{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0001","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-convex-domain-qualification；資料={\"interval\":[-2,3]}。請依 [interval,ordered,convex,universal_certificate] 欄序回答。","answer":{"kind":"expression","value":"([-2, 3], 是, 是, all segment points remain in the interval)","explanation_zh":"收束波核對=([-2, 3], 是, 是, all segment points remain in the interval)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-convex-domain-qualification","seed":202610252001,"parameters":{"interval":[-2,3],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-convex-domain-qualification","semantic_template":"qf-mod25-convexfn-convex-domain-qualification/{\"interval\":[-2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0002","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-convex-domain-qualification；資料={\"interval\":[-3,4]}。請依 [interval,ordered,convex,universal_certificate] 欄序回答。","answer":{"kind":"expression","value":"([-3, 4], 是, 是, all segment points remain in the interval)","explanation_zh":"收束波核對=([-3, 4], 是, 是, all segment points remain in the interval)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-convex-domain-qualification","seed":202610252002,"parameters":{"interval":[-3,4],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-convex-domain-qualification","semantic_template":"qf-mod25-convexfn-convex-domain-qualification/{\"interval\":[-3,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0003","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-convex-domain-qualification；資料={\"interval\":[-4,5]}。請依 [interval,ordered,convex,universal_certificate] 欄序回答。","answer":{"kind":"expression","value":"([-4, 5], 是, 是, all segment points remain in the interval)","explanation_zh":"收束波核對=([-4, 5], 是, 是, all segment points remain in the interval)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-convex-domain-qualification","seed":202610252003,"parameters":{"interval":[-4,5],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-convex-domain-qualification","semantic_template":"qf-mod25-convexfn-convex-domain-qualification/{\"interval\":[-4,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0004","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-convex-domain-qualification；資料={\"interval\":[-5,6]}。請依 [interval,ordered,convex,universal_certificate] 欄序回答。","answer":{"kind":"expression","value":"([-5, 6], 是, 是, all segment points remain in the interval)","explanation_zh":"收束波核對=([-5, 6], 是, 是, all segment points remain in the interval)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-convex-domain-qualification","seed":202610252004,"parameters":{"interval":[-5,6],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-convex-domain-qualification","semantic_template":"qf-mod25-convexfn-convex-domain-qualification/{\"interval\":[-5,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0005","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-convex-domain-qualification；資料={\"interval\":[-6,7]}。請依 [interval,ordered,convex,universal_certificate] 欄序回答。","answer":{"kind":"expression","value":"([-6, 7], 是, 是, all segment points remain in the interval)","explanation_zh":"收束波核對=([-6, 7], 是, 是, all segment points remain in the interval)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-convex-domain-qualification","seed":202610252005,"parameters":{"interval":[-6,7],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-convex-domain-qualification","semantic_template":"qf-mod25-convexfn-convex-domain-qualification/{\"interval\":[-6,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
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{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0007","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-convex-domain-qualification；資料={\"interval\":[-8,9]}。請依 [interval,ordered,convex,universal_certificate] 欄序回答。","answer":{"kind":"expression","value":"([-8, 9], 是, 是, all segment points remain in the interval)","explanation_zh":"收束波核對=([-8, 9], 是, 是, all segment points remain in the interval)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-convex-domain-qualification","seed":202610252007,"parameters":{"interval":[-8,9],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-convex-domain-qualification","semantic_template":"qf-mod25-convexfn-convex-domain-qualification/{\"interval\":[-8,9]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0008","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-convex-domain-qualification；資料={\"interval\":[-9,10]}。請依 [interval,ordered,convex,universal_certificate] 欄序回答。","answer":{"kind":"expression","value":"([-9, 10], 是, 是, all segment points remain in the interval)","explanation_zh":"收束波核對=([-9, 10], 是, 是, all segment points remain in the interval)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-convex-domain-qualification","seed":202610252008,"parameters":{"interval":[-9,10],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-convex-domain-qualification","semantic_template":"qf-mod25-convexfn-convex-domain-qualification/{\"interval\":[-9,10]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0009","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-convex-domain-qualification；資料={\"interval\":[-10,11]}。請依 [interval,ordered,convex,universal_certificate] 欄序回答。","answer":{"kind":"expression","value":"([-10, 11], 是, 是, all segment points remain in the interval)","explanation_zh":"收束波核對=([-10, 11], 是, 是, all segment points remain in the interval)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-convex-domain-qualification","seed":202610252009,"parameters":{"interval":[-10,11],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-convex-domain-qualification","semantic_template":"qf-mod25-convexfn-convex-domain-qualification/{\"interval\":[-10,11]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0010","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-convex-domain-qualification；資料={\"interval\":[-11,12]}。請依 [interval,ordered,convex,universal_certificate] 欄序回答。","answer":{"kind":"expression","value":"([-11, 12], 是, 是, all segment points remain in the interval)","explanation_zh":"收束波核對=([-11, 12], 是, 是, all segment points remain in the interval)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-convex-domain-qualification","seed":202610252010,"parameters":{"interval":[-11,12],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-convex-domain-qualification","semantic_template":"qf-mod25-convexfn-convex-domain-qualification/{\"interval\":[-11,12]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0011","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-convex-domain-qualification；資料={\"interval\":[-12,13]}。請依 [interval,ordered,convex,universal_certificate] 欄序回答。","answer":{"kind":"expression","value":"([-12, 13], 是, 是, all segment points remain in the interval)","explanation_zh":"收束波核對=([-12, 13], 是, 是, all segment points remain in the interval)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-convex-domain-qualification","seed":202610252011,"parameters":{"interval":[-12,13],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-convex-domain-qualification","semantic_template":"qf-mod25-convexfn-convex-domain-qualification/{\"interval\":[-12,13]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0012","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-convex-domain-qualification；資料={\"interval\":[-13,14]}。請依 [interval,ordered,convex,universal_certificate] 欄序回答。","answer":{"kind":"expression","value":"([-13, 14], 是, 是, all segment points remain in the interval)","explanation_zh":"收束波核對=([-13, 14], 是, 是, all segment points remain in the interval)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-convex-domain-qualification","seed":202610252012,"parameters":{"interval":[-13,14],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-convex-domain-qualification","semantic_template":"qf-mod25-convexfn-convex-domain-qualification/{\"interval\":[-13,14]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0013","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-convex-domain-qualification；資料={\"interval\":[-14,15]}。請依 [interval,ordered,convex,universal_certificate] 欄序回答。","answer":{"kind":"expression","value":"([-14, 15], 是, 是, all segment points remain in the interval)","explanation_zh":"收束波核對=([-14, 15], 是, 是, all segment points remain in the interval)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-convex-domain-qualification","seed":202610252013,"parameters":{"interval":[-14,15],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-convex-domain-qualification","semantic_template":"qf-mod25-convexfn-convex-domain-qualification/{\"interval\":[-14,15]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0014","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-convex-domain-qualification；資料={\"interval\":[-15,16]}。請依 [interval,ordered,convex,universal_certificate] 欄序回答。","answer":{"kind":"expression","value":"([-15, 16], 是, 是, all segment points remain in the interval)","explanation_zh":"收束波核對=([-15, 16], 是, 是, all segment points remain in the interval)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-convex-domain-qualification","seed":202610252014,"parameters":{"interval":[-15,16],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-convex-domain-qualification","semantic_template":"qf-mod25-convexfn-convex-domain-qualification/{\"interval\":[-15,16]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0015","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-convex-domain-qualification；資料={\"interval\":[-16,17]}。請依 [interval,ordered,convex,universal_certificate] 欄序回答。","answer":{"kind":"expression","value":"([-16, 17], 是, 是, all segment points remain in the interval)","explanation_zh":"收束波核對=([-16, 17], 是, 是, all segment points remain in the interval)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-convex-domain-qualification","seed":202610252015,"parameters":{"interval":[-16,17],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-convex-domain-qualification","semantic_template":"qf-mod25-convexfn-convex-domain-qualification/{\"interval\":[-16,17]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0016","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-convex-domain-qualification；資料={\"interval\":[-17,18]}。