{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r07-convex-set-0001","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-interval-certificate；資料={\"lower\":-2,\"upper\":3}。請依 [lower,upper,universal_scope,convex] 欄序回答。","answer":{"kind":"expression","value":"(-2, 3, all x,y in interval and all lambda in [0,1], 是)","explanation_zh":"依賴結構核對=(-2, 3, all x,y in interval and all lambda in [0,1], 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-interval-certificate","seed":202610247001,"parameters":{"lower":-2,"upper":3,"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-interval-certificate","semantic_template":"qf-mod24-convexset-interval-certificate/{\"lower\":-2,\"upper\":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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0002","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-interval-certificate；資料={\"lower\":-3,\"upper\":4}。請依 [lower,upper,universal_scope,convex] 欄序回答。","answer":{"kind":"expression","value":"(-3, 4, all x,y in interval and all lambda in [0,1], 是)","explanation_zh":"依賴結構核對=(-3, 4, all x,y in interval and all lambda in [0,1], 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-interval-certificate","seed":202610247002,"parameters":{"lower":-3,"upper":4,"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-interval-certificate","semantic_template":"qf-mod24-convexset-interval-certificate/{\"lower\":-3,\"upper\":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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0003","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-interval-certificate；資料={\"lower\":-4,\"upper\":5}。請依 [lower,upper,universal_scope,convex] 欄序回答。","answer":{"kind":"expression","value":"(-4, 5, all x,y in interval and all lambda in [0,1], 是)","explanation_zh":"依賴結構核對=(-4, 5, all x,y in interval and all lambda in [0,1], 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-interval-certificate","seed":202610247003,"parameters":{"lower":-4,"upper":5,"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-interval-certificate","semantic_template":"qf-mod24-convexset-interval-certificate/{\"lower\":-4,\"upper\":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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0004","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-interval-certificate；資料={\"lower\":-5,\"upper\":6}。請依 [lower,upper,universal_scope,convex] 欄序回答。","answer":{"kind":"expression","value":"(-5, 6, all x,y in interval and all lambda in [0,1], 是)","explanation_zh":"依賴結構核對=(-5, 6, all x,y in interval and all lambda in [0,1], 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-interval-certificate","seed":202610247004,"parameters":{"lower":-5,"upper":6,"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-interval-certificate","semantic_template":"qf-mod24-convexset-interval-certificate/{\"lower\":-5,\"upper\":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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0005","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-interval-certificate；資料={\"lower\":-6,\"upper\":7}。請依 [lower,upper,universal_scope,convex] 欄序回答。","answer":{"kind":"expression","value":"(-6, 7, all x,y in interval and all lambda in [0,1], 是)","explanation_zh":"依賴結構核對=(-6, 7, all x,y in interval and all lambda in [0,1], 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-interval-certificate","seed":202610247005,"parameters":{"lower":-6,"upper":7,"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-interval-certificate","semantic_template":"qf-mod24-convexset-interval-certificate/{\"lower\":-6,\"upper\":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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0006","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-interval-certificate；資料={\"lower\":-7,\"upper\":8}。請依 [lower,upper,universal_scope,convex] 欄序回答。","answer":{"kind":"expression","value":"(-7, 8, all x,y in interval and all lambda in [0,1], 是)","explanation_zh":"依賴結構核對=(-7, 8, all x,y in interval and all lambda in [0,1], 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-interval-certificate","seed":202610247006,"parameters":{"lower":-7,"upper":8,"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-interval-certificate","semantic_template":"qf-mod24-convexset-interval-certificate/{\"lower\":-7,\"upper\":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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0007","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-interval-certificate；資料={\"lower\":-8,\"upper\":9}。請依 [lower,upper,universal_scope,convex] 欄序回答。","answer":{"kind":"expression","value":"(-8, 9, all x,y in interval and all lambda in [0,1], 是)","explanation_zh":"依賴結構核對=(-8, 9, all x,y in interval and all lambda in [0,1], 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-interval-certificate","seed":202610247007,"parameters":{"lower":-8,"upper":9,"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-interval-certificate","semantic_template":"qf-mod24-convexset-interval-certificate/{\"lower\":-8,\"upper\":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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0008","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-interval-certificate；資料={\"lower\":-9,\"upper\":10}。