{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0001","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-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],[0,1,2,3,4],[0,2,4,1,3],[0,3,1,4,2],[0,4,3,2,1]]}。請依 [n,zero,one,missing_inverses,field] 欄序回答。","answer":{"kind":"expression","value":"(5, 0, 1, [], 是)","explanation_zh":"Wave4第四層核對=(5, 0, 1, [], 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-qualification","seed":202610301001,"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_field_table","operation":"field-qualification","semantic_template":"qf-mod30-field-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],[0,1,2,3,4],[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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0002","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-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,missing_inverses,field] 欄序回答。","answer":{"kind":"expression","value":"(5, 0, 1, [], 是)","explanation_zh":"Wave4第四層核對=(5, 0, 1, [], 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-qualification","seed":202610301002,"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_field_table","operation":"field-qualification","semantic_template":"qf-mod30-field-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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0003","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-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,missing_inverses,field] 欄序回答。","answer":{"kind":"expression","value":"(5, 0, 1, [], 是)","explanation_zh":"Wave4第四層核對=(5, 0, 1, [], 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-qualification","seed":202610301003,"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_field_table","operation":"field-qualification","semantic_template":"qf-mod30-field-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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0004","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-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,missing_inverses,field] 欄序回答。","answer":{"kind":"expression","value":"(5, 0, 1, [], 是)","explanation_zh":"Wave4第四層核對=(5, 0, 1, [], 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-qualification","seed":202610301004,"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_field_table","operation":"field-qualification","semantic_template":"qf-mod30-field-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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0005","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-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,missing_inverses,field] 欄序回答。","answer":{"kind":"expression","value":"(5, 0, 1, [], 是)","explanation_zh":"Wave4第四層核對=(5, 0, 1, [], 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-qualification","seed":202610301005,"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_field_table","operation":"field-qualification","semantic_template":"qf-mod30-field-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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0006","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-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],[0,1,2,3,4],[0,2,1,4,3],[0,3,4,2,1],[0,4,3,1,2]]}。請依 [n,zero,one,missing_inverses,field] 欄序回答。","answer":{"kind":"expression","value":"(5, 0, 1, [], 是)","explanation_zh":"Wave4第四層核對=(5, 0, 1, [], 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-qualification","seed":202610301006,"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_field_table","operation":"field-qualification","semantic_template":"qf-mod30-field-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],[0,1,2,3,4],[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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0007","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-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,missing_inverses,field] 欄序回答。","answer":{"kind":"expression","value":"(5, 0, 2, [], 是)","explanation_zh":"Wave4第四層核對=(5, 0, 2, [], 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-qualification","seed":202610301007,"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_field_table","operation":"field-qualification","semantic_template":"qf-mod30-field-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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0008","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-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,missing_inverses,field] 欄序回答。","answer":{"kind":"expression","value":"(5, 0, 2, [], 是)","explanation_zh":"Wave4第四層核對=(5, 0, 2, [], 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-qualification","seed":202610301008,"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_field_table","operation":"field-qualification","semantic_template":"qf-mod30-field-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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0009","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-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,missing_inverses,field] 欄序回答。","answer":{"kind":"expression","value":"(5, 0, 2, [], 是)","explanation_zh":"Wave4第四層核對=(5, 0, 2, [], 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-qualification","seed":202610301009,"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_field_table","operation":"field-qualification","semantic_template":"qf-mod30-field-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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0010","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-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,missing_inverses,field] 欄序回答。","answer":{"kind":"expression","value":"(5, 0, 2, [], 是)","explanation_zh":"Wave4第四層核對=(5, 0, 2, [], 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-qualification","seed":202610301010,"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_field_table","operation":"field-qualification","semantic_template":"qf-mod30-field-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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0011","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-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],[0,3,1,4,2],[0,1,2,3,4],[0,4,3,2,1],[0,2,4,1,3]]}。請依 [n,zero,one,missing_inverses,field] 欄序回答。","answer":{"kind":"expression","value":"(5, 0, 2, [], 是)","explanation_zh":"Wave4第四層核對=(5, 0, 2, [], 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-qualification","seed":202610301011,"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_field_table","operation":"field-qualification","semantic_template":"qf-mod30-field-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],[0,3,1,4,2],[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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0012","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-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,missing_inverses,field] 欄序回答。","answer":{"kind":"expression","value":"(5, 0, 2, [], 是)","explanation_zh":"Wave4第四層核對=(5, 0, 2, [], 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-qualification","seed":202610301012,"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_field_table","operation":"field-qualification","semantic_template":"qf-mod30-field-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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0013","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-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,missing_inverses,field] 欄序回答。","answer":{"kind":"expression","value":"(5, 0, 3, [], 是)","explanation_zh":"Wave4第四層核對=(5, 0, 3, [], 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-qualification","seed":202610301013,"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_field_table","operation":"field-qualification","semantic_template":"qf-mod30-field-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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0014","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-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,missing_inverses,field] 欄序回答。","answer":{"kind":"expression","value":"(5, 0, 3, [], 是)","explanation_zh":"Wave4第四層核對=(5, 0, 3, [], 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-qualification","seed":202610301014,"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_field_table","operation":"field-qualification","semantic_template":"qf-mod30-field-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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0015","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-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,missing_inverses,field] 欄序回答。","answer":{"kind":"expression","value":"(5, 0, 3, [], 是)","explanation_zh":"Wave4第四層核對=(5, 0, 3, [], 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-qualification","seed":202610301015,"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_field_table","operation":"field-qualification","semantic_template":"qf-mod30-field-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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0016","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-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],[0,3,4,1,2],[0,4,1,2,3],[0,1,2,3,4],[0,2,3,4,1]]}。