請依 [interval,ordered,convex,universal_certificate] 欄序回答。","answer":{"kind":"expression","value":"([-17, 18], 是, 是, all segment points remain in the interval)","explanation_zh":"收束波核對=([-17, 18], 是, 是, all segment points remain in the interval)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-convex-domain-qualification","seed":202610252016,"parameters":{"interval":[-17,18],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-convex-domain-qualification","semantic_template":"qf-mod25-convexfn-convex-domain-qualification/{\"interval\":[-17,18]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0017","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-convex-domain-qualification；資料={\"interval\":[-18,19]}。請依 [interval,ordered,convex,universal_certificate] 欄序回答。","answer":{"kind":"expression","value":"([-18, 19], 是, 是, all segment points remain in the interval)","explanation_zh":"收束波核對=([-18, 19], 是, 是, all segment points remain in the interval)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-convex-domain-qualification","seed":202610252017,"parameters":{"interval":[-18,19],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-convex-domain-qualification","semantic_template":"qf-mod25-convexfn-convex-domain-qualification/{\"interval\":[-18,19]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0018","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-convex-domain-qualification；資料={\"interval\":[-19,20]}。請依 [interval,ordered,convex,universal_certificate] 欄序回答。","answer":{"kind":"expression","value":"([-19, 20], 是, 是, all segment points remain in the interval)","explanation_zh":"收束波核對=([-19, 20], 是, 是, all segment points remain in the interval)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-convex-domain-qualification","seed":202610252018,"parameters":{"interval":[-19,20],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-convex-domain-qualification","semantic_template":"qf-mod25-convexfn-convex-domain-qualification/{\"interval\":[-19,20]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0019","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-convex-domain-qualification；資料={\"interval\":[-20,21]}。請依 [interval,ordered,convex,universal_certificate] 欄序回答。","answer":{"kind":"expression","value":"([-20, 21], 是, 是, all segment points remain in the interval)","explanation_zh":"收束波核對=([-20, 21], 是, 是, all segment points remain in the interval)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-convex-domain-qualification","seed":202610252019,"parameters":{"interval":[-20,21],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-convex-domain-qualification","semantic_template":"qf-mod25-convexfn-convex-domain-qualification/{\"interval\":[-20,21]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0020","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-convex-domain-qualification；資料={\"interval\":[-21,22]}。請依 [interval,ordered,convex,universal_certificate] 欄序回答。","answer":{"kind":"expression","value":"([-21, 22], 是, 是, all segment points remain in the interval)","explanation_zh":"收束波核對=([-21, 22], 是, 是, all segment points remain in the interval)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-convex-domain-qualification","seed":202610252020,"parameters":{"interval":[-21,22],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-convex-domain-qualification","semantic_template":"qf-mod25-convexfn-convex-domain-qualification/{\"interval\":[-21,22]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0021","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-convex-domain-qualification；資料={\"interval\":[-22,23]}。請依 [interval,ordered,convex,universal_certificate] 欄序回答。","answer":{"kind":"expression","value":"([-22, 23], 是, 是, all segment points remain in the interval)","explanation_zh":"收束波核對=([-22, 23], 是, 是, all segment points remain in the interval)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-convex-domain-qualification","seed":202610252021,"parameters":{"interval":[-22,23],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-convex-domain-qualification","semantic_template":"qf-mod25-convexfn-convex-domain-qualification/{\"interval\":[-22,23]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0022","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-convex-domain-qualification；資料={\"interval\":[-23,24]}。請依 [interval,ordered,convex,universal_certificate] 欄序回答。","answer":{"kind":"expression","value":"([-23, 24], 是, 是, all segment points remain in the interval)","explanation_zh":"收束波核對=([-23, 24], 是, 是, all segment points remain in the interval)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-convex-domain-qualification","seed":202610252022,"parameters":{"interval":[-23,24],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-convex-domain-qualification","semantic_template":"qf-mod25-convexfn-convex-domain-qualification/{\"interval\":[-23,24]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0023","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-convex-domain-qualification；資料={\"interval\":[-24,25]}。請依 [interval,ordered,convex,universal_certificate] 欄序回答。","answer":{"kind":"expression","value":"([-24, 25], 是, 是, all segment points remain in the interval)","explanation_zh":"收束波核對=([-24, 25], 是, 是, all segment points remain in the interval)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-convex-domain-qualification","seed":202610252023,"parameters":{"interval":[-24,25],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-convex-domain-qualification","semantic_template":"qf-mod25-convexfn-convex-domain-qualification/{\"interval\":[-24,25]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0024","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-convex-domain-qualification；資料={\"interval\":[-25,26]}。請依 [interval,ordered,convex,universal_certificate] 欄序回答。","answer":{"kind":"expression","value":"([-25, 26], 是, 是, all segment points remain in the interval)","explanation_zh":"收束波核對=([-25, 26], 是, 是, all segment points remain in the interval)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-convex-domain-qualification","seed":202610252024,"parameters":{"interval":[-25,26],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-convex-domain-qualification","semantic_template":"qf-mod25-convexfn-convex-domain-qualification/{\"interval\":[-25,26]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0025","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-convex-domain-qualification；資料={\"interval\":[-26,27]}。請依 [interval,ordered,convex,universal_certificate] 欄序回答。","answer":{"kind":"expression","value":"([-26, 27], 是, 是, all segment points remain in the interval)","explanation_zh":"收束波核對=([-26, 27], 是, 是, all segment points remain in the interval)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-convex-domain-qualification","seed":202610252025,"parameters":{"interval":[-26,27],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-convex-domain-qualification","semantic_template":"qf-mod25-convexfn-convex-domain-qualification/{\"interval\":[-26,27]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0026","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-affine-universal-certificate；資料={\"coefficients\":[-12,-2],\"interval\":[-2,3]}。請依 [coefficients,interval,domain_convex,jensen_identity,convex] 欄序回答。","answer":{"kind":"expression","value":"([-12, -2], [-2, 3], 是, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), 是)","explanation_zh":"收束波核對=([-12, -2], [-2, 3], 是, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-affine-universal-certificate","seed":202610252026,"parameters":{"coefficients":[-12,-2],"interval":[-2,3],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-affine-universal-certificate","semantic_template":"qf-mod25-convexfn-affine-universal-certificate/{\"coefficients\":[-12,-2],\"interval\":[-2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0027","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-affine-universal-certificate；資料={\"coefficients\":[-11,-1],\"interval\":[-3,4]}。請依 [coefficients,interval,domain_convex,jensen_identity,convex] 欄序回答。","answer":{"kind":"expression","value":"([-11, -1], [-3, 4], 是, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), 是)","explanation_zh":"收束波核對=([-11, -1], [-3, 4], 是, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-affine-universal-certificate","seed":202610252027,"parameters":{"coefficients":[-11,-1],"interval":[-3,4],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-affine-universal-certificate","semantic_template":"qf-mod25-convexfn-affine-universal-certificate/{\"coefficients\":[-11,-1],\"interval\":[-3,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0028","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-affine-universal-certificate；資料={\"coefficients\":[-10,0],\"interval\":[-4,5]}。請依 [coefficients,interval,domain_convex,jensen_identity,convex] 欄序回答。","answer":{"kind":"expression","value":"([-10, 0], [-4, 5], 是, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), 是)","explanation_zh":"收束波核對=([-10, 0], [-4, 5], 是, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-affine-universal-certificate","seed":202610252028,"parameters":{"coefficients":[-10,0],"interval":[-4,5],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-affine-universal-certificate","semantic_template":"qf-mod25-convexfn-affine-universal-certificate/{\"coefficients\":[-10,0],\"interval\":[-4,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0029","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-affine-universal-certificate；資料={\"coefficients\":[-9,1],\"interval\":[-5,6]}。請依 [coefficients,interval,domain_convex,jensen_identity,convex] 欄序回答。","answer":{"kind":"expression","value":"([-9, 1], [-5, 6], 是, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), 是)","explanation_zh":"收束波核對=([-9, 1], [-5, 6], 是, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-affine-universal-certificate","seed":202610252029,"parameters":{"coefficients":[-9,1],"interval":[-5,6],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-affine-universal-certificate","semantic_template":"qf-mod25-convexfn-affine-universal-certificate/{\"coefficients\":[-9,1],\"interval\":[-5,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0030","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-affine-universal-certificate；資料={\"coefficients\":[-8,2],\"interval\":[-6,7]}。