請依 [lower,upper,universal_scope,convex] 欄序回答。","answer":{"kind":"expression","value":"(-9, 10, all x,y in interval and all lambda in [0,1], 是)","explanation_zh":"依賴結構核對=(-9, 10, all x,y in interval and all lambda in [0,1], 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-interval-certificate","seed":202610247008,"parameters":{"lower":-9,"upper":10,"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-interval-certificate","semantic_template":"qf-mod24-convexset-interval-certificate/{\"lower\":-9,\"upper\":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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0009","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-interval-certificate；資料={\"lower\":-10,\"upper\":11}。請依 [lower,upper,universal_scope,convex] 欄序回答。","answer":{"kind":"expression","value":"(-10, 11, all x,y in interval and all lambda in [0,1], 是)","explanation_zh":"依賴結構核對=(-10, 11, all x,y in interval and all lambda in [0,1], 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-interval-certificate","seed":202610247009,"parameters":{"lower":-10,"upper":11,"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-interval-certificate","semantic_template":"qf-mod24-convexset-interval-certificate/{\"lower\":-10,\"upper\":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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0010","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-interval-certificate；資料={\"lower\":-11,\"upper\":12}。請依 [lower,upper,universal_scope,convex] 欄序回答。","answer":{"kind":"expression","value":"(-11, 12, all x,y in interval and all lambda in [0,1], 是)","explanation_zh":"依賴結構核對=(-11, 12, all x,y in interval and all lambda in [0,1], 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-interval-certificate","seed":202610247010,"parameters":{"lower":-11,"upper":12,"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-interval-certificate","semantic_template":"qf-mod24-convexset-interval-certificate/{\"lower\":-11,\"upper\":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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0011","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-interval-certificate；資料={\"lower\":-12,\"upper\":13}。請依 [lower,upper,universal_scope,convex] 欄序回答。","answer":{"kind":"expression","value":"(-12, 13, all x,y in interval and all lambda in [0,1], 是)","explanation_zh":"依賴結構核對=(-12, 13, all x,y in interval and all lambda in [0,1], 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-interval-certificate","seed":202610247011,"parameters":{"lower":-12,"upper":13,"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-interval-certificate","semantic_template":"qf-mod24-convexset-interval-certificate/{\"lower\":-12,\"upper\":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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0012","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-interval-certificate；資料={\"lower\":-13,\"upper\":14}。請依 [lower,upper,universal_scope,convex] 欄序回答。","answer":{"kind":"expression","value":"(-13, 14, all x,y in interval and all lambda in [0,1], 是)","explanation_zh":"依賴結構核對=(-13, 14, all x,y in interval and all lambda in [0,1], 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-interval-certificate","seed":202610247012,"parameters":{"lower":-13,"upper":14,"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-interval-certificate","semantic_template":"qf-mod24-convexset-interval-certificate/{\"lower\":-13,\"upper\":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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0013","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-interval-certificate；資料={\"lower\":-14,\"upper\":15}。請依 [lower,upper,universal_scope,convex] 欄序回答。","answer":{"kind":"expression","value":"(-14, 15, all x,y in interval and all lambda in [0,1], 是)","explanation_zh":"依賴結構核對=(-14, 15, all x,y in interval and all lambda in [0,1], 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-interval-certificate","seed":202610247013,"parameters":{"lower":-14,"upper":15,"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-interval-certificate","semantic_template":"qf-mod24-convexset-interval-certificate/{\"lower\":-14,\"upper\":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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0014","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-interval-certificate；資料={\"lower\":-15,\"upper\":16}。請依 [lower,upper,universal_scope,convex] 欄序回答。","answer":{"kind":"expression","value":"(-15, 16, all x,y in interval and all lambda in [0,1], 是)","explanation_zh":"依賴結構核對=(-15, 16, all x,y in interval and all lambda in [0,1], 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-interval-certificate","seed":202610247014,"parameters":{"lower":-15,"upper":16,"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-interval-certificate","semantic_template":"qf-mod24-convexset-interval-certificate/{\"lower\":-15,\"upper\":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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0015","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-interval-certificate；資料={\"lower\":-16,\"upper\":17}。請依 [lower,upper,universal_scope,convex] 欄序回答。","answer":{"kind":"expression","value":"(-16, 17, all x,y in interval and all lambda in [0,1], 是)","explanation_zh":"依賴結構核對=(-16, 17, all x,y in interval and all lambda in [0,1], 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-interval-certificate","seed":202610247015,"parameters":{"lower":-16,"upper":17,"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-interval-certificate","semantic_template":"qf-mod24-convexset-interval-certificate/{\"lower\":-16,\"upper\":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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0016","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-interval-certificate；資料={\"lower\":-17,\"upper\":18}。