":2}。請依 [element,one,inverse,found,finite_theorem_step] 欄序回答。","answer":{"kind":"expression","value":"(2, 1, 3, 是, finite_injective_multiplication_implies_surjective)","explanation_zh":"Wave4第三層核對=(2, 1, 3, 是, finite_injective_multiplication_implies_surjective)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-finite-inverse","seed":202610291078,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],"multiplication":[[0,0,0,0,0],[0,1,2,3,4],[0,2,4,1,3],[0,3,1,4,2],[0,4,3,2,1]],"element":2,"domain":"Complete_finite_integral_domain","operation":"integraldomain-finite-inverse","semantic_template":"qf-mod29-integraldomain-finite-inverse/{\"addition\":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],\"element\":2,\"multiplication\":[[0,0,0,0,0],[0,1,2,3,4],[0,2,4,1,3],[0,3,1,4,2],[0,4,3,2,1]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0079","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-finite-inverse；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,3,0,4,2],[2,0,4,1,3],[3,4,1,2,0],[4,2,3,0,1]],\"multiplication\":[[0,0,0,0,0],[0,1,2,3,4],[0,2,1,4,3],[0,3,4,2,1],[0,4,3,1,2]],\"element\":2}。請依 [element,one,inverse,found,finite_theorem_step] 欄序回答。","answer":{"kind":"expression","value":"(2, 1, 2, 是, finite_injective_multiplication_implies_surjective)","explanation_zh":"Wave4第三層核對=(2, 1, 2, 是, finite_injective_multiplication_implies_surjective)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-finite-inverse","seed":202610291079,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,3,0,4,2],[2,0,4,1,3],[3,4,1,2,0],[4,2,3,0,1]],"multiplication":[[0,0,0,0,0],[0,1,2,3,4],[0,2,1,4,3],[0,3,4,2,1],[0,4,3,1,2]],"element":2,"domain":"Complete_finite_integral_domain","operation":"integraldomain-finite-inverse","semantic_template":"qf-mod29-integraldomain-finite-inverse/{\"addition\":[[0,1,2,3,4],[1,3,0,4,2],[2,0,4,1,3],[3,4,1,2,0],[4,2,3,0,1]],\"element\":2,\"multiplication\":[[0,0,0,0,0],[0,1,2,3,4],[0,2,1,4,3],[0,3,4,2,1],[0,4,3,1,2]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0080","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-finite-inverse；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,4,3,0,2],[2,3,1,4,0],[3,0,4,2,1],[4,2,0,1,3]],\"multiplication\":[[0,0,0,0,0],[0,1,2,3,4],[0,2,3,4,1],[0,3,4,1,2],[0,4,1,2,3]],\"element\":1}。請依 [element,one,inverse,found,finite_theorem_step] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, 1, 是, finite_injective_multiplication_implies_surjective)","explanation_zh":"Wave4第三層核對=(1, 1, 1, 是, finite_injective_multiplication_implies_surjective)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-finite-inverse","seed":202610291080,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,4,3,0,2],[2,3,1,4,0],[3,0,4,2,1],[4,2,0,1,3]],"multiplication":[[0,0,0,0,0],[0,1,2,3,4],[0,2,3,4,1],[0,3,4,1,2],[0,4,1,2,3]],"element":1,"domain":"Complete_finite_integral_domain","operation":"integraldomain-finite-inverse","semantic_template":"qf-mod29-integraldomain-finite-inverse/{\"addition\":[[0,1,2,3,4],[1,4,3,0,2],[2,3,1,4,0],[3,0,4,2,1],[4,2,0,1,3]],\"element\":1,\"multiplication\":[[0,0,0,0,0],[0,1,2,3,4],[0,2,3,4,1],[0,3,4,1,2],[0,4,1,2,3]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0081","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-finite-inverse；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,4,0,2,3],[2,0,3,4,1],[3,2,4,1,0],[4,3,1,0,2]],\"multiplication\":[[0,0,0,0,0],[0,1,2,3,4],[0,2,1,4,3],[0,3,4,2,1],[0,4,3,1,2]],\"element\":4}。