請依 [coefficients,interval,domain_convex,jensen_identity,convex] 欄序回答。","answer":{"kind":"expression","value":"([-8, 2], [-6, 7], 是, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), 是)","explanation_zh":"收束波核對=([-8, 2], [-6, 7], 是, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-affine-universal-certificate","seed":202610252030,"parameters":{"coefficients":[-8,2],"interval":[-6,7],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-affine-universal-certificate","semantic_template":"qf-mod25-convexfn-affine-universal-certificate/{\"coefficients\":[-8,2],\"interval\":[-6,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0031","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-affine-universal-certificate；資料={\"coefficients\":[-7,-2],\"interval\":[-7,8]}。請依 [coefficients,interval,domain_convex,jensen_identity,convex] 欄序回答。","answer":{"kind":"expression","value":"([-7, -2], [-7, 8], 是, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), 是)","explanation_zh":"收束波核對=([-7, -2], [-7, 8], 是, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-affine-universal-certificate","seed":202610252031,"parameters":{"coefficients":[-7,-2],"interval":[-7,8],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-affine-universal-certificate","semantic_template":"qf-mod25-convexfn-affine-universal-certificate/{\"coefficients\":[-7,-2],\"interval\":[-7,8]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0032","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-affine-universal-certificate；資料={\"coefficients\":[-6,-1],\"interval\":[-8,9]}。請依 [coefficients,interval,domain_convex,jensen_identity,convex] 欄序回答。","answer":{"kind":"expression","value":"([-6, -1], [-8, 9], 是, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), 是)","explanation_zh":"收束波核對=([-6, -1], [-8, 9], 是, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-affine-universal-certificate","seed":202610252032,"parameters":{"coefficients":[-6,-1],"interval":[-8,9],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-affine-universal-certificate","semantic_template":"qf-mod25-convexfn-affine-universal-certificate/{\"coefficients\":[-6,-1],\"interval\":[-8,9]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0033","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-affine-universal-certificate；資料={\"coefficients\":[-5,0],\"interval\":[-9,10]}。請依 [coefficients,interval,domain_convex,jensen_identity,convex] 欄序回答。","answer":{"kind":"expression","value":"([-5, 0], [-9, 10], 是, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), 是)","explanation_zh":"收束波核對=([-5, 0], [-9, 10], 是, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-affine-universal-certificate","seed":202610252033,"parameters":{"coefficients":[-5,0],"interval":[-9,10],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-affine-universal-certificate","semantic_template":"qf-mod25-convexfn-affine-universal-certificate/{\"coefficients\":[-5,0],\"interval\":[-9,10]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0034","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-affine-universal-certificate；資料={\"coefficients\":[-4,1],\"interval\":[-10,11]}。請依 [coefficients,interval,domain_convex,jensen_identity,convex] 欄序回答。","answer":{"kind":"expression","value":"([-4, 1], [-10, 11], 是, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), 是)","explanation_zh":"收束波核對=([-4, 1], [-10, 11], 是, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-affine-universal-certificate","seed":202610252034,"parameters":{"coefficients":[-4,1],"interval":[-10,11],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-affine-universal-certificate","semantic_template":"qf-mod25-convexfn-affine-universal-certificate/{\"coefficients\":[-4,1],\"interval\":[-10,11]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0035","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-affine-universal-certificate；資料={\"coefficients\":[-3,2],\"interval\":[-11,12]}。請依 [coefficients,interval,domain_convex,jensen_identity,convex] 欄序回答。","answer":{"kind":"expression","value":"([-3, 2], [-11, 12], 是, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), 是)","explanation_zh":"收束波核對=([-3, 2], [-11, 12], 是, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-affine-universal-certificate","seed":202610252035,"parameters":{"coefficients":[-3,2],"interval":[-11,12],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-affine-universal-certificate","semantic_template":"qf-mod25-convexfn-affine-universal-certificate/{\"coefficients\":[-3,2],\"interval\":[-11,12]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0036","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-affine-universal-certificate；資料={\"coefficients\":[-2,-2],\"interval\":[-12,13]}。請依 [coefficients,interval,domain_convex,jensen_identity,convex] 欄序回答。","answer":{"kind":"expression","value":"([-2, -2], [-12, 13], 是, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), 是)","explanation_zh":"收束波核對=([-2, -2], [-12, 13], 是, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-affine-universal-certificate","seed":202610252036,"parameters":{"coefficients":[-2,-2],"interval":[-12,13],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-affine-universal-certificate","semantic_template":"qf-mod25-convexfn-affine-universal-certificate/{\"coefficients\":[-2,-2],\"interval\":[-12,13]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0037","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-affine-universal-certificate；資料={\"coefficients\":[-1,-1],\"interval\":[-13,14]}。請依 [coefficients,interval,domain_convex,jensen_identity,convex] 欄序回答。","answer":{"kind":"expression","value":"([-1, -1], [-13, 14], 是, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), 是)","explanation_zh":"收束波核對=([-1, -1], [-13, 14], 是, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-affine-universal-certificate","seed":202610252037,"parameters":{"coefficients":[-1,-1],"interval":[-13,14],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-affine-universal-certificate","semantic_template":"qf-mod25-convexfn-affine-universal-certificate/{\"coefficients\":[-1,-1],\"interval\":[-13,14]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0038","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-affine-universal-certificate；資料={\"coefficients\":[0,0],\"interval\":[-14,15]}。請依 [coefficients,interval,domain_convex,jensen_identity,convex] 欄序回答。","answer":{"kind":"expression","value":"([0, 0], [-14, 15], 是, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), 是)","explanation_zh":"收束波核對=([0, 0], [-14, 15], 是, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-affine-universal-certificate","seed":202610252038,"parameters":{"coefficients":[0,0],"interval":[-14,15],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-affine-universal-certificate","semantic_template":"qf-mod25-convexfn-affine-universal-certificate/{\"coefficients\":[0,0],\"interval\":[-14,15]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0039","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-affine-universal-certificate；資料={\"coefficients\":[1,1],\"interval\":[-15,16]}。請依 [coefficients,interval,domain_convex,jensen_identity,convex] 欄序回答。","answer":{"kind":"expression","value":"([1, 1], [-15, 16], 是, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), 是)","explanation_zh":"收束波核對=([1, 1], [-15, 16], 是, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-affine-universal-certificate","seed":202610252039,"parameters":{"coefficients":[1,1],"interval":[-15,16],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-affine-universal-certificate","semantic_template":"qf-mod25-convexfn-affine-universal-certificate/{\"coefficients\":[1,1],\"interval\":[-15,16]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0040","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-affine-universal-certificate；資料={\"coefficients\":[2,2],\"interval\":[-16,17]}。請依 [coefficients,interval,domain_convex,jensen_identity,convex] 欄序回答。","answer":{"kind":"expression","value":"([2, 2], [-16, 17], 是, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), 是)","explanation_zh":"收束波核對=([2, 2], [-16, 17], 是, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-affine-universal-certificate","seed":202610252040,"parameters":{"coefficients":[2,2],"interval":[-16,17],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-affine-universal-certificate","semantic_template":"qf-mod25-convexfn-affine-universal-certificate/{\"coefficients\":[2,2],\"interval\":[-16,17]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0041","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-affine-universal-certificate；資料={\"coefficients\":[3,-2],\"interval\":[-17,18]}。請依 [coefficients,interval,domain_convex,jensen_identity,convex] 欄序回答。","answer":{"kind":"expression","value":"([3, -2], [-17, 18], 是, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), 是)","explanation_zh":"收束波核對=([3, -2], [-17, 18], 是, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-affine-universal-certificate","seed":202610252041,"parameters":{"coefficients":[3,-2],"interval":[-17,18],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-affine-universal-certificate","semantic_template":"qf-mod25-convexfn-affine-universal-certificate/{\"coefficients\":[3,-2],\"interval\":[-17,18]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0042","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-affine-universal-certificate；資料={\"coefficients\":[4,-1],\"interval\":[-18,19]}。請依 [coefficients,interval,domain_convex,jensen_identity,convex] 欄序回答。","answer":{"kind":"expression","value":"([4, -1], [-18, 19], 是, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), 是)","explanation_zh":"收束波核對=([4, -1], [-18, 19], 是, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-affine-universal-certificate","seed":202610252042,"parameters":{"coefficients":[4,-1],"interval":[-18,19],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-affine-universal-certificate","semantic_template":"qf-mod25-convexfn-affine-universal-certificate/{\"coefficients\":[4,-1],\"interval\":[-18,19]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0043","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-affine-universal-certificate；資料={\"coefficients\":[5,0],\"interval\":[-19,20]}。