請依 [lower,upper,universal_scope,convex] 欄序回答。","answer":{"kind":"expression","value":"(-17, 18, all x,y in interval and all lambda in [0,1], 是)","explanation_zh":"依賴結構核對=(-17, 18, all x,y in interval and all lambda in [0,1], 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-interval-certificate","seed":202610247016,"parameters":{"lower":-17,"upper":18,"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-interval-certificate","semantic_template":"qf-mod24-convexset-interval-certificate/{\"lower\":-17,\"upper\":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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0017","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-interval-certificate；資料={\"lower\":-18,\"upper\":19}。請依 [lower,upper,universal_scope,convex] 欄序回答。","answer":{"kind":"expression","value":"(-18, 19, all x,y in interval and all lambda in [0,1], 是)","explanation_zh":"依賴結構核對=(-18, 19, all x,y in interval and all lambda in [0,1], 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-interval-certificate","seed":202610247017,"parameters":{"lower":-18,"upper":19,"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-interval-certificate","semantic_template":"qf-mod24-convexset-interval-certificate/{\"lower\":-18,\"upper\":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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0018","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-interval-certificate；資料={\"lower\":-19,\"upper\":20}。請依 [lower,upper,universal_scope,convex] 欄序回答。","answer":{"kind":"expression","value":"(-19, 20, all x,y in interval and all lambda in [0,1], 是)","explanation_zh":"依賴結構核對=(-19, 20, all x,y in interval and all lambda in [0,1], 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-interval-certificate","seed":202610247018,"parameters":{"lower":-19,"upper":20,"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-interval-certificate","semantic_template":"qf-mod24-convexset-interval-certificate/{\"lower\":-19,\"upper\":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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0019","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-interval-certificate；資料={\"lower\":-20,\"upper\":21}。請依 [lower,upper,universal_scope,convex] 欄序回答。","answer":{"kind":"expression","value":"(-20, 21, all x,y in interval and all lambda in [0,1], 是)","explanation_zh":"依賴結構核對=(-20, 21, all x,y in interval and all lambda in [0,1], 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-interval-certificate","seed":202610247019,"parameters":{"lower":-20,"upper":21,"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-interval-certificate","semantic_template":"qf-mod24-convexset-interval-certificate/{\"lower\":-20,\"upper\":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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0020","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-interval-certificate；資料={\"lower\":-21,\"upper\":22}。請依 [lower,upper,universal_scope,convex] 欄序回答。","answer":{"kind":"expression","value":"(-21, 22, all x,y in interval and all lambda in [0,1], 是)","explanation_zh":"依賴結構核對=(-21, 22, all x,y in interval and all lambda in [0,1], 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-interval-certificate","seed":202610247020,"parameters":{"lower":-21,"upper":22,"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-interval-certificate","semantic_template":"qf-mod24-convexset-interval-certificate/{\"lower\":-21,\"upper\":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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0021","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-interval-certificate；資料={\"lower\":-22,\"upper\":23}。請依 [lower,upper,universal_scope,convex] 欄序回答。","answer":{"kind":"expression","value":"(-22, 23, all x,y in interval and all lambda in [0,1], 是)","explanation_zh":"依賴結構核對=(-22, 23, all x,y in interval and all lambda in [0,1], 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-interval-certificate","seed":202610247021,"parameters":{"lower":-22,"upper":23,"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-interval-certificate","semantic_template":"qf-mod24-convexset-interval-certificate/{\"lower\":-22,\"upper\":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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0022","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-interval-certificate；資料={\"lower\":-23,\"upper\":24}。請依 [lower,upper,universal_scope,convex] 欄序回答。","answer":{"kind":"expression","value":"(-23, 24, all x,y in interval and all lambda in [0,1], 是)","explanation_zh":"依賴結構核對=(-23, 24, all x,y in interval and all lambda in [0,1], 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-interval-certificate","seed":202610247022,"parameters":{"lower":-23,"upper":24,"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-interval-certificate","semantic_template":"qf-mod24-convexset-interval-certificate/{\"lower\":-23,\"upper\":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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0023","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-interval-certificate；資料={\"lower\":-24,\"upper\":25}。請依 [lower,upper,universal_scope,convex] 欄序回答。","answer":{"kind":"expression","value":"(-24, 25, all x,y in interval and all lambda in [0,1], 是)","explanation_zh":"依賴結構核對=(-24, 25, all x,y in interval and all lambda in [0,1], 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-interval-certificate","seed":202610247023,"parameters":{"lower":-24,"upper":25,"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-interval-certificate","semantic_template":"qf-mod24-convexset-interval-certificate/{\"lower\":-24,\"upper\":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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0024","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-interval-certificate；資料={\"lower\":-25,\"upper\":26}。