請依 [element,one,inverse,found,finite_theorem_step] 欄序回答。","answer":{"kind":"expression","value":"(4, 1, 3, 是, finite_injective_multiplication_implies_surjective)","explanation_zh":"Wave4第三層核對=(4, 1, 3, 是, finite_injective_multiplication_implies_surjective)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-finite-inverse","seed":202610291081,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,4,0,2,3],[2,0,3,4,1],[3,2,4,1,0],[4,3,1,0,2]],"multiplication":[[0,0,0,0,0],[0,1,2,3,4],[0,2,1,4,3],[0,3,4,2,1],[0,4,3,1,2]],"element":4,"domain":"Complete_finite_integral_domain","operation":"integraldomain-finite-inverse","semantic_template":"qf-mod29-integraldomain-finite-inverse/{\"addition\":[[0,1,2,3,4],[1,4,0,2,3],[2,0,3,4,1],[3,2,4,1,0],[4,3,1,0,2]],\"element\":4,\"multiplication\":[[0,0,0,0,0],[0,1,2,3,4],[0,2,1,4,3],[0,3,4,2,1],[0,4,3,1,2]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0082","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-finite-inverse；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,4,3,0,2],[2,3,1,4,0],[3,0,4,2,1],[4,2,0,1,3]],\"multiplication\":[[0,0,0,0,0],[0,4,1,2,3],[0,1,2,3,4],[0,2,3,4,1],[0,3,4,1,2]],\"element\":3}。請依 [element,one,inverse,found,finite_theorem_step] 欄序回答。","answer":{"kind":"expression","value":"(3, 2, 1, 是, finite_injective_multiplication_implies_surjective)","explanation_zh":"Wave4第三層核對=(3, 2, 1, 是, finite_injective_multiplication_implies_surjective)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-finite-inverse","seed":202610291082,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,4,3,0,2],[2,3,1,4,0],[3,0,4,2,1],[4,2,0,1,3]],"multiplication":[[0,0,0,0,0],[0,4,1,2,3],[0,1,2,3,4],[0,2,3,4,1],[0,3,4,1,2]],"element":3,"domain":"Complete_finite_integral_domain","operation":"integraldomain-finite-inverse","semantic_template":"qf-mod29-integraldomain-finite-inverse/{\"addition\":[[0,1,2,3,4],[1,4,3,0,2],[2,3,1,4,0],[3,0,4,2,1],[4,2,0,1,3]],\"element\":3,\"multiplication\":[[0,0,0,0,0],[0,4,1,2,3],[0,1,2,3,4],[0,2,3,4,1],[0,3,4,1,2]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0083","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-finite-inverse；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],\"multiplication\":[[0,0,0,0,0],[0,3,1,4,2],[0,1,2,3,4],[0,4,3,2,1],[0,2,4,1,3]],\"element\":3}。請依 [element,one,inverse,found,finite_theorem_step] 欄序回答。","answer":{"kind":"expression","value":"(3, 2, 3, 是, finite_injective_multiplication_implies_surjective)","explanation_zh":"Wave4第三層核對=(3, 2, 3, 是, finite_injective_multiplication_implies_surjective)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-finite-inverse","seed":202610291083,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],"multiplication":[[0,0,0,0,0],[0,3,1,4,2],[0,1,2,3,4],[0,4,3,2,1],[0,2,4,1,3]],"element":3,"domain":"Complete_finite_integral_domain","operation":"integraldomain-finite-inverse","semantic_template":"qf-mod29-integraldomain-finite-inverse/{\"addition\":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],\"element\":3,\"multiplication\":[[0,0,0,0,0],[0,3,1,4,2],[0,1,2,3,4],[0,4,3,2,1],[0,2,4,1,3]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0084","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-finite-inverse；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,2,4,0,3],[2,4,3,1,0],[3,0,1,4,2],[4,3,0,2,1]],\"multiplication\":[[0,0,0,0,0],[0,4,1,2,3],[0,1,2,3,4],[0,2,3,4,1],[0,3,4,1,2]],\"element\":2}。