請依 [n,zero,one,missing_inverses,field] 欄序回答。","answer":{"kind":"expression","value":"(5, 0, 3, [], 是)","explanation_zh":"Wave4第四層核對=(5, 0, 3, [], 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-qualification","seed":202610301016,"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_field_table","operation":"field-qualification","semantic_template":"qf-mod30-field-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],[0,3,4,1,2],[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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0017","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-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,missing_inverses,field] 欄序回答。","answer":{"kind":"expression","value":"(5, 0, 3, [], 是)","explanation_zh":"Wave4第四層核對=(5, 0, 3, [], 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-qualification","seed":202610301017,"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_field_table","operation":"field-qualification","semantic_template":"qf-mod30-field-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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0018","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-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,missing_inverses,field] 欄序回答。","answer":{"kind":"expression","value":"(5, 0, 3, [], 是)","explanation_zh":"Wave4第四層核對=(5, 0, 3, [], 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-qualification","seed":202610301018,"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_field_table","operation":"field-qualification","semantic_template":"qf-mod30-field-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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0019","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-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,missing_inverses,field] 欄序回答。","answer":{"kind":"expression","value":"(5, 0, 4, [], 是)","explanation_zh":"Wave4第四層核對=(5, 0, 4, [], 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-qualification","seed":202610301019,"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_field_table","operation":"field-qualification","semantic_template":"qf-mod30-field-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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0020","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-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,missing_inverses,field] 欄序回答。","answer":{"kind":"expression","value":"(5, 0, 4, [], 是)","explanation_zh":"Wave4第四層核對=(5, 0, 4, [], 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-qualification","seed":202610301020,"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_field_table","operation":"field-qualification","semantic_template":"qf-mod30-field-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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0021","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-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],[0,3,4,2,1],[0,4,3,1,2],[0,2,1,4,3],[0,1,2,3,4]]}。請依 [n,zero,one,missing_inverses,field] 欄序回答。","answer":{"kind":"expression","value":"(5, 0, 4, [], 是)","explanation_zh":"Wave4第四層核對=(5, 0, 4, [], 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-qualification","seed":202610301021,"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_field_table","operation":"field-qualification","semantic_template":"qf-mod30-field-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],[0,3,4,2,1],[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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0022","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-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,missing_inverses,field] 欄序回答。","answer":{"kind":"expression","value":"(5, 0, 4, [], 是)","explanation_zh":"Wave4第四層核對=(5, 0, 4, [], 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-qualification","seed":202610301022,"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_field_table","operation":"field-qualification","semantic_template":"qf-mod30-field-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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0023","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-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,missing_inverses,field] 欄序回答。","answer":{"kind":"expression","value":"(5, 0, 4, [], 是)","explanation_zh":"Wave4第四層核對=(5, 0, 4, [], 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-qualification","seed":202610301023,"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_field_table","operation":"field-qualification","semantic_template":"qf-mod30-field-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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0024","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-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,missing_inverses,field] 欄序回答。","answer":{"kind":"expression","value":"(5, 0, 4, [], 是)","explanation_zh":"Wave4第四層核對=(5, 0, 4, [], 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-qualification","seed":202610301024,"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_field_table","operation":"field-qualification","semantic_template":"qf-mod30-field-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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0025","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-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,missing_inverses,field] 欄序回答。","answer":{"kind":"expression","value":"(5, 1, 0, [], 是)","explanation_zh":"Wave4第四層核對=(5, 1, 0, [], 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-qualification","seed":202610301025,"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_field_table","operation":"field-qualification","semantic_template":"qf-mod30-field-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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0026","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-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] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, 1, 是)","explanation_zh":"Wave4第四層核對=(1, 1, 1, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-inverse","seed":202610301026,"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_field_table","operation":"field-inverse","semantic_template":"qf-mod30-field-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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0027","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-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] 欄序回答。","answer":{"kind":"expression","value":"(2, 1, 4, 是)","explanation_zh":"Wave4第四層核對=(2, 1, 4, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-inverse","seed":202610301027,"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_field_table","operation":"field-inverse","semantic_template":"qf-mod30-field-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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0028","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-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] 欄序回答。","answer":{"kind":"expression","value":"(2, 1, 3, 是)","explanation_zh":"Wave4第四層核對=(2, 1, 3, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-inverse","seed":202610301028,"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_field_table","operation":"field-inverse","semantic_template":"qf-mod30-field-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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0029","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-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] 欄序回答。","answer":{"kind":"expression","value":"(2, 1, 2, 是)","explanation_zh":"Wave4第四層核對=(2, 1, 2, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-inverse","seed":202610301029,"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_field_table","operation":"field-inverse","semantic_template":"qf-mod30-field-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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0030","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-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] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, 