請依 [lower,upper,universal_scope,convex] 欄序回答。","answer":{"kind":"expression","value":"(-25, 26, all x,y in interval and all lambda in [0,1], 是)","explanation_zh":"依賴結構核對=(-25, 26, all x,y in interval and all lambda in [0,1], 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-interval-certificate","seed":202610247024,"parameters":{"lower":-25,"upper":26,"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-interval-certificate","semantic_template":"qf-mod24-convexset-interval-certificate/{\"lower\":-25,\"upper\":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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0025","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-interval-certificate；資料={\"lower\":-26,\"upper\":27}。請依 [lower,upper,universal_scope,convex] 欄序回答。","answer":{"kind":"expression","value":"(-26, 27, all x,y in interval and all lambda in [0,1], 是)","explanation_zh":"依賴結構核對=(-26, 27, all x,y in interval and all lambda in [0,1], 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-interval-certificate","seed":202610247025,"parameters":{"lower":-26,"upper":27,"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-interval-certificate","semantic_template":"qf-mod24-convexset-interval-certificate/{\"lower\":-26,\"upper\":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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0026","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-box-certificate；資料={\"bounds\":[-2,3,-1,4]}。請依 [bounds,universal_scope,convex] 欄序回答。","answer":{"kind":"expression","value":"([-2, 3, -1, 4], coordinatewise interval certificate for every lambda in [0,1], 是)","explanation_zh":"依賴結構核對=([-2, 3, -1, 4], coordinatewise interval certificate for every lambda in [0,1], 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-box-certificate","seed":202610247026,"parameters":{"bounds":[-2,3,-1,4],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-box-certificate","semantic_template":"qf-mod24-convexset-box-certificate/{\"bounds\":[-2,3,-1,4]}"}},"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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0027","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-box-certificate；資料={\"bounds\":[-3,4,-2,5]}。請依 [bounds,universal_scope,convex] 欄序回答。","answer":{"kind":"expression","value":"([-3, 4, -2, 5], coordinatewise interval certificate for every lambda in [0,1], 是)","explanation_zh":"依賴結構核對=([-3, 4, -2, 5], coordinatewise interval certificate for every lambda in [0,1], 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-box-certificate","seed":202610247027,"parameters":{"bounds":[-3,4,-2,5],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-box-certificate","semantic_template":"qf-mod24-convexset-box-certificate/{\"bounds\":[-3,4,-2,5]}"}},"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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0028","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-box-certificate；資料={\"bounds\":[-4,5,-3,6]}。請依 [bounds,universal_scope,convex] 欄序回答。","answer":{"kind":"expression","value":"([-4, 5, -3, 6], coordinatewise interval certificate for every lambda in [0,1], 是)","explanation_zh":"依賴結構核對=([-4, 5, -3, 6], coordinatewise interval certificate for every lambda in [0,1], 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-box-certificate","seed":202610247028,"parameters":{"bounds":[-4,5,-3,6],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-box-certificate","semantic_template":"qf-mod24-convexset-box-certificate/{\"bounds\":[-4,5,-3,6]}"}},"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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0029","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-box-certificate；資料={\"bounds\":[-5,6,-4,7]}。請依 [bounds,universal_scope,convex] 欄序回答。","answer":{"kind":"expression","value":"([-5, 6, -4, 7], coordinatewise interval certificate for every lambda in [0,1], 是)","explanation_zh":"依賴結構核對=([-5, 6, -4, 7], coordinatewise interval certificate for every lambda in [0,1], 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-box-certificate","seed":202610247029,"parameters":{"bounds":[-5,6,-4,7],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-box-certificate","semantic_template":"qf-mod24-convexset-box-certificate/{\"bounds\":[-5,6,-4,7]}"}},"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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0030","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-box-certificate；資料={\"bounds\":[-6,7,-5,8]}。請依 [bounds,universal_scope,convex] 欄序回答。","answer":{"kind":"expression","value":"([-6, 7, -5, 8], coordinatewise interval certificate for every lambda in [0,1], 是)","explanation_zh":"依賴結構核對=([-6, 7, -5, 8], coordinatewise interval certificate for every lambda in [0,1], 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-box-certificate","seed":202610247030,"parameters":{"bounds":[-6,7,-5,8],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-box-certificate","semantic_template":"qf-mod24-convexset-box-certificate/{\"bounds\":[-6,7,-5,8]}"}},"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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0031","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-box-certificate；資料={\"bounds\":[-7,8,-6,9]}。