請依 [element,one,inverse,found,finite_theorem_step] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 2, 是, finite_injective_multiplication_implies_surjective)","explanation_zh":"Wave4第三層核對=(2, 2, 2, 是, finite_injective_multiplication_implies_surjective)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-finite-inverse","seed":202610291084,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,2,4,0,3],[2,4,3,1,0],[3,0,1,4,2],[4,3,0,2,1]],"multiplication":[[0,0,0,0,0],[0,4,1,2,3],[0,1,2,3,4],[0,2,3,4,1],[0,3,4,1,2]],"element":2,"domain":"Complete_finite_integral_domain","operation":"integraldomain-finite-inverse","semantic_template":"qf-mod29-integraldomain-finite-inverse/{\"addition\":[[0,1,2,3,4],[1,2,4,0,3],[2,4,3,1,0],[3,0,1,4,2],[4,3,0,2,1]],\"element\":2,\"multiplication\":[[0,0,0,0,0],[0,4,1,2,3],[0,1,2,3,4],[0,2,3,4,1],[0,3,4,1,2]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0085","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-finite-inverse；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,4,0,2,3],[2,0,3,4,1],[3,2,4,1,0],[4,3,1,0,2]],\"multiplication\":[[0,0,0,0,0],[0,2,1,4,3],[0,1,2,3,4],[0,4,3,1,2],[0,3,4,2,1]],\"element\":3}。請依 [element,one,inverse,found,finite_theorem_step] 欄序回答。","answer":{"kind":"expression","value":"(3, 2, 4, 是, finite_injective_multiplication_implies_surjective)","explanation_zh":"Wave4第三層核對=(3, 2, 4, 是, finite_injective_multiplication_implies_surjective)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-finite-inverse","seed":202610291085,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,4,0,2,3],[2,0,3,4,1],[3,2,4,1,0],[4,3,1,0,2]],"multiplication":[[0,0,0,0,0],[0,2,1,4,3],[0,1,2,3,4],[0,4,3,1,2],[0,3,4,2,1]],"element":3,"domain":"Complete_finite_integral_domain","operation":"integraldomain-finite-inverse","semantic_template":"qf-mod29-integraldomain-finite-inverse/{\"addition\":[[0,1,2,3,4],[1,4,0,2,3],[2,0,3,4,1],[3,2,4,1,0],[4,3,1,0,2]],\"element\":3,\"multiplication\":[[0,0,0,0,0],[0,2,1,4,3],[0,1,2,3,4],[0,4,3,1,2],[0,3,4,2,1]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0086","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-finite-inverse；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"multiplication\":[[0,0,0,0,0],[0,3,1,4,2],[0,1,2,3,4],[0,4,3,2,1],[0,2,4,1,3]],\"element\":1}。請依 [element,one,inverse,found,finite_theorem_step] 欄序回答。","answer":{"kind":"expression","value":"(1, 2, 4, 是, finite_injective_multiplication_implies_surjective)","explanation_zh":"Wave4第三層核對=(1, 2, 4, 是, finite_injective_multiplication_implies_surjective)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-finite-inverse","seed":202610291086,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"multiplication":[[0,0,0,0,0],[0,3,1,4,2],[0,1,2,3,4],[0,4,3,2,1],[0,2,4,1,3]],"element":1,"domain":"Complete_finite_integral_domain","operation":"integraldomain-finite-inverse","semantic_template":"qf-mod29-integraldomain-finite-inverse/{\"addition\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"element\":1,\"multiplication\":[[0,0,0,0,0],[0,3,1,4,2],[0,1,2,3,4],[0,4,3,2,1],[0,2,4,1,3]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0087","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-finite-inverse；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,3,0,4,2],[2,0,4,1,3],[3,4,1,2,0],[4,2,3,0,1]],\"multiplication\":[[0,0,0,0,0],[0,2,1,4,3],[0,1,2,3,4],[0,4,3,1,2],[0,3,4,2,1]],\"element\":1}。