1, 是)","explanation_zh":"Wave4第四層核對=(1, 1, 1, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-inverse","seed":202610301030,"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_field_table","operation":"field-inverse","semantic_template":"qf-mod30-field-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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0031","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-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] 欄序回答。","answer":{"kind":"expression","value":"(4, 1, 3, 是)","explanation_zh":"Wave4第四層核對=(4, 1, 3, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-inverse","seed":202610301031,"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_field_table","operation":"field-inverse","semantic_template":"qf-mod30-field-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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0032","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-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] 欄序回答。","answer":{"kind":"expression","value":"(3, 2, 1, 是)","explanation_zh":"Wave4第四層核對=(3, 2, 1, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-inverse","seed":202610301032,"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_field_table","operation":"field-inverse","semantic_template":"qf-mod30-field-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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0033","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-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] 欄序回答。","answer":{"kind":"expression","value":"(3, 2, 3, 是)","explanation_zh":"Wave4第四層核對=(3, 2, 3, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-inverse","seed":202610301033,"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_field_table","operation":"field-inverse","semantic_template":"qf-mod30-field-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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0034","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-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] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 2, 是)","explanation_zh":"Wave4第四層核對=(2, 2, 2, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-inverse","seed":202610301034,"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_field_table","operation":"field-inverse","semantic_template":"qf-mod30-field-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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0035","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-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] 欄序回答。","answer":{"kind":"expression","value":"(3, 2, 4, 是)","explanation_zh":"Wave4第四層核對=(3, 2, 4, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-inverse","seed":202610301035,"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_field_table","operation":"field-inverse","semantic_template":"qf-mod30-field-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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0036","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-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] 欄序回答。","answer":{"kind":"expression","value":"(1, 2, 4, 是)","explanation_zh":"Wave4第四層核對=(1, 2, 4, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-inverse","seed":202610301036,"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_field_table","operation":"field-inverse","semantic_template":"qf-mod30-field-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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0037","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-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] 欄序回答。","answer":{"kind":"expression","value":"(1, 2, 1, 是)","explanation_zh":"Wave4第四層核對=(1, 2, 1, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-inverse","seed":202610301037,"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_field_table","operation":"field-inverse","semantic_template":"qf-mod30-field-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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0038","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-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] 欄序回答。","answer":{"kind":"expression","value":"(3, 3, 3, 是)","explanation_zh":"Wave4第四層核對=(3, 3, 3, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-inverse","seed":202610301038,"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_field_table","operation":"field-inverse","semantic_template":"qf-mod30-field-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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0039","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-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] 欄序回答。","answer":{"kind":"expression","value":"(1, 3, 4, 是)","explanation_zh":"Wave4第四層核對=(1, 3, 4, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-inverse","seed":202610301039,"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_field_table","operation":"field-inverse","semantic_template":"qf-mod30-field-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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0040","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-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] 欄序回答。","answer":{"kind":"expression","value":"(1, 3, 2, 是)","explanation_zh":"Wave4第四層核對=(1, 3, 2, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-inverse","seed":202610301040,"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_field_table","operation":"field-inverse","semantic_template":"qf-mod30-field-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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0041","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-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] 欄序回答。","answer":{"kind":"expression","value":"(1, 3, 1, 是)","explanation_zh":"Wave4第四層核對=(1, 3, 1, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-inverse","seed":202610301041,"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_field_table","operation":"field-inverse","semantic_template":"qf-mod30-field-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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0042","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-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] 欄序回答。","answer":{"kind":"expression","value":"(3, 3, 3, 是)","explanation_zh":"Wave4第四層核對=(3, 3, 3, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-inverse","seed":202610301042,"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_field_table","operation":"field-inverse","semantic_template":"qf-mod30-field-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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0043","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-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] 欄序回答。","answer":{"kind":"expression","value":"(4, 3, 2, 是)","explanation_zh":"Wave4第四層核對=(4, 3, 2, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-inverse","seed":202610301043,"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_field_table","operation":"field-inverse","semantic_template":"qf-mod30-field-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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0044","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-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] 欄序回答。","answer":{"kind":"expression","value":"(2, 4, 1, 是)","explanation_zh":"Wave4第四層核對=(2, 4, 1, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-inverse","seed":202610301044,"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_field_table","operation":"field-inverse","semantic_template":"qf-mod30-field-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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0045","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-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] 欄序回答。","answer":{"kind":"expression","value":"(2, 4, 2, 是)","explanation_zh":"Wave4第四層核對=(2, 4, 2, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-inverse","seed":202610301045,"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_field_table","operation":"field-inverse","semantic_template":"qf-mod30-field-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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0046","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-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] 欄序回答。","answer":{"kind":"expression","value":"(4, 4, 4, 是)","explanation_zh":"Wave4第四層核對=(4, 4, 4, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-inverse","seed":202610301046,"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_field_table","operation":"field-inverse","semantic_template":"qf-mod30-field-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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0047","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-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] 欄序回答。","answer":{"kind":"expression","value":"(2, 4, 3, 是)","explanation_zh":"Wave4第四層核對=(2, 4, 3, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-inverse","seed":202610301047,"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_field_table","operation":"field-inverse","semantic_template":"qf-mod30-field-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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0048","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-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] 欄序回答。","answer":{"kind":"expression","value":"(1, 4, 3, 是)","explanation_zh":"Wave4第四層核對=(1, 4, 3, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-inverse","seed":202610301048,"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_field_table","operation":"field-inverse","semantic_template":"qf-mod30-field-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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0049","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-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] 欄序回答。","answer":{"kind":"expression","value":"(1, 4, 1, 是)","explanation_zh":"Wave4第四層核對=(1, 4, 1, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-inverse","seed":202610301049,"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_field_table","operation":"field-inverse","semantic_template":"qf-mod30-field-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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0050","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-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] 欄序回答。","answer":{"kind":"expression","value":"(0, 0, 0, 是)","explanation_zh":"Wave4第四層核對=(0, 0, 0, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-inverse","seed":202610301050,"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_field_table","operation":"field-inverse","semantic_template":"qf-mod30-field-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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0051","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-division；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]],\"numerator\":1,\"denominator\":1}。請依 [numerator,denominator,inverse,quotient,check,pass] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, 1, 1, 1, 是)","explanation_zh":"Wave4第四層核對=(1, 1, 1, 1, 1, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-division","seed":202610301051,"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]],"numerator":1,"denominator":1,"domain":"Complete_finite_field_table","operation":"field-division","semantic_template":"qf-mod30-field-division/{\"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]],\"denominator\":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,\"numerator\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0052","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-division；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]],\"numerator\":2,\"denominator\":2}。請依 [numerator,denominator,inverse,quotient,check,pass] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 4, 1, 2, 是)","explanation_zh":"Wave4第四層核對=(2, 2, 4, 1, 2, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-division","seed":202610301052,"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]],"numerator":2,"denominator":2,"domain":"Complete_finite_field_table","operation":"field-division","semantic_template":"qf-mod30-field-division/{\"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]],\"denominator\":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,\"numerator\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0053","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-division；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]],\"numerator\":2,\"denominator\":2}。請依 [numerator,denominator,inverse,quotient,check,pass] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 3, 1, 2, 是)","explanation_zh":"Wave4第四層核對=(2, 2, 3, 1, 2, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-division","seed":202610301053,"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]],"numerator":2,"denominator":2,"domain":"Complete_finite_field_table","operation":"field-division","semantic_template":"qf-mod30-field-division/{\"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]],\"denominator\":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,\"numerator\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0054","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-division；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]],\"numerator\":2,\"denominator\":2}。請依 [numerator,denominator,inverse,quotient,check,pass] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 2, 1, 2, 是)","explanation_zh":"Wave4第四層核對=(2, 2, 2, 1, 2, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-division","seed":202610301054,"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]],"numerator":2,"denominator":2,"domain":"Complete_finite_field_table","operation":"field-division","semantic_template":"qf-mod30-field-division/{\"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]],\"denominator\":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,\"numerator\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0055","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-division；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]],\"numerator\":0,\"denominator\":1}。請依 [numerator,denominator,inverse,quotient,check,pass] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 1, 0, 0, 是)","explanation_zh":"Wave4第四層核對=(0, 1, 1, 0, 0, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-division","seed":202610301055,"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]],"numerator":0,"denominator":1,"domain":"Complete_finite_field_table","operation":"field-division","semantic_template":"qf-mod30-field-division/{\"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]],\"denominator\":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,\"numerator\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0056","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-division；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]],\"numerator\":1,\"denominator\":4}。請依 [numerator,denominator,inverse,quotient,check,pass] 欄序回答。","answer":{"kind":"expression","value":"(1, 4, 3, 3, 1, 是)","explanation_zh":"Wave4第四層核對=(1, 4, 3, 3, 1, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-division","seed":202610301056,"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]],"numerator":1,"denominator":4,"domain":"Complete_finite_field_table","operation":"field-division","semantic_template":"qf-mod30-field-division/{\"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]],\"denominator\":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,\"numerator\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0057","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-division；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]],\"numerator\":1,\"denominator\":3}。請依 [numerator,denominator,inverse,quotient,check,pass] 欄序回答。","answer":{"kind":"expression","value":"(1, 3, 1, 4, 1, 是)","explanation_zh":"Wave4第四層核對=(1, 3, 1, 4, 1, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-division","seed":202610301057,"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]],"numerator":1,"denominator":3,"domain":"Complete_finite_field_table","operation":"field-division","semantic_template":"qf-mod30-field-division/{\"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]],\"denominator\":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,\"numerator\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0058","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-division；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]],\"numerator\":4,\"denominator\":3}。請依 [numerator,denominator,inverse,quotient,check,pass] 欄序回答。","answer":{"kind":"expression","value":"(4, 3, 3, 1, 4, 是)","explanation_zh":"Wave4第四層核對=(4, 3, 3, 1, 4, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-division","seed":202610301058,"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]],"numerator":4,"denominator":3,"domain":"Complete_finite_field_table","operation":"field-division","semantic_template":"qf-mod30-field-division/{\"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]],\"denominator\":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,\"numerator\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0059","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-division；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]],\"numerator\":4,\"denominator\":2}。