請依 [bounds,universal_scope,convex] 欄序回答。","answer":{"kind":"expression","value":"([-7, 8, -6, 9], coordinatewise interval certificate for every lambda in [0,1], 是)","explanation_zh":"依賴結構核對=([-7, 8, -6, 9], coordinatewise interval certificate for every lambda in [0,1], 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-box-certificate","seed":202610247031,"parameters":{"bounds":[-7,8,-6,9],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-box-certificate","semantic_template":"qf-mod24-convexset-box-certificate/{\"bounds\":[-7,8,-6,9]}"}},"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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0032","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-box-certificate；資料={\"bounds\":[-8,9,-7,10]}。請依 [bounds,universal_scope,convex] 欄序回答。","answer":{"kind":"expression","value":"([-8, 9, -7, 10], coordinatewise interval certificate for every lambda in [0,1], 是)","explanation_zh":"依賴結構核對=([-8, 9, -7, 10], coordinatewise interval certificate for every lambda in [0,1], 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-box-certificate","seed":202610247032,"parameters":{"bounds":[-8,9,-7,10],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-box-certificate","semantic_template":"qf-mod24-convexset-box-certificate/{\"bounds\":[-8,9,-7,10]}"}},"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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0033","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-box-certificate；資料={\"bounds\":[-9,10,-8,11]}。請依 [bounds,universal_scope,convex] 欄序回答。","answer":{"kind":"expression","value":"([-9, 10, -8, 11], coordinatewise interval certificate for every lambda in [0,1], 是)","explanation_zh":"依賴結構核對=([-9, 10, -8, 11], coordinatewise interval certificate for every lambda in [0,1], 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-box-certificate","seed":202610247033,"parameters":{"bounds":[-9,10,-8,11],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-box-certificate","semantic_template":"qf-mod24-convexset-box-certificate/{\"bounds\":[-9,10,-8,11]}"}},"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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0034","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-box-certificate；資料={\"bounds\":[-10,11,-9,12]}。請依 [bounds,universal_scope,convex] 欄序回答。","answer":{"kind":"expression","value":"([-10, 11, -9, 12], coordinatewise interval certificate for every lambda in [0,1], 是)","explanation_zh":"依賴結構核對=([-10, 11, -9, 12], coordinatewise interval certificate for every lambda in [0,1], 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-box-certificate","seed":202610247034,"parameters":{"bounds":[-10,11,-9,12],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-box-certificate","semantic_template":"qf-mod24-convexset-box-certificate/{\"bounds\":[-10,11,-9,12]}"}},"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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0035","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-box-certificate；資料={\"bounds\":[-11,12,-10,13]}。請依 [bounds,universal_scope,convex] 欄序回答。","answer":{"kind":"expression","value":"([-11, 12, -10, 13], coordinatewise interval certificate for every lambda in [0,1], 是)","explanation_zh":"依賴結構核對=([-11, 12, -10, 13], coordinatewise interval certificate for every lambda in [0,1], 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-box-certificate","seed":202610247035,"parameters":{"bounds":[-11,12,-10,13],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-box-certificate","semantic_template":"qf-mod24-convexset-box-certificate/{\"bounds\":[-11,12,-10,13]}"}},"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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0036","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-box-certificate；資料={\"bounds\":[-12,13,-11,14]}。請依 [bounds,universal_scope,convex] 欄序回答。","answer":{"kind":"expression","value":"([-12, 13, -11, 14], coordinatewise interval certificate for every lambda in [0,1], 是)","explanation_zh":"依賴結構核對=([-12, 13, -11, 14], coordinatewise interval certificate for every lambda in [0,1], 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-box-certificate","seed":202610247036,"parameters":{"bounds":[-12,13,-11,14],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-box-certificate","semantic_template":"qf-mod24-convexset-box-certificate/{\"bounds\":[-12,13,-11,14]}"}},"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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0037","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-box-certificate；資料={\"bounds\":[-13,14,-12,15]}。請依 [bounds,universal_scope,convex] 欄序回答。","answer":{"kind":"expression","value":"([-13, 14, -12, 15], coordinatewise interval certificate for every lambda in [0,1], 是)","explanation_zh":"依賴結構核對=([-13, 14, -12, 15], coordinatewise interval certificate for every lambda in [0,1], 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-box-certificate","seed":202610247037,"parameters":{"bounds":[-13,14,-12,15],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-box-certificate","semantic_template":"qf-mod24-convexset-box-certificate/{\"bounds\":[-13,14,-12,15]}"}},"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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0038","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-box-certificate；資料={\"bounds\":[-14,15,-13,16]}。