請依 [element,one,inverse,found,finite_theorem_step] 欄序回答。","answer":{"kind":"expression","value":"(1, 2, 1, 是, finite_injective_multiplication_implies_surjective)","explanation_zh":"Wave4第三層核對=(1, 2, 1, 是, finite_injective_multiplication_implies_surjective)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-finite-inverse","seed":202610291087,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,3,0,4,2],[2,0,4,1,3],[3,4,1,2,0],[4,2,3,0,1]],"multiplication":[[0,0,0,0,0],[0,2,1,4,3],[0,1,2,3,4],[0,4,3,1,2],[0,3,4,2,1]],"element":1,"domain":"Complete_finite_integral_domain","operation":"integraldomain-finite-inverse","semantic_template":"qf-mod29-integraldomain-finite-inverse/{\"addition\":[[0,1,2,3,4],[1,3,0,4,2],[2,0,4,1,3],[3,4,1,2,0],[4,2,3,0,1]],\"element\":1,\"multiplication\":[[0,0,0,0,0],[0,2,1,4,3],[0,1,2,3,4],[0,4,3,1,2],[0,3,4,2,1]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0088","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-finite-inverse；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,4,0,2,3],[2,0,3,4,1],[3,2,4,1,0],[4,3,1,0,2]],\"multiplication\":[[0,0,0,0,0],[0,4,3,1,2],[0,3,4,2,1],[0,1,2,3,4],[0,2,1,4,3]],\"element\":3}。請依 [element,one,inverse,found,finite_theorem_step] 欄序回答。","answer":{"kind":"expression","value":"(3, 3, 3, 是, finite_injective_multiplication_implies_surjective)","explanation_zh":"Wave4第三層核對=(3, 3, 3, 是, finite_injective_multiplication_implies_surjective)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-finite-inverse","seed":202610291088,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,4,0,2,3],[2,0,3,4,1],[3,2,4,1,0],[4,3,1,0,2]],"multiplication":[[0,0,0,0,0],[0,4,3,1,2],[0,3,4,2,1],[0,1,2,3,4],[0,2,1,4,3]],"element":3,"domain":"Complete_finite_integral_domain","operation":"integraldomain-finite-inverse","semantic_template":"qf-mod29-integraldomain-finite-inverse/{\"addition\":[[0,1,2,3,4],[1,4,0,2,3],[2,0,3,4,1],[3,2,4,1,0],[4,3,1,0,2]],\"element\":3,\"multiplication\":[[0,0,0,0,0],[0,4,3,1,2],[0,3,4,2,1],[0,1,2,3,4],[0,2,1,4,3]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0089","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-finite-inverse；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"multiplication\":[[0,0,0,0,0],[0,2,4,1,3],[0,4,3,2,1],[0,1,2,3,4],[0,3,1,4,2]],\"element\":1}。請依 [element,one,inverse,found,finite_theorem_step] 欄序回答。","answer":{"kind":"expression","value":"(1, 3, 4, 是, finite_injective_multiplication_implies_surjective)","explanation_zh":"Wave4第三層核對=(1, 3, 4, 是, finite_injective_multiplication_implies_surjective)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-finite-inverse","seed":202610291089,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"multiplication":[[0,0,0,0,0],[0,2,4,1,3],[0,4,3,2,1],[0,1,2,3,4],[0,3,1,4,2]],"element":1,"domain":"Complete_finite_integral_domain","operation":"integraldomain-finite-inverse","semantic_template":"qf-mod29-integraldomain-finite-inverse/{\"addition\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"element\":1,\"multiplication\":[[0,0,0,0,0],[0,2,4,1,3],[0,4,3,2,1],[0,1,2,3,4],[0,3,1,4,2]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0090","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-finite-inverse；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,3,0,4,2],[2,0,4,1,3],[3,4,1,2,0],[4,2,3,0,1]],\"multiplication\":[[0,0,0,0,0],[0,4,3,1,2],[0,3,4,2,1],[0,1,2,3,4],[0,2,1,4,3]],\"element\":1}。