請依 [numerator,denominator,inverse,quotient,check,pass] 欄序回答。","answer":{"kind":"expression","value":"(4, 2, 2, 4, 4, 是)","explanation_zh":"Wave4第四層核對=(4, 2, 2, 4, 4, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-division","seed":202610301059,"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]],"numerator":4,"denominator":2,"domain":"Complete_finite_field_table","operation":"field-division","semantic_template":"qf-mod30-field-division/{\"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]],\"denominator\":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,\"numerator\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0060","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-division；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]],\"numerator\":0,\"denominator\":3}。請依 [numerator,denominator,inverse,quotient,check,pass] 欄序回答。","answer":{"kind":"expression","value":"(0, 3, 4, 0, 0, 是)","explanation_zh":"Wave4第四層核對=(0, 3, 4, 0, 0, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-division","seed":202610301060,"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]],"numerator":0,"denominator":3,"domain":"Complete_finite_field_table","operation":"field-division","semantic_template":"qf-mod30-field-division/{\"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]],\"denominator\":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,\"numerator\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0061","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-division；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]],\"numerator\":2,\"denominator\":1}。請依 [numerator,denominator,inverse,quotient,check,pass] 欄序回答。","answer":{"kind":"expression","value":"(2, 1, 4, 4, 2, 是)","explanation_zh":"Wave4第四層核對=(2, 1, 4, 4, 2, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-division","seed":202610301061,"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]],"numerator":2,"denominator":1,"domain":"Complete_finite_field_table","operation":"field-division","semantic_template":"qf-mod30-field-division/{\"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]],\"denominator\":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,\"numerator\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0062","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-division；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]],\"numerator\":4,\"denominator\":1}。請依 [numerator,denominator,inverse,quotient,check,pass] 欄序回答。","answer":{"kind":"expression","value":"(4, 1, 1, 3, 4, 是)","explanation_zh":"Wave4第四層核對=(4, 1, 1, 3, 4, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-division","seed":202610301062,"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]],"numerator":4,"denominator":1,"domain":"Complete_finite_field_table","operation":"field-division","semantic_template":"qf-mod30-field-division/{\"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]],\"denominator\":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,\"numerator\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0063","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-division；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]],\"numerator\":2,\"denominator\":3}。請依 [numerator,denominator,inverse,quotient,check,pass] 欄序回答。","answer":{"kind":"expression","value":"(2, 3, 3, 2, 2, 是)","explanation_zh":"Wave4第四層核對=(2, 3, 3, 2, 2, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-division","seed":202610301063,"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]],"numerator":2,"denominator":3,"domain":"Complete_finite_field_table","operation":"field-division","semantic_template":"qf-mod30-field-division/{\"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]],\"denominator\":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,\"numerator\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0064","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-division；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]],\"numerator\":2,\"denominator\":1}。請依 [numerator,denominator,inverse,quotient,check,pass] 欄序回答。","answer":{"kind":"expression","value":"(2, 1, 4, 1, 2, 是)","explanation_zh":"Wave4第四層核對=(2, 1, 4, 1, 2, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-division","seed":202610301064,"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]],"numerator":2,"denominator":1,"domain":"Complete_finite_field_table","operation":"field-division","semantic_template":"qf-mod30-field-division/{\"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]],\"denominator\":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,\"numerator\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0065","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-division；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]],\"numerator\":0,\"denominator\":1}。請依 [numerator,denominator,inverse,quotient,check,pass] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 2, 0, 0, 是)","explanation_zh":"Wave4第四層核對=(0, 1, 2, 0, 0, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-division","seed":202610301065,"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]],"numerator":0,"denominator":1,"domain":"Complete_finite_field_table","operation":"field-division","semantic_template":"qf-mod30-field-division/{\"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]],\"denominator\":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,\"numerator\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0066","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-division；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]],\"numerator\":3,\"denominator\":1}。請依 [numerator,denominator,inverse,quotient,check,pass] 欄序回答。","answer":{"kind":"expression","value":"(3, 1, 1, 1, 3, 是)","explanation_zh":"Wave4第四層核對=(3, 1, 1, 1, 3, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-division","seed":202610301066,"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]],"numerator":3,"denominator":1,"domain":"Complete_finite_field_table","operation":"field-division","semantic_template":"qf-mod30-field-division/{\"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]],\"denominator\":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,\"numerator\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0067","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-division；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]],\"numerator\":4,\"denominator\":3}。請依 [numerator,denominator,inverse,quotient,check,pass] 欄序回答。","answer":{"kind":"expression","value":"(4, 3, 3, 4, 4, 是)","explanation_zh":"Wave4第四層核對=(4, 3, 3, 4, 4, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-division","seed":202610301067,"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]],"numerator":4,"denominator":3,"domain":"Complete_finite_field_table","operation":"field-division","semantic_template":"qf-mod30-field-division/{\"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]],\"denominator\":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,\"numerator\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0068","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-division；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]],\"numerator\":2,\"denominator\":4}。請依 [numerator,denominator,inverse,quotient,check,pass] 欄序回答。","answer":{"kind":"expression","value":"(2, 4, 2, 1, 2, 是)","explanation_zh":"Wave4第四層核對=(2, 4, 2, 1, 2, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-division","seed":202610301068,"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]],"numerator":2,"denominator":4,"domain":"Complete_finite_field_table","operation":"field-division","semantic_template":"qf-mod30-field-division/{\"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]],\"denominator\":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,\"numerator\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0069","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-division；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]],\"numerator\":3,\"denominator\":2}。