請依 [bounds,universal_scope,convex] 欄序回答。","answer":{"kind":"expression","value":"([-14, 15, -13, 16], coordinatewise interval certificate for every lambda in [0,1], 是)","explanation_zh":"依賴結構核對=([-14, 15, -13, 16], coordinatewise interval certificate for every lambda in [0,1], 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-box-certificate","seed":202610247038,"parameters":{"bounds":[-14,15,-13,16],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-box-certificate","semantic_template":"qf-mod24-convexset-box-certificate/{\"bounds\":[-14,15,-13,16]}"}},"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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0039","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-box-certificate；資料={\"bounds\":[-15,16,-14,17]}。請依 [bounds,universal_scope,convex] 欄序回答。","answer":{"kind":"expression","value":"([-15, 16, -14, 17], coordinatewise interval certificate for every lambda in [0,1], 是)","explanation_zh":"依賴結構核對=([-15, 16, -14, 17], coordinatewise interval certificate for every lambda in [0,1], 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-box-certificate","seed":202610247039,"parameters":{"bounds":[-15,16,-14,17],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-box-certificate","semantic_template":"qf-mod24-convexset-box-certificate/{\"bounds\":[-15,16,-14,17]}"}},"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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0040","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-box-certificate；資料={\"bounds\":[-16,17,-15,18]}。請依 [bounds,universal_scope,convex] 欄序回答。","answer":{"kind":"expression","value":"([-16, 17, -15, 18], coordinatewise interval certificate for every lambda in [0,1], 是)","explanation_zh":"依賴結構核對=([-16, 17, -15, 18], coordinatewise interval certificate for every lambda in [0,1], 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-box-certificate","seed":202610247040,"parameters":{"bounds":[-16,17,-15,18],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-box-certificate","semantic_template":"qf-mod24-convexset-box-certificate/{\"bounds\":[-16,17,-15,18]}"}},"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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0041","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-box-certificate；資料={\"bounds\":[-17,18,-16,19]}。請依 [bounds,universal_scope,convex] 欄序回答。","answer":{"kind":"expression","value":"([-17, 18, -16, 19], coordinatewise interval certificate for every lambda in [0,1], 是)","explanation_zh":"依賴結構核對=([-17, 18, -16, 19], coordinatewise interval certificate for every lambda in [0,1], 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-box-certificate","seed":202610247041,"parameters":{"bounds":[-17,18,-16,19],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-box-certificate","semantic_template":"qf-mod24-convexset-box-certificate/{\"bounds\":[-17,18,-16,19]}"}},"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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0042","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-box-certificate；資料={\"bounds\":[-18,19,-17,20]}。請依 [bounds,universal_scope,convex] 欄序回答。","answer":{"kind":"expression","value":"([-18, 19, -17, 20], coordinatewise interval certificate for every lambda in [0,1], 是)","explanation_zh":"依賴結構核對=([-18, 19, -17, 20], coordinatewise interval certificate for every lambda in [0,1], 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-box-certificate","seed":202610247042,"parameters":{"bounds":[-18,19,-17,20],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-box-certificate","semantic_template":"qf-mod24-convexset-box-certificate/{\"bounds\":[-18,19,-17,20]}"}},"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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0043","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-box-certificate；資料={\"bounds\":[-19,20,-18,21]}。請依 [bounds,universal_scope,convex] 欄序回答。","answer":{"kind":"expression","value":"([-19, 20, -18, 21], coordinatewise interval certificate for every lambda in [0,1], 是)","explanation_zh":"依賴結構核對=([-19, 20, -18, 21], coordinatewise interval certificate for every lambda in [0,1], 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-box-certificate","seed":202610247043,"parameters":{"bounds":[-19,20,-18,21],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-box-certificate","semantic_template":"qf-mod24-convexset-box-certificate/{\"bounds\":[-19,20,-18,21]}"}},"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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0044","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-box-certificate；資料={\"bounds\":[-20,21,-19,22]}。請依 [bounds,universal_scope,convex] 欄序回答。","answer":{"kind":"expression","value":"([-20, 21, -19, 22], coordinatewise interval certificate for every lambda in [0,1], 是)","explanation_zh":"依賴結構核對=([-20, 21, -19, 22], coordinatewise interval certificate for every lambda in [0,1], 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-box-certificate","seed":202610247044,"parameters":{"bounds":[-20,21,-19,22],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-box-certificate","semantic_template":"qf-mod24-convexset-box-certificate/{\"bounds\":[-20,21,-19,22]}"}},"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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0045","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-box-certificate；資料={\"bounds\":[-21,22,-20,23]}。請依 [bounds,universal_scope,convex] 欄序回答。","answer":{"kind":"expression","value":"([-21, 22, -20, 23], coordinatewise interval certificate for every lambda in [0,1], 是)","explanation_zh":"依賴結構核對=([-21, 22, -20, 23], coordinatewise interval certificate for every lambda in [0,1], 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-box-certificate","seed":202610247045,"parameters":{"bounds":[-21,22,-20,23],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-box-certificate","semantic_template":"qf-mod24-convexset-box-certificate/{\"bounds\":[-21,22,-20,23]}"}},"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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0046","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-box-certificate；資料={\"bounds\":[-22,23,-21,24]}。