請依 [coefficients,interval,domain_convex,jensen_identity,convex] 欄序回答。","answer":{"kind":"expression","value":"([5, 0], [-19, 20], 是, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), 是)","explanation_zh":"收束波核對=([5, 0], [-19, 20], 是, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-affine-universal-certificate","seed":202610252043,"parameters":{"coefficients":[5,0],"interval":[-19,20],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-affine-universal-certificate","semantic_template":"qf-mod25-convexfn-affine-universal-certificate/{\"coefficients\":[5,0],\"interval\":[-19,20]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0044","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-affine-universal-certificate；資料={\"coefficients\":[6,1],\"interval\":[-20,21]}。請依 [coefficients,interval,domain_convex,jensen_identity,convex] 欄序回答。","answer":{"kind":"expression","value":"([6, 1], [-20, 21], 是, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), 是)","explanation_zh":"收束波核對=([6, 1], [-20, 21], 是, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-affine-universal-certificate","seed":202610252044,"parameters":{"coefficients":[6,1],"interval":[-20,21],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-affine-universal-certificate","semantic_template":"qf-mod25-convexfn-affine-universal-certificate/{\"coefficients\":[6,1],\"interval\":[-20,21]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0045","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-affine-universal-certificate；資料={\"coefficients\":[7,2],\"interval\":[-21,22]}。請依 [coefficients,interval,domain_convex,jensen_identity,convex] 欄序回答。","answer":{"kind":"expression","value":"([7, 2], [-21, 22], 是, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), 是)","explanation_zh":"收束波核對=([7, 2], [-21, 22], 是, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-affine-universal-certificate","seed":202610252045,"parameters":{"coefficients":[7,2],"interval":[-21,22],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-affine-universal-certificate","semantic_template":"qf-mod25-convexfn-affine-universal-certificate/{\"coefficients\":[7,2],\"interval\":[-21,22]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0046","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-affine-universal-certificate；資料={\"coefficients\":[8,-2],\"interval\":[-22,23]}。請依 [coefficients,interval,domain_convex,jensen_identity,convex] 欄序回答。","answer":{"kind":"expression","value":"([8, -2], [-22, 23], 是, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), 是)","explanation_zh":"收束波核對=([8, -2], [-22, 23], 是, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-affine-universal-certificate","seed":202610252046,"parameters":{"coefficients":[8,-2],"interval":[-22,23],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-affine-universal-certificate","semantic_template":"qf-mod25-convexfn-affine-universal-certificate/{\"coefficients\":[8,-2],\"interval\":[-22,23]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0047","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-affine-universal-certificate；資料={\"coefficients\":[9,-1],\"interval\":[-23,24]}。請依 [coefficients,interval,domain_convex,jensen_identity,convex] 欄序回答。","answer":{"kind":"expression","value":"([9, -1], [-23, 24], 是, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), 是)","explanation_zh":"收束波核對=([9, -1], [-23, 24], 是, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-affine-universal-certificate","seed":202610252047,"parameters":{"coefficients":[9,-1],"interval":[-23,24],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-affine-universal-certificate","semantic_template":"qf-mod25-convexfn-affine-universal-certificate/{\"coefficients\":[9,-1],\"interval\":[-23,24]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0048","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-affine-universal-certificate；資料={\"coefficients\":[10,0],\"interval\":[-24,25]}。請依 [coefficients,interval,domain_convex,jensen_identity,convex] 欄序回答。","answer":{"kind":"expression","value":"([10, 0], [-24, 25], 是, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), 是)","explanation_zh":"收束波核對=([10, 0], [-24, 25], 是, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-affine-universal-certificate","seed":202610252048,"parameters":{"coefficients":[10,0],"interval":[-24,25],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-affine-universal-certificate","semantic_template":"qf-mod25-convexfn-affine-universal-certificate/{\"coefficients\":[10,0],\"interval\":[-24,25]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0049","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-affine-universal-certificate；資料={\"coefficients\":[11,1],\"interval\":[-25,26]}。請依 [coefficients,interval,domain_convex,jensen_identity,convex] 欄序回答。","answer":{"kind":"expression","value":"([11, 1], [-25, 26], 是, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), 是)","explanation_zh":"收束波核對=([11, 1], [-25, 26], 是, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-affine-universal-certificate","seed":202610252049,"parameters":{"coefficients":[11,1],"interval":[-25,26],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-affine-universal-certificate","semantic_template":"qf-mod25-convexfn-affine-universal-certificate/{\"coefficients\":[11,1],\"interval\":[-25,26]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0050","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-affine-universal-certificate；資料={\"coefficients\":[12,2],\"interval\":[-26,27]}。請依 [coefficients,interval,domain_convex,jensen_identity,convex] 欄序回答。","answer":{"kind":"expression","value":"([12, 2], [-26, 27], 是, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), 是)","explanation_zh":"收束波核對=([12, 2], [-26, 27], 是, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-affine-universal-certificate","seed":202610252050,"parameters":{"coefficients":[12,2],"interval":[-26,27],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-affine-universal-certificate","semantic_template":"qf-mod25-convexfn-affine-universal-certificate/{\"coefficients\":[12,2],\"interval\":[-26,27]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0051","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-quadratic-universal-certificate；資料={\"coefficients\":[-1,-12,-3],\"interval\":[-2,3]}。請依 [coefficients,interval,nondegenerate,leading_coefficient,jensen_gap_formula,convex] 欄序回答。","answer":{"kind":"expression","value":"([-1, -12, -3], [-2, 3], 是, -1, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, 否)","explanation_zh":"收束波核對=([-1, -12, -3], [-2, 3], 是, -1, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-quadratic-universal-certificate","seed":202610252051,"parameters":{"coefficients":[-1,-12,-3],"interval":[-2,3],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-quadratic-universal-certificate","semantic_template":"qf-mod25-convexfn-quadratic-universal-certificate/{\"coefficients\":[-1,-12,-3],\"interval\":[-2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0052","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-quadratic-universal-certificate；資料={\"coefficients\":[1,-11,-2],\"interval\":[-3,4]}。請依 [coefficients,interval,nondegenerate,leading_coefficient,jensen_gap_formula,convex] 欄序回答。","answer":{"kind":"expression","value":"([1, -11, -2], [-3, 4], 是, 1, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, 是)","explanation_zh":"收束波核對=([1, -11, -2], [-3, 4], 是, 1, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-quadratic-universal-certificate","seed":202610252052,"parameters":{"coefficients":[1,-11,-2],"interval":[-3,4],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-quadratic-universal-certificate","semantic_template":"qf-mod25-convexfn-quadratic-universal-certificate/{\"coefficients\":[1,-11,-2],\"interval\":[-3,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0053","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-quadratic-universal-certificate；資料={\"coefficients\":[2,-10,-1],\"interval\":[-4,5]}。請依 [coefficients,interval,nondegenerate,leading_coefficient,jensen_gap_formula,convex] 欄序回答。","answer":{"kind":"expression","value":"([2, -10, -1], [-4, 5], 是, 2, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, 是)","explanation_zh":"收束波核對=([2, -10, -1], [-4, 5], 是, 2, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-quadratic-universal-certificate","seed":202610252053,"parameters":{"coefficients":[2,-10,-1],"interval":[-4,5],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-quadratic-universal-certificate","semantic_template":"qf-mod25-convexfn-quadratic-universal-certificate/{\"coefficients\":[2,-10,-1],\"interval\":[-4,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0054","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-quadratic-universal-certificate；資料={\"coefficients\":[3,-9,0],\"interval\":[-5,6]}。請依 [coefficients,interval,nondegenerate,leading_coefficient,jensen_gap_formula,convex] 欄序回答。","answer":{"kind":"expression","value":"([3, -9, 0], [-5, 6], 是, 3, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, 是)","explanation_zh":"收束波核對=([3, -9, 0], [-5, 6], 是, 3, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-quadratic-universal-certificate","seed":202610252054,"parameters":{"coefficients":[3,-9,0],"interval":[-5,6],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-quadratic-universal-certificate","semantic_template":"qf-mod25-convexfn-quadratic-universal-certificate/{\"coefficients\":[3,-9,0],\"interval\":[-5,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0055","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-quadratic-universal-certificate；資料={\"coefficients\":[0,-8,1],\"interval\":[-6,7]}。請依 [coefficients,interval,nondegenerate,leading_coefficient,jensen_gap_formula,convex] 欄序回答。","answer":{"kind":"expression","value":"([0, -8, 1], [-6, 7], 是, 0, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, 是)","explanation_zh":"收束波核對=([0, -8, 1], [-6, 7], 是, 0, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-quadratic-universal-certificate","seed":202610252055,"parameters":{"coefficients":[0,-8,1],"interval":[-6,7],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-quadratic-universal-certificate","semantic_template":"qf-mod25-convexfn-quadratic-universal-certificate/{\"coefficients\":[0,-8,1],\"interval\":[-6,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0056","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-quadratic-universal-certificate；資料={\"coefficients\":[-1,-7,2],\"interval\":[-7,8]}。