請依 [element,one,inverse,found,finite_theorem_step] 欄序回答。","answer":{"kind":"expression","value":"(1, 3, 2, 是, finite_injective_multiplication_implies_surjective)","explanation_zh":"Wave4第三層核對=(1, 3, 2, 是, finite_injective_multiplication_implies_surjective)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-finite-inverse","seed":202610291090,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,3,0,4,2],[2,0,4,1,3],[3,4,1,2,0],[4,2,3,0,1]],"multiplication":[[0,0,0,0,0],[0,4,3,1,2],[0,3,4,2,1],[0,1,2,3,4],[0,2,1,4,3]],"element":1,"domain":"Complete_finite_integral_domain","operation":"integraldomain-finite-inverse","semantic_template":"qf-mod29-integraldomain-finite-inverse/{\"addition\":[[0,1,2,3,4],[1,3,0,4,2],[2,0,4,1,3],[3,4,1,2,0],[4,2,3,0,1]],\"element\":1,\"multiplication\":[[0,0,0,0,0],[0,4,3,1,2],[0,3,4,2,1],[0,1,2,3,4],[0,2,1,4,3]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0091","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-finite-inverse；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,4,3,0,2],[2,3,1,4,0],[3,0,4,2,1],[4,2,0,1,3]],\"multiplication\":[[0,0,0,0,0],[0,3,4,1,2],[0,4,1,2,3],[0,1,2,3,4],[0,2,3,4,1]],\"element\":1}。請依 [element,one,inverse,found,finite_theorem_step] 欄序回答。","answer":{"kind":"expression","value":"(1, 3, 1, 是, finite_injective_multiplication_implies_surjective)","explanation_zh":"Wave4第三層核對=(1, 3, 1, 是, finite_injective_multiplication_implies_surjective)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-finite-inverse","seed":202610291091,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,4,3,0,2],[2,3,1,4,0],[3,0,4,2,1],[4,2,0,1,3]],"multiplication":[[0,0,0,0,0],[0,3,4,1,2],[0,4,1,2,3],[0,1,2,3,4],[0,2,3,4,1]],"element":1,"domain":"Complete_finite_integral_domain","operation":"integraldomain-finite-inverse","semantic_template":"qf-mod29-integraldomain-finite-inverse/{\"addition\":[[0,1,2,3,4],[1,4,3,0,2],[2,3,1,4,0],[3,0,4,2,1],[4,2,0,1,3]],\"element\":1,\"multiplication\":[[0,0,0,0,0],[0,3,4,1,2],[0,4,1,2,3],[0,1,2,3,4],[0,2,3,4,1]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0092","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-finite-inverse；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],\"multiplication\":[[0,0,0,0,0],[0,2,4,1,3],[0,4,3,2,1],[0,1,2,3,4],[0,3,1,4,2]],\"element\":3}。請依 [element,one,inverse,found,finite_theorem_step] 欄序回答。","answer":{"kind":"expression","value":"(3, 3, 3, 是, finite_injective_multiplication_implies_surjective)","explanation_zh":"Wave4第三層核對=(3, 3, 3, 