請依 [bounds,universal_scope,convex] 欄序回答。","answer":{"kind":"expression","value":"([-22, 23, -21, 24], coordinatewise interval certificate for every lambda in [0,1], 是)","explanation_zh":"依賴結構核對=([-22, 23, -21, 24], coordinatewise interval certificate for every lambda in [0,1], 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-box-certificate","seed":202610247046,"parameters":{"bounds":[-22,23,-21,24],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-box-certificate","semantic_template":"qf-mod24-convexset-box-certificate/{\"bounds\":[-22,23,-21,24]}"}},"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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0047","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-box-certificate；資料={\"bounds\":[-23,24,-22,25]}。請依 [bounds,universal_scope,convex] 欄序回答。","answer":{"kind":"expression","value":"([-23, 24, -22, 25], coordinatewise interval certificate for every lambda in [0,1], 是)","explanation_zh":"依賴結構核對=([-23, 24, -22, 25], coordinatewise interval certificate for every lambda in [0,1], 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-box-certificate","seed":202610247047,"parameters":{"bounds":[-23,24,-22,25],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-box-certificate","semantic_template":"qf-mod24-convexset-box-certificate/{\"bounds\":[-23,24,-22,25]}"}},"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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0048","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-box-certificate；資料={\"bounds\":[-24,25,-23,26]}。請依 [bounds,universal_scope,convex] 欄序回答。","answer":{"kind":"expression","value":"([-24, 25, -23, 26], coordinatewise interval certificate for every lambda in [0,1], 是)","explanation_zh":"依賴結構核對=([-24, 25, -23, 26], coordinatewise interval certificate for every lambda in [0,1], 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-box-certificate","seed":202610247048,"parameters":{"bounds":[-24,25,-23,26],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-box-certificate","semantic_template":"qf-mod24-convexset-box-certificate/{\"bounds\":[-24,25,-23,26]}"}},"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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0049","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-box-certificate；資料={\"bounds\":[-25,26,-24,27]}。請依 [bounds,universal_scope,convex] 欄序回答。","answer":{"kind":"expression","value":"([-25, 26, -24, 27], coordinatewise interval certificate for every lambda in [0,1], 是)","explanation_zh":"依賴結構核對=([-25, 26, -24, 27], coordinatewise interval certificate for every lambda in [0,1], 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-box-certificate","seed":202610247049,"parameters":{"bounds":[-25,26,-24,27],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-box-certificate","semantic_template":"qf-mod24-convexset-box-certificate/{\"bounds\":[-25,26,-24,27]}"}},"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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0050","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-box-certificate；資料={\"bounds\":[-26,27,-25,28]}。請依 [bounds,universal_scope,convex] 欄序回答。","answer":{"kind":"expression","value":"([-26, 27, -25, 28], coordinatewise interval certificate for every lambda in [0,1], 是)","explanation_zh":"依賴結構核對=([-26, 27, -25, 28], coordinatewise interval certificate for every lambda in [0,1], 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-box-certificate","seed":202610247050,"parameters":{"bounds":[-26,27,-25,28],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-box-certificate","semantic_template":"qf-mod24-convexset-box-certificate/{\"bounds\":[-26,27,-25,28]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0051","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-halfspace-combination；資料={\"normal\":[1,1],\"bound\":2,\"left\":[0,0],\"right\":[1,0],\"lambda\":[1,3]}。請依 [normal,bound,left,right,left_dot,right_dot,lambda,point,point_dot,inside] 欄序回答。","answer":{"kind":"expression","value":"([1, 1], 2, [0, 0], [1, 0], 0, 1, 1/3, [2/3, 0], 2/3, 是)","explanation_zh":"依賴結構核對=([1, 1], 2, [0, 0], [1, 0], 0, 1, 1/3, [2/3, 0], 2/3, 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-halfspace-combination","seed":202610247051,"parameters":{"normal":[1,1],"bound":2,"left":[0,0],"right":[1,0],"lambda":[1,3],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-halfspace-combination","semantic_template":"qf-mod24-convexset-halfspace-combination/{\"bound\":2,\"lambda\":[1,3],\"left\":[0,0],\"normal\":[1,1],\"right\":[1,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0052","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-halfspace-combination；資料={\"normal\":[1,1],\"bound\":3,\"left\":[0,0],\"right\":[2,0],\"lambda\":[2,4]}。請依 [normal,bound,left,right,left_dot,right_dot,lambda,point,point_dot,inside] 欄序回答。","answer":{"kind":"expression","value":"([1, 1], 3, [0, 0], [2, 0], 0, 2, 1/2, [1, 0], 1, 是)","explanation_zh":"依賴結構核對=([1, 1], 3, [0, 0], [2, 0], 0, 2, 1/2, [1, 0], 1, 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-halfspace-combination","seed":202610247052,"parameters":{"normal":[1,1],"bound":3,"left":[0,0],"right":[2,0],"lambda":[2,4],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-halfspace-combination","semantic_template":"qf-mod24-convexset-halfspace-combination/{\"bound\":3,\"lambda\":[2,4],\"left\":[0,0],\"normal\":[1,1],\"right\":[2,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0053","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-halfspace-combination；資料={\"normal\":[1,1],\"bound\":4,\"left\":[0,0],\"right\":[3,0],\"lambda\":[3,5]}。