請依 [coefficients,interval,nondegenerate,leading_coefficient,jensen_gap_formula,convex] 欄序回答。","answer":{"kind":"expression","value":"([-1, -7, 2], [-7, 8], 是, -1, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, 否)","explanation_zh":"收束波核對=([-1, -7, 2], [-7, 8], 是, -1, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-quadratic-universal-certificate","seed":202610252056,"parameters":{"coefficients":[-1,-7,2],"interval":[-7,8],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-quadratic-universal-certificate","semantic_template":"qf-mod25-convexfn-quadratic-universal-certificate/{\"coefficients\":[-1,-7,2],\"interval\":[-7,8]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0057","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-quadratic-universal-certificate；資料={\"coefficients\":[2,-6,3],\"interval\":[-8,9]}。請依 [coefficients,interval,nondegenerate,leading_coefficient,jensen_gap_formula,convex] 欄序回答。","answer":{"kind":"expression","value":"([2, -6, 3], [-8, 9], 是, 2, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, 是)","explanation_zh":"收束波核對=([2, -6, 3], [-8, 9], 是, 2, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-quadratic-universal-certificate","seed":202610252057,"parameters":{"coefficients":[2,-6,3],"interval":[-8,9],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-quadratic-universal-certificate","semantic_template":"qf-mod25-convexfn-quadratic-universal-certificate/{\"coefficients\":[2,-6,3],\"interval\":[-8,9]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0058","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-quadratic-universal-certificate；資料={\"coefficients\":[3,-5,-3],\"interval\":[-9,10]}。請依 [coefficients,interval,nondegenerate,leading_coefficient,jensen_gap_formula,convex] 欄序回答。","answer":{"kind":"expression","value":"([3, -5, -3], [-9, 10], 是, 3, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, 是)","explanation_zh":"收束波核對=([3, -5, -3], [-9, 10], 是, 3, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-quadratic-universal-certificate","seed":202610252058,"parameters":{"coefficients":[3,-5,-3],"interval":[-9,10],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-quadratic-universal-certificate","semantic_template":"qf-mod25-convexfn-quadratic-universal-certificate/{\"coefficients\":[3,-5,-3],\"interval\":[-9,10]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0059","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-quadratic-universal-certificate；資料={\"coefficients\":[0,-4,-2],\"interval\":[-10,11]}。請依 [coefficients,interval,nondegenerate,leading_coefficient,jensen_gap_formula,convex] 欄序回答。","answer":{"kind":"expression","value":"([0, -4, -2], [-10, 11], 是, 0, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, 是)","explanation_zh":"收束波核對=([0, -4, -2], [-10, 11], 是, 0, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-quadratic-universal-certificate","seed":202610252059,"parameters":{"coefficients":[0,-4,-2],"interval":[-10,11],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-quadratic-universal-certificate","semantic_template":"qf-mod25-convexfn-quadratic-universal-certificate/{\"coefficients\":[0,-4,-2],\"interval\":[-10,11]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0060","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-quadratic-universal-certificate；資料={\"coefficients\":[1,-3,-1],\"interval\":[-11,12]}。請依 [coefficients,interval,nondegenerate,leading_coefficient,jensen_gap_formula,convex] 欄序回答。","answer":{"kind":"expression","value":"([1, -3, -1], [-11, 12], 是, 1, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, 是)","explanation_zh":"收束波核對=([1, -3, -1], [-11, 12], 是, 1, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-quadratic-universal-certificate","seed":202610252060,"parameters":{"coefficients":[1,-3,-1],"interval":[-11,12],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-quadratic-universal-certificate","semantic_template":"qf-mod25-convexfn-quadratic-universal-certificate/{\"coefficients\":[1,-3,-1],\"interval\":[-11,12]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0061","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-quadratic-universal-certificate；資料={\"coefficients\":[-1,-2,0],\"interval\":[-12,13]}。請依 [coefficients,interval,nondegenerate,leading_coefficient,jensen_gap_formula,convex] 欄序回答。","answer":{"kind":"expression","value":"([-1, -2, 0], [-12, 13], 是, -1, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, 否)","explanation_zh":"收束波核對=([-1, -2, 0], [-12, 13], 是, -1, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-quadratic-universal-certificate","seed":202610252061,"parameters":{"coefficients":[-1,-2,0],"interval":[-12,13],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-quadratic-universal-certificate","semantic_template":"qf-mod25-convexfn-quadratic-universal-certificate/{\"coefficients\":[-1,-2,0],\"interval\":[-12,13]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0062","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-quadratic-universal-certificate；資料={\"coefficients\":[3,-1,1],\"interval\":[-13,14]}。請依 [coefficients,interval,nondegenerate,leading_coefficient,jensen_gap_formula,convex] 欄序回答。","answer":{"kind":"expression","value":"([3, -1, 1], [-13, 14], 是, 3, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, 是)","explanation_zh":"收束波核對=([3, -1, 1], [-13, 14], 是, 3, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-quadratic-universal-certificate","seed":202610252062,"parameters":{"coefficients":[3,-1,1],"interval":[-13,14],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-quadratic-universal-certificate","semantic_template":"qf-mod25-convexfn-quadratic-universal-certificate/{\"coefficients\":[3,-1,1],\"interval\":[-13,14]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0063","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-quadratic-universal-certificate；資料={\"coefficients\":[0,0,2],\"interval\":[-14,15]}。請依 [coefficients,interval,nondegenerate,leading_coefficient,jensen_gap_formula,convex] 欄序回答。","answer":{"kind":"expression","value":"([0, 0, 2], [-14, 15], 是, 0, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, 是)","explanation_zh":"收束波核對=([0, 0, 2], [-14, 15], 是, 0, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-quadratic-universal-certificate","seed":202610252063,"parameters":{"coefficients":[0,0,2],"interval":[-14,15],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-quadratic-universal-certificate","semantic_template":"qf-mod25-convexfn-quadratic-universal-certificate/{\"coefficients\":[0,0,2],\"interval\":[-14,15]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0064","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-quadratic-universal-certificate；資料={\"coefficients\":[1,1,3],\"interval\":[-15,16]}。請依 [coefficients,interval,nondegenerate,leading_coefficient,jensen_gap_formula,convex] 欄序回答。","answer":{"kind":"expression","value":"([1, 1, 3], [-15, 16], 是, 1, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, 是)","explanation_zh":"收束波核對=([1, 1, 3], [-15, 16], 是, 1, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-quadratic-universal-certificate","seed":202610252064,"parameters":{"coefficients":[1,1,3],"interval":[-15,16],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-quadratic-universal-certificate","semantic_template":"qf-mod25-convexfn-quadratic-universal-certificate/{\"coefficients\":[1,1,3],\"interval\":[-15,16]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0065","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-quadratic-universal-certificate；資料={\"coefficients\":[2,2,-3],\"interval\":[-16,17]}。請依 [coefficients,interval,nondegenerate,leading_coefficient,jensen_gap_formula,convex] 欄序回答。","answer":{"kind":"expression","value":"([2, 2, -3], [-16, 17], 是, 2, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, 是)","explanation_zh":"收束波核對=([2, 2, -3], [-16, 17], 是, 2, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-quadratic-universal-certificate","seed":202610252065,"parameters":{"coefficients":[2,2,-3],"interval":[-16,17],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-quadratic-universal-certificate","semantic_template":"qf-mod25-convexfn-quadratic-universal-certificate/{\"coefficients\":[2,2,-3],\"interval\":[-16,17]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0066","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-quadratic-universal-certificate；資料={\"coefficients\":[-1,3,-2],\"interval\":[-17,18]}。請依 [coefficients,interval,nondegenerate,leading_coefficient,jensen_gap_formula,convex] 欄序回答。","answer":{"kind":"expression","value":"([-1, 3, -2], [-17, 18], 是, -1, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, 否)","explanation_zh":"收束波核對=([-1, 3, -2], [-17, 18], 是, -1, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-quadratic-universal-certificate","seed":202610252066,"parameters":{"coefficients":[-1,3,-2],"interval":[-17,18],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-quadratic-universal-certificate","semantic_template":"qf-mod25-convexfn-quadratic-universal-certificate/{\"coefficients\":[-1,3,-2],\"interval\":[-17,18]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0067","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-quadratic-universal-certificate；資料={\"coefficients\":[0,4,-1],\"interval\":[-18,19]}。請依 [coefficients,interval,nondegenerate,leading_coefficient,jensen_gap_formula,convex] 欄序回答。","answer":{"kind":"expression","value":"([0, 4, -1], [-18, 19], 是, 0, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, 是)","explanation_zh":"收束波核對=([0, 4, -1], [-18, 19], 是, 0, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-quadratic-universal-certificate","seed":202610252067,"parameters":{"coefficients":[0,4,-1],"interval":[-18,19],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-quadratic-universal-certificate","semantic_template":"qf-mod25-convexfn-quadratic-universal-certificate/{\"coefficients\":[0,4,-1],\"interval\":[-18,19]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0068","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-quadratic-universal-certificate；資料={\"coefficients\":[1,5,0],\"interval\":[-19,20]}。