是, finite_injective_multiplication_implies_surjective)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-finite-inverse","seed":202610291092,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],"multiplication":[[0,0,0,0,0],[0,2,4,1,3],[0,4,3,2,1],[0,1,2,3,4],[0,3,1,4,2]],"element":3,"domain":"Complete_finite_integral_domain","operation":"integraldomain-finite-inverse","semantic_template":"qf-mod29-integraldomain-finite-inverse/{\"addition\":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],\"element\":3,\"multiplication\":[[0,0,0,0,0],[0,2,4,1,3],[0,4,3,2,1],[0,1,2,3,4],[0,3,1,4,2]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0093","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-finite-inverse；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,2,4,0,3],[2,4,3,1,0],[3,0,1,4,2],[4,3,0,2,1]],\"multiplication\":[[0,0,0,0,0],[0,3,4,1,2],[0,4,1,2,3],[0,1,2,3,4],[0,2,3,4,1]],\"element\":4}。請依 [element,one,inverse,found,finite_theorem_step] 欄序回答。","answer":{"kind":"expression","value":"(4, 3, 2, 是, finite_injective_multiplication_implies_surjective)","explanation_zh":"Wave4第三層核對=(4, 3, 2, 是, finite_injective_multiplication_implies_surjective)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-finite-inverse","seed":202610291093,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,2,4,0,3],[2,4,3,1,0],[3,0,1,4,2],[4,3,0,2,1]],"multiplication":[[0,0,0,0,0],[0,3,4,1,2],[0,4,1,2,3],[0,1,2,3,4],[0,2,3,4,1]],"element":4,"domain":"Complete_finite_integral_domain","operation":"integraldomain-finite-inverse","semantic_template":"qf-mod29-integraldomain-finite-inverse/{\"addition\":[[0,1,2,3,4],[1,2,4,0,3],[2,4,3,1,0],[3,0,1,4,2],[4,3,0,2,1]],\"element\":4,\"multiplication\":[[0,0,0,0,0],[0,3,4,1,2],[0,4,1,2,3],[0,1,2,3,4],[0,2,3,4,1]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0094","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-finite-inverse；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,3,0,4,2],[2,0,4,1,3],[3,4,1,2,0],[4,2,3,0,1]],\"multiplication\":[[0,0,0,0,0],[0,3,4,2,1],[0,4,3,1,2],[0,2,1,4,3],[0,1,2,3,4]],\"element\":2}。請依 [element,one,inverse,found,finite_theorem_step] 欄序回答。","answer":{"kind":"expression","value":"(2, 4, 1, 是, finite_injective_multiplication_implies_surjective)","explanation_zh":"Wave4第三層核對=(2, 4, 1, 是, finite_injective_multiplication_implies_surjective)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-finite-inverse","seed":202610291094,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,3,0,4,2],[2,0,4,1,3],[3,4,1,2,0],[4,2,3,0,1]],"multiplication":[[0,0,0,0,0],[0,3,4,2,1],[0,4,3,1,2],[0,2,1,4,3],[0,1,2,3,4]],"element":2,"domain":"Complete_finite_integral_domain","operation":"integraldomain-finite-inverse","semantic_template":"qf-mod29-integraldomain-finite-inverse/{\"addition\":[[0,1,2,3,4],[1,3,0,4,2],[2,0,4,1,3],[3,4,1,2,0],[4,2,3,0,1]],\"element\":2,\"multiplication\":[[0,0,0,0,0],[0,3,4,2,1],[0,4,3,1,2],[0,2,1,4,3],[0,1,2,3,4]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0095","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-finite-inverse；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,2,4,0,3],[2,4,3,1,0],[3,0,1,4,2],[4,3,0,2,1]],\"multiplication\":[[0,0,0,0,0],[0,2,3,4,1],[0,3,4,1,2],[0,4,1,2,3],[0,1,2,3,4]],\"element\":2}。