請依 [numerator,denominator,inverse,quotient,check,pass] 欄序回答。","answer":{"kind":"expression","value":"(3, 2, 1, 2, 3, 是)","explanation_zh":"Wave4第四層核對=(3, 2, 1, 2, 3, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-division","seed":202610301069,"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]],"numerator":3,"denominator":2,"domain":"Complete_finite_field_table","operation":"field-division","semantic_template":"qf-mod30-field-division/{\"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]],\"denominator\":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,\"numerator\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0070","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-division；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]],\"numerator\":0,\"denominator\":2}。請依 [numerator,denominator,inverse,quotient,check,pass] 欄序回答。","answer":{"kind":"expression","value":"(0, 2, 2, 0, 0, 是)","explanation_zh":"Wave4第四層核對=(0, 2, 2, 0, 0, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-division","seed":202610301070,"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]],"numerator":0,"denominator":2,"domain":"Complete_finite_field_table","operation":"field-division","semantic_template":"qf-mod30-field-division/{\"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]],\"denominator\":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,\"numerator\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0071","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-division；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]],\"numerator\":4,\"denominator\":4}。請依 [numerator,denominator,inverse,quotient,check,pass] 欄序回答。","answer":{"kind":"expression","value":"(4, 4, 4, 4, 4, 是)","explanation_zh":"Wave4第四層核對=(4, 4, 4, 4, 4, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-division","seed":202610301071,"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]],"numerator":4,"denominator":4,"domain":"Complete_finite_field_table","operation":"field-division","semantic_template":"qf-mod30-field-division/{\"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]],\"denominator\":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,\"numerator\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0072","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-division；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]],\"numerator\":2,\"denominator\":2}。請依 [numerator,denominator,inverse,quotient,check,pass] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 3, 4, 2, 是)","explanation_zh":"Wave4第四層核對=(2, 2, 3, 4, 2, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-division","seed":202610301072,"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]],"numerator":2,"denominator":2,"domain":"Complete_finite_field_table","operation":"field-division","semantic_template":"qf-mod30-field-division/{\"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]],\"denominator\":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,\"numerator\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0073","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-division；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]],\"numerator\":1,\"denominator\":1}。請依 [numerator,denominator,inverse,quotient,check,pass] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, 3, 4, 1, 是)","explanation_zh":"Wave4第四層核對=(1, 1, 3, 4, 1, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-division","seed":202610301073,"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]],"numerator":1,"denominator":1,"domain":"Complete_finite_field_table","operation":"field-division","semantic_template":"qf-mod30-field-division/{\"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]],\"denominator\":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,\"numerator\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0074","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-division；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]],\"numerator\":1,\"denominator\":1}。請依 [numerator,denominator,inverse,quotient,check,pass] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, 1, 4, 1, 是)","explanation_zh":"Wave4第四層核對=(1, 1, 1, 4, 1, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-division","seed":202610301074,"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]],"numerator":1,"denominator":1,"domain":"Complete_finite_field_table","operation":"field-division","semantic_template":"qf-mod30-field-division/{\"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]],\"denominator\":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,\"numerator\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0075","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-division；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]],\"numerator\":1,\"denominator\":0}。請依 [numerator,denominator,inverse,quotient,check,pass] 欄序回答。","answer":{"kind":"expression","value":"(1, 0, 0, 1, 1, 是)","explanation_zh":"Wave4第四層核對=(1, 0, 0, 1, 1, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-division","seed":202610301075,"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]],"numerator":1,"denominator":0,"domain":"Complete_finite_field_table","operation":"field-division","semantic_template":"qf-mod30-field-division/{\"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]],\"denominator\":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,\"numerator\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0076","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-finite-domain-theorem；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 [n,missing,field,theorem_step,order_boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, [], 是, finite_domain_injective_to_surjective_theorem, field_order_prime_power_not_necessarily_prime)","explanation_zh":"Wave4第四層核對=(5, [], 是, finite_domain_injective_to_surjective_theorem, field_order_prime_power_not_necessarily_prime)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-finite-domain-theorem","seed":202610301076,"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_field_table","operation":"field-finite-domain-theorem","semantic_template":"qf-mod30-field-finite-domain-theorem/{\"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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0077","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-finite-domain-theorem；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,missing,field,theorem_step,order_boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, [], 是, finite_domain_injective_to_surjective_theorem, field_order_prime_power_not_necessarily_prime)","explanation_zh":"Wave4第四層核對=(5, [], 是, finite_domain_injective_to_surjective_theorem, field_order_prime_power_not_necessarily_prime)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-finite-domain-theorem","seed":202610301077,"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_field_table","operation":"field-finite-domain-theorem","semantic_template":"qf-mod30-field-finite-domain-theorem/{\"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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0078","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-finite-domain-theorem；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,missing,field,theorem_step,order_boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, [], 是, finite_domain_injective_to_surjective_theorem, field_order_prime_power_not_necessarily_prime)","explanation_zh":"Wave4第四層核對=(5, [], 是, finite_domain_injective_to_surjective_theorem, field_order_prime_power_not_necessarily_prime)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-finite-domain-theorem","seed":202610301078,"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_field_table","operation":"field-finite-domain-theorem","semantic_template":"qf-mod30-field-finite-domain-theorem/{\"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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0079","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-finite-domain-theorem；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,missing,field,theorem_step,order_boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, [], 是, finite_domain_injective_to_surjective_theorem, field_order_prime_power_not_necessarily_prime)","explanation_zh":"Wave4第四層核對=(5, [], 是, finite_domain_injective_to_surjective_theorem, field_order_prime_power_not_necessarily_prime)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-finite-domain-theorem","seed":202610301079,"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_field_table","operation":"field-finite-domain-theorem","semantic_template":"qf-mod30-field-finite-domain-theorem/{\"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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0080","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-finite-domain-theorem；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,missing,field,theorem_step,order_boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, [], 是, finite_domain_injective_to_surjective_theorem, field_order_prime_power_not_necessarily_prime)","explanation_zh":"Wave4第四層核對=(5, [], 