請依 [coefficients,interval,nondegenerate,leading_coefficient,jensen_gap_formula,convex] 欄序回答。","answer":{"kind":"expression","value":"([1, 5, 0], [-19, 20], 是, 1, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, 是)","explanation_zh":"收束波核對=([1, 5, 0], [-19, 20], 是, 1, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-quadratic-universal-certificate","seed":202610252068,"parameters":{"coefficients":[1,5,0],"interval":[-19,20],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-quadratic-universal-certificate","semantic_template":"qf-mod25-convexfn-quadratic-universal-certificate/{\"coefficients\":[1,5,0],\"interval\":[-19,20]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0069","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-quadratic-universal-certificate；資料={\"coefficients\":[2,6,1],\"interval\":[-20,21]}。請依 [coefficients,interval,nondegenerate,leading_coefficient,jensen_gap_formula,convex] 欄序回答。","answer":{"kind":"expression","value":"([2, 6, 1], [-20, 21], 是, 2, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, 是)","explanation_zh":"收束波核對=([2, 6, 1], [-20, 21], 是, 2, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-quadratic-universal-certificate","seed":202610252069,"parameters":{"coefficients":[2,6,1],"interval":[-20,21],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-quadratic-universal-certificate","semantic_template":"qf-mod25-convexfn-quadratic-universal-certificate/{\"coefficients\":[2,6,1],\"interval\":[-20,21]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0070","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-quadratic-universal-certificate；資料={\"coefficients\":[3,7,2],\"interval\":[-21,22]}。請依 [coefficients,interval,nondegenerate,leading_coefficient,jensen_gap_formula,convex] 欄序回答。","answer":{"kind":"expression","value":"([3, 7, 2], [-21, 22], 是, 3, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, 是)","explanation_zh":"收束波核對=([3, 7, 2], [-21, 22], 是, 3, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-quadratic-universal-certificate","seed":202610252070,"parameters":{"coefficients":[3,7,2],"interval":[-21,22],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-quadratic-universal-certificate","semantic_template":"qf-mod25-convexfn-quadratic-universal-certificate/{\"coefficients\":[3,7,2],\"interval\":[-21,22]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0071","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-quadratic-universal-certificate；資料={\"coefficients\":[-1,8,3],\"interval\":[-22,23]}。請依 [coefficients,interval,nondegenerate,leading_coefficient,jensen_gap_formula,convex] 欄序回答。","answer":{"kind":"expression","value":"([-1, 8, 3], [-22, 23], 是, -1, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, 否)","explanation_zh":"收束波核對=([-1, 8, 3], [-22, 23], 是, -1, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-quadratic-universal-certificate","seed":202610252071,"parameters":{"coefficients":[-1,8,3],"interval":[-22,23],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-quadratic-universal-certificate","semantic_template":"qf-mod25-convexfn-quadratic-universal-certificate/{\"coefficients\":[-1,8,3],\"interval\":[-22,23]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0072","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-quadratic-universal-certificate；資料={\"coefficients\":[1,9,-3],\"interval\":[-23,24]}。請依 [coefficients,interval,nondegenerate,leading_coefficient,jensen_gap_formula,convex] 欄序回答。","answer":{"kind":"expression","value":"([1, 9, -3], [-23, 24], 是, 1, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, 是)","explanation_zh":"收束波核對=([1, 9, -3], [-23, 24], 是, 1, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-quadratic-universal-certificate","seed":202610252072,"parameters":{"coefficients":[1,9,-3],"interval":[-23,24],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-quadratic-universal-certificate","semantic_template":"qf-mod25-convexfn-quadratic-universal-certificate/{\"coefficients\":[1,9,-3],\"interval\":[-23,24]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0073","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-quadratic-universal-certificate；資料={\"coefficients\":[2,10,-2],\"interval\":[-24,25]}。請依 [coefficients,interval,nondegenerate,leading_coefficient,jensen_gap_formula,convex] 欄序回答。","answer":{"kind":"expression","value":"([2, 10, -2], [-24, 25], 是, 2, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, 是)","explanation_zh":"收束波核對=([2, 10, -2], [-24, 25], 是, 2, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-quadratic-universal-certificate","seed":202610252073,"parameters":{"coefficients":[2,10,-2],"interval":[-24,25],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-quadratic-universal-certificate","semantic_template":"qf-mod25-convexfn-quadratic-universal-certificate/{\"coefficients\":[2,10,-2],\"interval\":[-24,25]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0074","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-quadratic-universal-certificate；資料={\"coefficients\":[3,11,-1],\"interval\":[-25,26]}。請依 [coefficients,interval,nondegenerate,leading_coefficient,jensen_gap_formula,convex] 欄序回答。","answer":{"kind":"expression","value":"([3, 11, -1], [-25, 26], 是, 3, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, 是)","explanation_zh":"收束波核對=([3, 11, -1], [-25, 26], 是, 3, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-quadratic-universal-certificate","seed":202610252074,"parameters":{"coefficients":[3,11,-1],"interval":[-25,26],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-quadratic-universal-certificate","semantic_template":"qf-mod25-convexfn-quadratic-universal-certificate/{\"coefficients\":[3,11,-1],\"interval\":[-25,26]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0075","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-quadratic-universal-certificate；資料={\"coefficients\":[0,12,0],\"interval\":[-26,27]}。請依 [coefficients,interval,nondegenerate,leading_coefficient,jensen_gap_formula,convex] 欄序回答。","answer":{"kind":"expression","value":"([0, 12, 0], [-26, 27], 是, 0, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, 是)","explanation_zh":"收束波核對=([0, 12, 0], [-26, 27], 是, 0, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-quadratic-universal-certificate","seed":202610252075,"parameters":{"coefficients":[0,12,0],"interval":[-26,27],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-quadratic-universal-certificate","semantic_template":"qf-mod25-convexfn-quadratic-universal-certificate/{\"coefficients\":[0,12,0],\"interval\":[-26,27]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0076","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-jensen-instance-audit；資料={\"coefficients\":[-1,-12,-3],\"interval\":[-5,5],\"x\":[-2,1],\"y\":[-3,1],\"lambda\":[1,3]}。請依 [coefficients,interval,x,y,lambda,point,lhs,rhs,gap,passes,scope] 欄序回答。","answer":{"kind":"expression","value":"([-1, -12, -3], [-5, 5], -2, -3, 1/3, -8/3, 197/9, 65/3, -2/9, 否, nonconvexity_witness)","explanation_zh":"收束波核對=([-1, -12, -3], [-5, 5], -2, -3, 1/3, -8/3, 197/9, 65/3, -2/9, 否, nonconvexity_witness)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-jensen-instance-audit","seed":202610252076,"parameters":{"coefficients":[-1,-12,-3],"interval":[-5,5],"x":[-2,1],"y":[-3,1],"lambda":[1,3],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-jensen-instance-audit","semantic_template":"qf-mod25-convexfn-jensen-instance-audit/{\"coefficients\":[-1,-12,-3],\"interval\":[-5,5],\"lambda\":[1,3],\"x\":[-2,1],\"y\":[-3,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0077","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-jensen-instance-audit；資料={\"coefficients\":[1,-11,-2],\"interval\":[-5,5],\"x\":[-1,1],\"y\":[-2,1],\"lambda\":[2,4]}。請依 [coefficients,interval,x,y,lambda,point,lhs,rhs,gap,passes,scope] 欄序回答。","answer":{"kind":"expression","value":"([1, -11, -2], [-5, 5], -1, -2, 1/2, -3/2, 67/4, 17, 1/4, 是, single_instance_only)","explanation_zh":"收束波核對=([1, -11, -2], [-5, 5], -1, -2, 1/2, -3/2, 67/4, 17, 1/4, 是, single_instance_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-jensen-instance-audit","seed":202610252077,"parameters":{"coefficients":[1,-11,-2],"interval":[-5,5],"x":[-1,1],"y":[-2,1],"lambda":[2,4],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-jensen-instance-audit","semantic_template":"qf-mod25-convexfn-jensen-instance-audit/{\"coefficients\":[1,-11,-2],\"interval\":[-5,5],\"lambda\":[2,4],\"x\":[-1,1],\"y\":[-2,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0078","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-jensen-instance-audit；資料={\"coefficients\":[2,-10,-1],\"interval\":[-5,5],\"x\":[0,1],\"y\":[-1,1],\"lambda\":[3,5]}。請依 [coefficients,interval,x,y,lambda,point,lhs,rhs,gap,passes,scope] 欄序回答。","answer":{"kind":"expression","value":"([2, -10, -1], [-5, 5], 0, -1, 3/5, -2/5, 83/25, 19/5, 12/25, 是, single_instance_only)","explanation_zh":"收束波核對=([2, -10, -1], [-5, 5], 0, -1, 3/5, -2/5, 83/25, 19/5, 12/25, 是, single_instance_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-jensen-instance-audit","seed":202610252078,"parameters":{"coefficients":[2,-10,-1],"interval":[-5,5],"x":[0,1],"y":[-1,1],"lambda":[3,5],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-jensen-instance-audit","semantic_template":"qf-mod25-convexfn-jensen-instance-audit/{\"coefficients\":[2,-10,-1],\"interval\":[-5,5],\"lambda\":[3,5],\"x\":[0,1],\"y\":[-1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0079","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-jensen-instance-audit；資料={\"coefficients\":[3,-9,0],\"interval\":[-5,5],\"x\":[1,1],\"y\":[0,1],\"lambda\":[4,6]}。