請依 [element,one,inverse,found,finite_theorem_step] 欄序回答。","answer":{"kind":"expression","value":"(2, 4, 2, 是, finite_injective_multiplication_implies_surjective)","explanation_zh":"Wave4第三層核對=(2, 4, 2, 是, finite_injective_multiplication_implies_surjective)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-finite-inverse","seed":202610291095,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,2,4,0,3],[2,4,3,1,0],[3,0,1,4,2],[4,3,0,2,1]],"multiplication":[[0,0,0,0,0],[0,2,3,4,1],[0,3,4,1,2],[0,4,1,2,3],[0,1,2,3,4]],"element":2,"domain":"Complete_finite_integral_domain","operation":"integraldomain-finite-inverse","semantic_template":"qf-mod29-integraldomain-finite-inverse/{\"addition\":[[0,1,2,3,4],[1,2,4,0,3],[2,4,3,1,0],[3,0,1,4,2],[4,3,0,2,1]],\"element\":2,\"multiplication\":[[0,0,0,0,0],[0,2,3,4,1],[0,3,4,1,2],[0,4,1,2,3],[0,1,2,3,4]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0096","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-finite-inverse；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,4,0,2,3],[2,0,3,4,1],[3,2,4,1,0],[4,3,1,0,2]],\"multiplication\":[[0,0,0,0,0],[0,3,4,2,1],[0,4,3,1,2],[0,2,1,4,3],[0,1,2,3,4]],\"element\":4}。請依 [element,one,inverse,found,finite_theorem_step] 欄序回答。","answer":{"kind":"expression","value":"(4, 4, 4, 是, finite_injective_multiplication_implies_surjective)","explanation_zh":"Wave4第三層核對=(4, 4, 4, 是, finite_injective_multiplication_implies_surjective)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-finite-inverse","seed":202610291096,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,4,0,2,3],[2,0,3,4,1],[3,2,4,1,0],[4,3,1,0,2]],"multiplication":[[0,0,0,0,0],[0,3,4,2,1],[0,4,3,1,2],[0,2,1,4,3],[0,1,2,3,4]],"element":4,"domain":"Complete_finite_integral_domain","operation":"integraldomain-finite-inverse","semantic_template":"qf-mod29-integraldomain-finite-inverse/{\"addition\":[[0,1,2,3,4],[1,4,0,2,3],[2,0,3,4,1],[3,2,4,1,0],[4,3,1,0,2]],\"element\":4,\"multiplication\":[[0,0,0,0,0],[0,3,4,2,1],[0,4,3,1,2],[0,2,1,4,3],[0,1,2,3,4]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0097","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-finite-inverse；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],\"multiplication\":[[0,0,0,0,0],[0,4,3,2,1],[0,3,1,4,2],[0,2,4,1,3],[0,1,2,3,4]],\"element\":2}。請依 [element,one,inverse,found,finite_theorem_step] 欄序回答。","answer":{"kind":"expression","value":"(2, 4, 3, 是, finite_injective_multiplication_implies_surjective)","explanation_zh":"Wave4第三層核對=(2, 4, 3, 是, finite_injective_multiplication_implies_surjective)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-finite-inverse","seed":202610291097,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],"multiplication":[[0,0,0,0,0],[0,4,3,2,1],[0,3,1,4,2],[0,2,4,1,3],[0,1,2,3,4]],"element":2,"domain":"Complete_finite_integral_domain","operation":"integraldomain-finite-inverse","semantic_template":"qf-mod29-integraldomain-finite-inverse/{\"addition\":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],\"element\":2,\"multiplication\":[[0,0,0,0,0],[0,4,3,2,1],[0,3,1,4,2],[0,2,4,1,3],[0,1,2,3,4]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0098","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-finite-inverse；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,4,3,0,2],[2,3,1,4,0],[3,0,4,2,1],[4,2,0,1,3]],\"multiplication\":[[0,0,0,0,0],[0,2,3,4,1],[0,3,4,1,2],[0,4,1,2,3],[0,1,2,3,4]],\"element\":1}。