是, finite_domain_injective_to_surjective_theorem, field_order_prime_power_not_necessarily_prime)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-finite-domain-theorem","seed":202610301080,"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_field_table","operation":"field-finite-domain-theorem","semantic_template":"qf-mod30-field-finite-domain-theorem/{\"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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0081","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-finite-domain-theorem；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 [n,missing,field,theorem_step,order_boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, [], 是, finite_domain_injective_to_surjective_theorem, field_order_prime_power_not_necessarily_prime)","explanation_zh":"Wave4第四層核對=(5, [], 是, finite_domain_injective_to_surjective_theorem, field_order_prime_power_not_necessarily_prime)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-finite-domain-theorem","seed":202610301081,"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_field_table","operation":"field-finite-domain-theorem","semantic_template":"qf-mod30-field-finite-domain-theorem/{\"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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0082","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-finite-domain-theorem；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,missing,field,theorem_step,order_boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, [], 是, finite_domain_injective_to_surjective_theorem, field_order_prime_power_not_necessarily_prime)","explanation_zh":"Wave4第四層核對=(5, [], 是, finite_domain_injective_to_surjective_theorem, field_order_prime_power_not_necessarily_prime)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-finite-domain-theorem","seed":202610301082,"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_field_table","operation":"field-finite-domain-theorem","semantic_template":"qf-mod30-field-finite-domain-theorem/{\"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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0083","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-finite-domain-theorem；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,missing,field,theorem_step,order_boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, [], 是, finite_domain_injective_to_surjective_theorem, field_order_prime_power_not_necessarily_prime)","explanation_zh":"Wave4第四層核對=(5, [], 是, finite_domain_injective_to_surjective_theorem, field_order_prime_power_not_necessarily_prime)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-finite-domain-theorem","seed":202610301083,"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_field_table","operation":"field-finite-domain-theorem","semantic_template":"qf-mod30-field-finite-domain-theorem/{\"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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0084","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-finite-domain-theorem；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,missing,field,theorem_step,order_boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, [], 是, finite_domain_injective_to_surjective_theorem, field_order_prime_power_not_necessarily_prime)","explanation_zh":"Wave4第四層核對=(5, [], 是, finite_domain_injective_to_surjective_theorem, field_order_prime_power_not_necessarily_prime)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-finite-domain-theorem","seed":202610301084,"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_field_table","operation":"field-finite-domain-theorem","semantic_template":"qf-mod30-field-finite-domain-theorem/{\"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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0085","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-finite-domain-theorem；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,missing,field,theorem_step,order_boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, [], 是, finite_domain_injective_to_surjective_theorem, field_order_prime_power_not_necessarily_prime)","explanation_zh":"Wave4第四層核對=(5, [], 是, finite_domain_injective_to_surjective_theorem, field_order_prime_power_not_necessarily_prime)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-finite-domain-theorem","seed":202610301085,"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_field_table","operation":"field-finite-domain-theorem","semantic_template":"qf-mod30-field-finite-domain-theorem/{\"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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0086","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-finite-domain-theorem；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 [n,missing,field,theorem_step,order_boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, [], 是, finite_domain_injective_to_surjective_theorem, field_order_prime_power_not_necessarily_prime)","explanation_zh":"Wave4第四層核對=(5, [], 是, finite_domain_injective_to_surjective_theorem, field_order_prime_power_not_necessarily_prime)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-finite-domain-theorem","seed":202610301086,"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_field_table","operation":"field-finite-domain-theorem","semantic_template":"qf-mod30-field-finite-domain-theorem/{\"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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0087","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-finite-domain-theorem；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,missing,field,theorem_step,order_boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, [], 是, finite_domain_injective_to_surjective_theorem, field_order_prime_power_not_necessarily_prime)","explanation_zh":"Wave4第四層核對=(5, [], 是, finite_domain_injective_to_surjective_theorem, field_order_prime_power_not_necessarily_prime)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-finite-domain-theorem","seed":202610301087,"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_field_table","operation":"field-finite-domain-theorem","semantic_template":"qf-mod30-field-finite-domain-theorem/{\"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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0088","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-finite-domain-theorem；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,missing,field,theorem_step,order_boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, [], 是, finite_domain_injective_to_surjective_theorem, field_order_prime_power_not_necessarily_prime)","explanation_zh":"Wave4第四層核對=(5, [], 是, finite_domain_injective_to_surjective_theorem, field_order_prime_power_not_necessarily_prime)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-finite-domain-theorem","seed":202610301088,"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_field_table","operation":"field-finite-domain-theorem","semantic_template":"qf-mod30-field-finite-domain-theorem/{\"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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0089","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-finite-domain-theorem；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,missing,field,theorem_step,order_boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, [], 是, finite_domain_injective_to_surjective_theorem, field_order_prime_power_not_necessarily_prime)","explanation_zh":"Wave4第四層核對=(5, [], 是, finite_domain_injective_to_surjective_theorem, field_order_prime_power_not_necessarily_prime)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-finite-domain-theorem","seed":202610301089,"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_field_table","operation":"field-finite-domain-theorem","semantic_template":"qf-mod30-field-finite-domain-theorem/{\"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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0090","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-finite-domain-theorem；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,missing,field,theorem_step,order_boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, [], 是, finite_domain_injective_to_surjective_theorem, field_order_prime_power_not_necessarily_prime)","explanation_zh":"Wave4第四層核對=(5, [], 