請依 [coefficients,interval,x,y,lambda,point,lhs,rhs,gap,passes,scope] 欄序回答。","answer":{"kind":"expression","value":"([3, -9, 0], [-5, 5], 1, 0, 2/3, 2/3, -14/3, -4, 2/3, 是, single_instance_only)","explanation_zh":"收束波核對=([3, -9, 0], [-5, 5], 1, 0, 2/3, 2/3, -14/3, -4, 2/3, 是, single_instance_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-jensen-instance-audit","seed":202610252079,"parameters":{"coefficients":[3,-9,0],"interval":[-5,5],"x":[1,1],"y":[0,1],"lambda":[4,6],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-jensen-instance-audit","semantic_template":"qf-mod25-convexfn-jensen-instance-audit/{\"coefficients\":[3,-9,0],\"interval\":[-5,5],\"lambda\":[4,6],\"x\":[1,1],\"y\":[0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0080","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-jensen-instance-audit；資料={\"coefficients\":[0,-8,1],\"interval\":[-5,5],\"x\":[2,1],\"y\":[1,1],\"lambda\":[5,7]}。請依 [coefficients,interval,x,y,lambda,point,lhs,rhs,gap,passes,scope] 欄序回答。","answer":{"kind":"expression","value":"([0, -8, 1], [-5, 5], 2, 1, 5/7, 12/7, -89/7, -89/7, 0, 是, single_instance_only)","explanation_zh":"收束波核對=([0, -8, 1], [-5, 5], 2, 1, 5/7, 12/7, -89/7, -89/7, 0, 是, single_instance_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-jensen-instance-audit","seed":202610252080,"parameters":{"coefficients":[0,-8,1],"interval":[-5,5],"x":[2,1],"y":[1,1],"lambda":[5,7],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-jensen-instance-audit","semantic_template":"qf-mod25-convexfn-jensen-instance-audit/{\"coefficients\":[0,-8,1],\"interval\":[-5,5],\"lambda\":[5,7],\"x\":[2,1],\"y\":[1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0081","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-jensen-instance-audit；資料={\"coefficients\":[1,-7,2],\"interval\":[-5,5],\"x\":[-2,1],\"y\":[2,1],\"lambda\":[6,8]}。請依 [coefficients,interval,x,y,lambda,point,lhs,rhs,gap,passes,scope] 欄序回答。","answer":{"kind":"expression","value":"([1, -7, 2], [-5, 5], -2, 2, 3/4, -1, 10, 13, 3, 是, single_instance_only)","explanation_zh":"收束波核對=([1, -7, 2], [-5, 5], -2, 2, 3/4, -1, 10, 13, 3, 是, single_instance_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-jensen-instance-audit","seed":202610252081,"parameters":{"coefficients":[1,-7,2],"interval":[-5,5],"x":[-2,1],"y":[2,1],"lambda":[6,8],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-jensen-instance-audit","semantic_template":"qf-mod25-convexfn-jensen-instance-audit/{\"coefficients\":[1,-7,2],\"interval\":[-5,5],\"lambda\":[6,8],\"x\":[-2,1],\"y\":[2,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0082","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-jensen-instance-audit；資料={\"coefficients\":[-1,-6,3],\"interval\":[-5,5],\"x\":[-1,1],\"y\":[3,1],\"lambda\":[7,9]}。請依 [coefficients,interval,x,y,lambda,point,lhs,rhs,gap,passes,scope] 欄序回答。","answer":{"kind":"expression","value":"([-1, -6, 3], [-5, 5], -1, 3, 7/9, -1/9, 296/81, 8/9, -224/81, 否, nonconvexity_witness)","explanation_zh":"收束波核對=([-1, -6, 3], [-5, 5], -1, 3, 7/9, -1/9, 296/81, 8/9, -224/81, 否, nonconvexity_witness)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-jensen-instance-audit","seed":202610252082,"parameters":{"coefficients":[-1,-6,3],"interval":[-5,5],"x":[-1,1],"y":[3,1],"lambda":[7,9],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-jensen-instance-audit","semantic_template":"qf-mod25-convexfn-jensen-instance-audit/{\"coefficients\":[-1,-6,3],\"interval\":[-5,5],\"lambda\":[7,9],\"x\":[-1,1],\"y\":[3,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0083","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-jensen-instance-audit；資料={\"coefficients\":[3,-5,-3],\"interval\":[-5,5],\"x\":[0,1],\"y\":[-3,1],\"lambda\":[8,10]}。請依 [coefficients,interval,x,y,lambda,point,lhs,rhs,gap,passes,scope] 欄序回答。","answer":{"kind":"expression","value":"([3, -5, -3], [-5, 5], 0, -3, 4/5, -3/5, 27/25, 27/5, 108/25, 是, single_instance_only)","explanation_zh":"收束波核對=([3, -5, -3], [-5, 5], 0, -3, 4/5, -3/5, 27/25, 27/5, 108/25, 是, single_instance_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-jensen-instance-audit","seed":202610252083,"parameters":{"coefficients":[3,-5,-3],"interval":[-5,5],"x":[0,1],"y":[-3,1],"lambda":[8,10],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-jensen-instance-audit","semantic_template":"qf-mod25-convexfn-jensen-instance-audit/{\"coefficients\":[3,-5,-3],\"interval\":[-5,5],\"lambda\":[8,10],\"x\":[0,1],\"y\":[-3,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0084","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-jensen-instance-audit；資料={\"coefficients\":[0,-4,-2],\"interval\":[-5,5],\"x\":[1,1],\"y\":[-2,1],\"lambda\":[9,11]}。請依 [coefficients,interval,x,y,lambda,point,lhs,rhs,gap,passes,scope] 欄序回答。","answer":{"kind":"expression","value":"([0, -4, -2], [-5, 5], 1, -2, 9/11, 5/11, -42/11, -42/11, 0, 是, single_instance_only)","explanation_zh":"收束波核對=([0, -4, -2], [-5, 5], 1, -2, 9/11, 5/11, -42/11, -42/11, 0, 是, single_instance_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-jensen-instance-audit","seed":202610252084,"parameters":{"coefficients":[0,-4,-2],"interval":[-5,5],"x":[1,1],"y":[-2,1],"lambda":[9,11],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-jensen-instance-audit","semantic_template":"qf-mod25-convexfn-jensen-instance-audit/{\"coefficients\":[0,-4,-2],\"interval\":[-5,5],\"lambda\":[9,11],\"x\":[1,1],\"y\":[-2,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0085","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-jensen-instance-audit；資料={\"coefficients\":[1,-3,-1],\"interval\":[-5,5],\"x\":[2,1],\"y\":[-1,1],\"lambda\":[10,12]}。請依 [coefficients,interval,x,y,lambda,point,lhs,rhs,gap,passes,scope] 欄序回答。","answer":{"kind":"expression","value":"([1, -3, -1], [-5, 5], 2, -1, 5/6, 3/2, -13/4, -2, 5/4, 是, single_instance_only)","explanation_zh":"收束波核對=([1, -3, -1], [-5, 5], 2, -1, 5/6, 3/2, -13/4, -2, 5/4, 是, single_instance_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-jensen-instance-audit","seed":202610252085,"parameters":{"coefficients":[1,-3,-1],"interval":[-5,5],"x":[2,1],"y":[-1,1],"lambda":[10,12],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-jensen-instance-audit","semantic_template":"qf-mod25-convexfn-jensen-instance-audit/{\"coefficients\":[1,-3,-1],\"interval\":[-5,5],\"lambda\":[10,12],\"x\":[2,1],\"y\":[-1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0086","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-jensen-instance-audit；資料={\"coefficients\":[2,-2,0],\"interval\":[-5,5],\"x\":[-2,1],\"y\":[0,1],\"lambda\":[11,13]}。請依 [coefficients,interval,x,y,lambda,point,lhs,rhs,gap,passes,scope] 欄序回答。","answer":{"kind":"expression","value":"([2, -2, 0], [-5, 5], -2, 0, 11/13, -22/13, 1540/169, 132/13, 176/169, 是, single_instance_only)","explanation_zh":"收束波核對=([2, -2, 0], [-5, 5], -2, 0, 11/13, -22/13, 1540/169, 132/13, 176/169, 是, single_instance_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-jensen-instance-audit","seed":202610252086,"parameters":{"coefficients":[2,-2,0],"interval":[-5,5],"x":[-2,1],"y":[0,1],"lambda":[11,13],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-jensen-instance-audit","semantic_template":"qf-mod25-convexfn-jensen-instance-audit/{\"coefficients\":[2,-2,0],\"interval\":[-5,5],\"lambda\":[11,13],\"x\":[-2,1],\"y\":[0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0087","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-jensen-instance-audit；資料={\"coefficients\":[3,-1,1],\"interval\":[-5,5],\"x\":[-1,1],\"y\":[1,1],\"lambda\":[12,14]}。請依 [coefficients,interval,x,y,lambda,point,lhs,rhs,gap,passes,scope] 欄序回答。","answer":{"kind":"expression","value":"([3, -1, 1], [-5, 5], -1, 1, 6/7, -5/7, 159/49, 33/7, 72/49, 是, single_instance_only)","explanation_zh":"收束波核對=([3, -1, 1], [-5, 5], -1, 1, 6/7, -5/7, 159/49, 33/7, 72/49, 是, single_instance_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-jensen-instance-audit","seed":202610252087,"parameters":{"coefficients":[3,-1,1],"interval":[-5,5],"x":[-1,1],"y":[1,1],"lambda":[12,14],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-jensen-instance-audit","semantic_template":"qf-mod25-convexfn-jensen-instance-audit/{\"coefficients\":[3,-1,1],\"interval\":[-5,5],\"lambda\":[12,14],\"x\":[-1,1],\"y\":[1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0088","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-jensen-instance-audit；資料={\"coefficients\":[-1,0,2],\"interval\":[-5,5],\"x\":[0,1],\"y\":[2,1],\"lambda\":[13,15]}。請依 [coefficients,interval,x,y,lambda,point,lhs,rhs,gap,passes,scope] 欄序回答。","answer":{"kind":"expression","value":"([-1, 0, 2], [-5, 5], 0, 2, 13/15, 4/15, 434/225, 22/15, -104/225, 否, nonconvexity_witness)","explanation_zh":"收束波核對=([-1, 0, 2], [-5, 5], 0, 2, 13/15, 4/15, 434/225, 22/15, -104/225, 否, nonconvexity_witness)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-jensen-instance-audit","seed":202610252088,"parameters":{"coefficients":[-1,0,2],"interval":[-5,5],"x":[0,1],"y":[2,1],"lambda":[13,15],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-jensen-instance-audit","semantic_template":"qf-mod25-convexfn-jensen-instance-audit/{\"coefficients\":[-1,0,2],\"interval\":[-5,5],\"lambda\":[13,15],\"x\":[0,1],\"y\":[2,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0089","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-jensen-instance-audit；資料={\"coefficients\":[1,1,3],\"interval\":[-5,5],\"x\":[1,1],\"y\":[3,1],\"lambda\":[14,16]}。請依 [coefficients,interval,x,y,lambda,point,lhs,rhs,gap,passes,scope] 欄序回答。","answer":{"kind":"expression","value":"([1, 1, 3], [-5, 5], 1, 3, 7/8, 5/4, 93/16, 25/4, 7/16, 是, single_instance_only)","explanation_zh":"收束波核對=([1, 1, 3], [-5, 5], 1, 3, 7/8, 5/4, 93/16, 25/4, 7/16, 是, single_instance_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-jensen-instance-audit","seed":202610252089,"parameters":{"coefficients":[1,1,3],"interval":[-5,5],"x":[1,1],"y":[3,1],"lambda":[14,16],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-jensen-instance-audit","semantic_template":"qf-mod25-convexfn-jensen-instance-audit/{\"coefficients\":[1,1,3],\"interval\":[-5,5],\"lambda\":[14,16],\"x\":[1,1],\"y\":[3,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0090","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-jensen-instance-audit；資料={\"coefficients\":[2,2,-3],\"interval\":[-5,5],\"x\":[2,1],\"y\":[-3,1],\"lambda\":[15,17]}。