請依 [element,one,inverse,found,finite_theorem_step] 欄序回答。","answer":{"kind":"expression","value":"(1, 4, 3, 是, finite_injective_multiplication_implies_surjective)","explanation_zh":"Wave4第三層核對=(1, 4, 3, 是, finite_injective_multiplication_implies_surjective)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-finite-inverse","seed":202610291098,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,4,3,0,2],[2,3,1,4,0],[3,0,4,2,1],[4,2,0,1,3]],"multiplication":[[0,0,0,0,0],[0,2,3,4,1],[0,3,4,1,2],[0,4,1,2,3],[0,1,2,3,4]],"element":1,"domain":"Complete_finite_integral_domain","operation":"integraldomain-finite-inverse","semantic_template":"qf-mod29-integraldomain-finite-inverse/{\"addition\":[[0,1,2,3,4],[1,4,3,0,2],[2,3,1,4,0],[3,0,4,2,1],[4,2,0,1,3]],\"element\":1,\"multiplication\":[[0,0,0,0,0],[0,2,3,4,1],[0,3,4,1,2],[0,4,1,2,3],[0,1,2,3,4]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0099","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-finite-inverse；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"multiplication\":[[0,0,0,0,0],[0,4,3,2,1],[0,3,1,4,2],[0,2,4,1,3],[0,1,2,3,4]],\"element\":1}。請依 [element,one,inverse,found,finite_theorem_step] 欄序回答。","answer":{"kind":"expression","value":"(1, 4, 1, 是, finite_injective_multiplication_implies_surjective)","explanation_zh":"Wave4第三層核對=(1, 4, 1, 是, finite_injective_multiplication_implies_surjective)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-finite-inverse","seed":202610291099,"parameters":{"n":5,"addition":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"multiplication":[[0,0,0,0,0],[0,4,3,2,1],[0,3,1,4,2],[0,2,4,1,3],[0,1,2,3,4]],"element":1,"domain":"Complete_finite_integral_domain","operation":"integraldomain-finite-inverse","semantic_template":"qf-mod29-integraldomain-finite-inverse/{\"addition\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"element\":1,\"multiplication\":[[0,0,0,0,0],[0,4,3,2,1],[0,3,1,4,2],[0,2,4,1,3],[0,1,2,3,4]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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-r01-finite-integral-domain-0100","batch_id":"batch-20261014-advance-20261029-r01-finite-integral-domain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-integral-domain"],"title_zh":"有限整環","stem_zh":"同一完整有限交換含一環表。操作=integraldomain-finite-inverse；資料={\"n\":5,\"addition\":[[2,0,3,4,1],[0,1,2,3,4],[3,2,4,1,0],[4,3,1,0,2],[1,4,0,2,3]],\"multiplication\":[[0,1,2,3,4],[1,1,1,1,1],[2,1,4,0,3],[3,1,0,4,2],[4,1,3,2,0]],\"element\":0}。請依 [element,one,inverse,found,finite_theorem_step] 欄序回答。","answer":{"kind":"expression","value":"(0, 0, 0, 是, finite_injective_multiplication_implies_surjective)","explanation_zh":"Wave4第三層核對=(0, 0, 0, 是, finite_injective_multiplication_implies_surjective)；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；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限整環"]},"variant":{"family_id":"qf-mod29-integraldomain-finite-inverse","seed":202610291100,"parameters":{"n":5,"addition":[[2,0,3,4,1],[0,1,2,3,4],[3,2,4,1,0],[4,3,1,0,2],[1,4,0,2,3]],"multiplication":[[0,1,2,3,4],[1,1,1,1,1],[2,1,4,0,3],[3,1,0,4,2],[4,1,3,2,0]],"element":0,"domain":"Complete_finite_integral_domain","operation":"integraldomain-finite-inverse","semantic_template":"qf-mod29-integraldomain-finite-inverse/{\"addition\":[[2,0,3,4,1],[0,1,2,3,4],[3,2,4,1,0],[4,3,1,0,2],[1,4,0,2,3]],\"element\":0,\"multiplication\":[[0,1,2,3,4],[1,1,1,1,1],[2,1,4,0,3],[3,1,0,4,2],[4,1,3,2,0]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、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"}}