是, finite_domain_injective_to_surjective_theorem, field_order_prime_power_not_necessarily_prime)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-finite-domain-theorem","seed":202610301090,"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_field_table","operation":"field-finite-domain-theorem","semantic_template":"qf-mod30-field-finite-domain-theorem/{\"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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0091","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-finite-domain-theorem；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 [n,missing,field,theorem_step,order_boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, [], 是, finite_domain_injective_to_surjective_theorem, field_order_prime_power_not_necessarily_prime)","explanation_zh":"Wave4第四層核對=(5, [], 是, finite_domain_injective_to_surjective_theorem, field_order_prime_power_not_necessarily_prime)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-finite-domain-theorem","seed":202610301091,"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_field_table","operation":"field-finite-domain-theorem","semantic_template":"qf-mod30-field-finite-domain-theorem/{\"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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0092","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-finite-domain-theorem；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,missing,field,theorem_step,order_boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, [], 是, finite_domain_injective_to_surjective_theorem, field_order_prime_power_not_necessarily_prime)","explanation_zh":"Wave4第四層核對=(5, [], 是, finite_domain_injective_to_surjective_theorem, field_order_prime_power_not_necessarily_prime)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-finite-domain-theorem","seed":202610301092,"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_field_table","operation":"field-finite-domain-theorem","semantic_template":"qf-mod30-field-finite-domain-theorem/{\"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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0093","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-finite-domain-theorem；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,missing,field,theorem_step,order_boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, [], 是, finite_domain_injective_to_surjective_theorem, field_order_prime_power_not_necessarily_prime)","explanation_zh":"Wave4第四層核對=(5, [], 是, finite_domain_injective_to_surjective_theorem, field_order_prime_power_not_necessarily_prime)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-finite-domain-theorem","seed":202610301093,"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_field_table","operation":"field-finite-domain-theorem","semantic_template":"qf-mod30-field-finite-domain-theorem/{\"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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0094","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-finite-domain-theorem；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,missing,field,theorem_step,order_boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, [], 是, finite_domain_injective_to_surjective_theorem, field_order_prime_power_not_necessarily_prime)","explanation_zh":"Wave4第四層核對=(5, [], 是, finite_domain_injective_to_surjective_theorem, field_order_prime_power_not_necessarily_prime)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-finite-domain-theorem","seed":202610301094,"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_field_table","operation":"field-finite-domain-theorem","semantic_template":"qf-mod30-field-finite-domain-theorem/{\"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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0095","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-finite-domain-theorem；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,missing,field,theorem_step,order_boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, [], 是, finite_domain_injective_to_surjective_theorem, field_order_prime_power_not_necessarily_prime)","explanation_zh":"Wave4第四層核對=(5, [], 是, finite_domain_injective_to_surjective_theorem, field_order_prime_power_not_necessarily_prime)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-finite-domain-theorem","seed":202610301095,"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_field_table","operation":"field-finite-domain-theorem","semantic_template":"qf-mod30-field-finite-domain-theorem/{\"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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0096","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-finite-domain-theorem；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 [n,missing,field,theorem_step,order_boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, [], 是, finite_domain_injective_to_surjective_theorem, field_order_prime_power_not_necessarily_prime)","explanation_zh":"Wave4第四層核對=(5, [], 是, finite_domain_injective_to_surjective_theorem, field_order_prime_power_not_necessarily_prime)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-finite-domain-theorem","seed":202610301096,"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_field_table","operation":"field-finite-domain-theorem","semantic_template":"qf-mod30-field-finite-domain-theorem/{\"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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0097","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-finite-domain-theorem；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,missing,field,theorem_step,order_boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, [], 是, finite_domain_injective_to_surjective_theorem, field_order_prime_power_not_necessarily_prime)","explanation_zh":"Wave4第四層核對=(5, [], 是, finite_domain_injective_to_surjective_theorem, field_order_prime_power_not_necessarily_prime)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-finite-domain-theorem","seed":202610301097,"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_field_table","operation":"field-finite-domain-theorem","semantic_template":"qf-mod30-field-finite-domain-theorem/{\"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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0098","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-finite-domain-theorem；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,missing,field,theorem_step,order_boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, [], 是, finite_domain_injective_to_surjective_theorem, field_order_prime_power_not_necessarily_prime)","explanation_zh":"Wave4第四層核對=(5, [], 是, finite_domain_injective_to_surjective_theorem, field_order_prime_power_not_necessarily_prime)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-finite-domain-theorem","seed":202610301098,"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_field_table","operation":"field-finite-domain-theorem","semantic_template":"qf-mod30-field-finite-domain-theorem/{\"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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0099","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-finite-domain-theorem；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,missing,field,theorem_step,order_boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, [], 是, finite_domain_injective_to_surjective_theorem, field_order_prime_power_not_necessarily_prime)","explanation_zh":"Wave4第四層核對=(5, [], 是, finite_domain_injective_to_surjective_theorem, field_order_prime_power_not_necessarily_prime)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-finite-domain-theorem","seed":202610301099,"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_field_table","operation":"field-finite-domain-theorem","semantic_template":"qf-mod30-field-finite-domain-theorem/{\"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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r01-finite-field-0100","batch_id":"batch-20261014-advance-20261030-r01-finite-field-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-field"],"title_zh":"有限域","stem_zh":"完整有限交換環雙表與全部非零逆元。操作=field-finite-domain-theorem；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,missing,field,theorem_step,order_boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, [], 是, finite_domain_injective_to_surjective_theorem, field_order_prime_power_not_necessarily_prime)","explanation_zh":"Wave4第四層核對=(5, [], 是, finite_domain_injective_to_surjective_theorem, field_order_prime_power_not_necessarily_prime)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域"]},"variant":{"family_id":"qf-mod30-field-finite-domain-theorem","seed":202610301100,"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_field_table","operation":"field-finite-domain-theorem","semantic_template":"qf-mod30-field-finite-domain-theorem/{\"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":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}