請依 [coefficients,interval,x,y,lambda,point,lhs,rhs,gap,passes,scope] 欄序回答。","answer":{"kind":"expression","value":"([2, 2, -3], [-5, 5], 2, -3, 15/17, 24/17, 1101/289, 9, 1500/289, 是, single_instance_only)","explanation_zh":"收束波核對=([2, 2, -3], [-5, 5], 2, -3, 15/17, 24/17, 1101/289, 9, 1500/289, 是, single_instance_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-jensen-instance-audit","seed":202610252090,"parameters":{"coefficients":[2,2,-3],"interval":[-5,5],"x":[2,1],"y":[-3,1],"lambda":[15,17],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-jensen-instance-audit","semantic_template":"qf-mod25-convexfn-jensen-instance-audit/{\"coefficients\":[2,2,-3],\"interval\":[-5,5],\"lambda\":[15,17],\"x\":[2,1],\"y\":[-3,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0091","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-jensen-instance-audit；資料={\"coefficients\":[3,3,-2],\"interval\":[-5,5],\"x\":[-2,1],\"y\":[-2,1],\"lambda\":[16,18]}。請依 [coefficients,interval,x,y,lambda,point,lhs,rhs,gap,passes,scope] 欄序回答。","answer":{"kind":"expression","value":"([3, 3, -2], [-5, 5], -2, -2, 8/9, -2, 4, 4, 0, 是, single_instance_only)","explanation_zh":"收束波核對=([3, 3, -2], [-5, 5], -2, -2, 8/9, -2, 4, 4, 0, 是, single_instance_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-jensen-instance-audit","seed":202610252091,"parameters":{"coefficients":[3,3,-2],"interval":[-5,5],"x":[-2,1],"y":[-2,1],"lambda":[16,18],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-jensen-instance-audit","semantic_template":"qf-mod25-convexfn-jensen-instance-audit/{\"coefficients\":[3,3,-2],\"interval\":[-5,5],\"lambda\":[16,18],\"x\":[-2,1],\"y\":[-2,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0092","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-jensen-instance-audit；資料={\"coefficients\":[0,4,-1],\"interval\":[-5,5],\"x\":[-1,1],\"y\":[-1,1],\"lambda\":[17,19]}。請依 [coefficients,interval,x,y,lambda,point,lhs,rhs,gap,passes,scope] 欄序回答。","answer":{"kind":"expression","value":"([0, 4, -1], [-5, 5], -1, -1, 17/19, -1, -5, -5, 0, 是, single_instance_only)","explanation_zh":"收束波核對=([0, 4, -1], [-5, 5], -1, -1, 17/19, -1, -5, -5, 0, 是, single_instance_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-jensen-instance-audit","seed":202610252092,"parameters":{"coefficients":[0,4,-1],"interval":[-5,5],"x":[-1,1],"y":[-1,1],"lambda":[17,19],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-jensen-instance-audit","semantic_template":"qf-mod25-convexfn-jensen-instance-audit/{\"coefficients\":[0,4,-1],\"interval\":[-5,5],\"lambda\":[17,19],\"x\":[-1,1],\"y\":[-1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0093","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-jensen-instance-audit；資料={\"coefficients\":[1,5,0],\"interval\":[-5,5],\"x\":[0,1],\"y\":[0,1],\"lambda\":[18,20]}。請依 [coefficients,interval,x,y,lambda,point,lhs,rhs,gap,passes,scope] 欄序回答。","answer":{"kind":"expression","value":"([1, 5, 0], [-5, 5], 0, 0, 9/10, 0, 0, 0, 0, 是, single_instance_only)","explanation_zh":"收束波核對=([1, 5, 0], [-5, 5], 0, 0, 9/10, 0, 0, 0, 0, 是, single_instance_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-jensen-instance-audit","seed":202610252093,"parameters":{"coefficients":[1,5,0],"interval":[-5,5],"x":[0,1],"y":[0,1],"lambda":[18,20],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-jensen-instance-audit","semantic_template":"qf-mod25-convexfn-jensen-instance-audit/{\"coefficients\":[1,5,0],\"interval\":[-5,5],\"lambda\":[18,20],\"x\":[0,1],\"y\":[0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0094","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-jensen-instance-audit；資料={\"coefficients\":[-1,6,1],\"interval\":[-5,5],\"x\":[1,1],\"y\":[1,1],\"lambda\":[19,21]}。請依 [coefficients,interval,x,y,lambda,point,lhs,rhs,gap,passes,scope] 欄序回答。","answer":{"kind":"expression","value":"([-1, 6, 1], [-5, 5], 1, 1, 19/21, 1, 6, 6, 0, 是, single_instance_only)","explanation_zh":"收束波核對=([-1, 6, 1], [-5, 5], 1, 1, 19/21, 1, 6, 6, 0, 是, single_instance_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-jensen-instance-audit","seed":202610252094,"parameters":{"coefficients":[-1,6,1],"interval":[-5,5],"x":[1,1],"y":[1,1],"lambda":[19,21],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-jensen-instance-audit","semantic_template":"qf-mod25-convexfn-jensen-instance-audit/{\"coefficients\":[-1,6,1],\"interval\":[-5,5],\"lambda\":[19,21],\"x\":[1,1],\"y\":[1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0095","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-jensen-instance-audit；資料={\"coefficients\":[3,7,2],\"interval\":[-5,5],\"x\":[2,1],\"y\":[2,1],\"lambda\":[20,22]}。請依 [coefficients,interval,x,y,lambda,point,lhs,rhs,gap,passes,scope] 欄序回答。","answer":{"kind":"expression","value":"([3, 7, 2], [-5, 5], 2, 2, 10/11, 2, 28, 28, 0, 是, single_instance_only)","explanation_zh":"收束波核對=([3, 7, 2], [-5, 5], 2, 2, 10/11, 2, 28, 28, 0, 是, single_instance_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-jensen-instance-audit","seed":202610252095,"parameters":{"coefficients":[3,7,2],"interval":[-5,5],"x":[2,1],"y":[2,1],"lambda":[20,22],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-jensen-instance-audit","semantic_template":"qf-mod25-convexfn-jensen-instance-audit/{\"coefficients\":[3,7,2],\"interval\":[-5,5],\"lambda\":[20,22],\"x\":[2,1],\"y\":[2,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0096","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-jensen-instance-audit；資料={\"coefficients\":[0,8,3],\"interval\":[-5,5],\"x\":[-2,1],\"y\":[3,1],\"lambda\":[21,23]}。請依 [coefficients,interval,x,y,lambda,point,lhs,rhs,gap,passes,scope] 欄序回答。","answer":{"kind":"expression","value":"([0, 8, 3], [-5, 5], -2, 3, 21/23, -36/23, -219/23, -219/23, 0, 是, single_instance_only)","explanation_zh":"收束波核對=([0, 8, 3], [-5, 5], -2, 3, 21/23, -36/23, -219/23, -219/23, 0, 是, single_instance_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-jensen-instance-audit","seed":202610252096,"parameters":{"coefficients":[0,8,3],"interval":[-5,5],"x":[-2,1],"y":[3,1],"lambda":[21,23],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-jensen-instance-audit","semantic_template":"qf-mod25-convexfn-jensen-instance-audit/{\"coefficients\":[0,8,3],\"interval\":[-5,5],\"lambda\":[21,23],\"x\":[-2,1],\"y\":[3,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0097","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-jensen-instance-audit；資料={\"coefficients\":[1,9,-3],\"interval\":[-5,5],\"x\":[-1,1],\"y\":[-3,1],\"lambda\":[22,24]}。請依 [coefficients,interval,x,y,lambda,point,lhs,rhs,gap,passes,scope] 欄序回答。","answer":{"kind":"expression","value":"([1, 9, -3], [-5, 5], -1, -3, 11/12, -7/6, -437/36, -71/6, 11/36, 是, single_instance_only)","explanation_zh":"收束波核對=([1, 9, -3], [-5, 5], -1, -3, 11/12, -7/6, -437/36, -71/6, 11/36, 是, single_instance_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-jensen-instance-audit","seed":202610252097,"parameters":{"coefficients":[1,9,-3],"interval":[-5,5],"x":[-1,1],"y":[-3,1],"lambda":[22,24],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-jensen-instance-audit","semantic_template":"qf-mod25-convexfn-jensen-instance-audit/{\"coefficients\":[1,9,-3],\"interval\":[-5,5],\"lambda\":[22,24],\"x\":[-1,1],\"y\":[-3,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0098","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-jensen-instance-audit；資料={\"coefficients\":[2,10,-2],\"interval\":[-5,5],\"x\":[0,1],\"y\":[-2,1],\"lambda\":[23,25]}。請依 [coefficients,interval,x,y,lambda,point,lhs,rhs,gap,passes,scope] 欄序回答。","answer":{"kind":"expression","value":"([2, 10, -2], [-5, 5], 0, -2, 23/25, -4/25, -2218/625, -74/25, 368/625, 是, single_instance_only)","explanation_zh":"收束波核對=([2, 10, -2], [-5, 5], 0, -2, 23/25, -4/25, -2218/625, -74/25, 368/625, 是, single_instance_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-jensen-instance-audit","seed":202610252098,"parameters":{"coefficients":[2,10,-2],"interval":[-5,5],"x":[0,1],"y":[-2,1],"lambda":[23,25],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-jensen-instance-audit","semantic_template":"qf-mod25-convexfn-jensen-instance-audit/{\"coefficients\":[2,10,-2],\"interval\":[-5,5],\"lambda\":[23,25],\"x\":[0,1],\"y\":[-2,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0099","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-jensen-instance-audit；資料={\"coefficients\":[3,11,-1],\"interval\":[-5,5],\"x\":[1,1],\"y\":[-1,1],\"lambda\":[24,26]}。請依 [coefficients,interval,x,y,lambda,point,lhs,rhs,gap,passes,scope] 欄序回答。","answer":{"kind":"expression","value":"([3, 11, -1], [-5, 5], 1, -1, 12/13, 11/13, 1767/169, 147/13, 144/169, 是, single_instance_only)","explanation_zh":"收束波核對=([3, 11, -1], [-5, 5], 1, -1, 12/13, 11/13, 1767/169, 147/13, 144/169, 是, single_instance_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-jensen-instance-audit","seed":202610252099,"parameters":{"coefficients":[3,11,-1],"interval":[-5,5],"x":[1,1],"y":[-1,1],"lambda":[24,26],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-jensen-instance-audit","semantic_template":"qf-mod25-convexfn-jensen-instance-audit/{\"coefficients\":[3,11,-1],\"interval\":[-5,5],\"lambda\":[24,26],\"x\":[1,1],\"y\":[-1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r02-convex-function-0100","batch_id":"batch-20261013-advance-20261025-r02-convex-function-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-function"],"title_zh":"凸函數","stem_zh":"函數必須配合明示凸定義域。操作=convexfn-jensen-instance-audit；資料={\"coefficients\":[-1,12,0],\"interval\":[-5,5],\"x\":[2,1],\"y\":[0,1],\"lambda\":[25,27]}。請依 [coefficients,interval,x,y,lambda,point,lhs,rhs,gap,passes,scope] 欄序回答。","answer":{"kind":"expression","value":"([-1, 12, 0], [-5, 5], 2, 0, 25/27, 50/27, 13700/729, 500/27, -200/729, 否, nonconvexity_witness)","explanation_zh":"收束波核對=([-1, 12, 0], [-5, 5], 2, 0, 25/27, 50/27, 13700/729, 500/27, -200/729, 否, nonconvexity_witness)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸函數"]},"variant":{"family_id":"qf-mod25-convexfn-jensen-instance-audit","seed":202610252100,"parameters":{"coefficients":[-1,12,0],"interval":[-5,5],"x":[2,1],"y":[0,1],"lambda":[25,27],"domain":"Exact_univariate_quadratic_convexity","operation":"convexfn-jensen-instance-audit","semantic_template":"qf-mod25-convexfn-jensen-instance-audit/{\"coefficients\":[-1,12,0],\"interval\":[-5,5],\"lambda\":[25,27],\"x\":[2,1],\"y\":[0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}