請依 [normal,bound,left,right,left_dot,right_dot,lambda,point,point_dot,inside] 欄序回答。","answer":{"kind":"expression","value":"([1, 1], 4, [0, 0], [3, 0], 0, 3, 3/5, [6/5, 0], 6/5, 是)","explanation_zh":"依賴結構核對=([1, 1], 4, [0, 0], [3, 0], 0, 3, 3/5, [6/5, 0], 6/5, 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-halfspace-combination","seed":202610247053,"parameters":{"normal":[1,1],"bound":4,"left":[0,0],"right":[3,0],"lambda":[3,5],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-halfspace-combination","semantic_template":"qf-mod24-convexset-halfspace-combination/{\"bound\":4,\"lambda\":[3,5],\"left\":[0,0],\"normal\":[1,1],\"right\":[3,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0054","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-halfspace-combination；資料={\"normal\":[1,1],\"bound\":5,\"left\":[0,0],\"right\":[4,0],\"lambda\":[4,6]}。請依 [normal,bound,left,right,left_dot,right_dot,lambda,point,point_dot,inside] 欄序回答。","answer":{"kind":"expression","value":"([1, 1], 5, [0, 0], [4, 0], 0, 4, 2/3, [4/3, 0], 4/3, 是)","explanation_zh":"依賴結構核對=([1, 1], 5, [0, 0], [4, 0], 0, 4, 2/3, [4/3, 0], 4/3, 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-halfspace-combination","seed":202610247054,"parameters":{"normal":[1,1],"bound":5,"left":[0,0],"right":[4,0],"lambda":[4,6],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-halfspace-combination","semantic_template":"qf-mod24-convexset-halfspace-combination/{\"bound\":5,\"lambda\":[4,6],\"left\":[0,0],\"normal\":[1,1],\"right\":[4,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0055","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-halfspace-combination；資料={\"normal\":[1,1],\"bound\":6,\"left\":[0,0],\"right\":[5,0],\"lambda\":[5,7]}。請依 [normal,bound,left,right,left_dot,right_dot,lambda,point,point_dot,inside] 欄序回答。","answer":{"kind":"expression","value":"([1, 1], 6, [0, 0], [5, 0], 0, 5, 5/7, [10/7, 0], 10/7, 是)","explanation_zh":"依賴結構核對=([1, 1], 6, [0, 0], [5, 0], 0, 5, 5/7, [10/7, 0], 10/7, 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-halfspace-combination","seed":202610247055,"parameters":{"normal":[1,1],"bound":6,"left":[0,0],"right":[5,0],"lambda":[5,7],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-halfspace-combination","semantic_template":"qf-mod24-convexset-halfspace-combination/{\"bound\":6,\"lambda\":[5,7],\"left\":[0,0],\"normal\":[1,1],\"right\":[5,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0056","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-halfspace-combination；資料={\"normal\":[1,1],\"bound\":7,\"left\":[0,0],\"right\":[6,0],\"lambda\":[6,8]}。請依 [normal,bound,left,right,left_dot,right_dot,lambda,point,point_dot,inside] 欄序回答。","answer":{"kind":"expression","value":"([1, 1], 7, [0, 0], [6, 0], 0, 6, 3/4, [3/2, 0], 3/2, 是)","explanation_zh":"依賴結構核對=([1, 1], 7, [0, 0], [6, 0], 0, 6, 3/4, [3/2, 0], 3/2, 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-halfspace-combination","seed":202610247056,"parameters":{"normal":[1,1],"bound":7,"left":[0,0],"right":[6,0],"lambda":[6,8],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-halfspace-combination","semantic_template":"qf-mod24-convexset-halfspace-combination/{\"bound\":7,\"lambda\":[6,8],\"left\":[0,0],\"normal\":[1,1],\"right\":[6,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0057","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-halfspace-combination；資料={\"normal\":[1,1],\"bound\":8,\"left\":[0,0],\"right\":[7,0],\"lambda\":[7,9]}。請依 [normal,bound,left,right,left_dot,right_dot,lambda,point,point_dot,inside] 欄序回答。","answer":{"kind":"expression","value":"([1, 1], 8, [0, 0], [7, 0], 0, 7, 7/9, [14/9, 0], 14/9, 是)","explanation_zh":"依賴結構核對=([1, 1], 8, [0, 0], [7, 0], 0, 7, 7/9, [14/9, 0], 14/9, 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-halfspace-combination","seed":202610247057,"parameters":{"normal":[1,1],"bound":8,"left":[0,0],"right":[7,0],"lambda":[7,9],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-halfspace-combination","semantic_template":"qf-mod24-convexset-halfspace-combination/{\"bound\":8,\"lambda\":[7,9],\"left\":[0,0],\"normal\":[1,1],\"right\":[7,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0058","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-halfspace-combination；資料={\"normal\":[1,1],\"bound\":9,\"left\":[0,0],\"right\":[8,0],\"lambda\":[8,10]}。請依 [normal,bound,left,right,left_dot,right_dot,lambda,point,point_dot,inside] 欄序回答。","answer":{"kind":"expression","value":"([1, 1], 9, [0, 0], [8, 0], 0, 8, 4/5, [8/5, 0], 8/5, 是)","explanation_zh":"依賴結構核對=([1, 1], 9, [0, 0], [8, 0], 0, 8, 4/5, [8/5, 0], 8/5, 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-halfspace-combination","seed":202610247058,"parameters":{"normal":[1,1],"bound":9,"left":[0,0],"right":[8,0],"lambda":[8,10],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-halfspace-combination","semantic_template":"qf-mod24-convexset-halfspace-combination/{\"bound\":9,\"lambda\":[8,10],\"left\":[0,0],\"normal\":[1,1],\"right\":[8,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0059","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-halfspace-combination；資料={\"normal\":[1,1],\"bound\":10,\"left\":[0,0],\"right\":[9,0],\"lambda\":[9,11]}。請依 [normal,bound,left,right,left_dot,right_dot,lambda,point,point_dot,inside] 欄序回答。","answer":{"kind":"expression","value":"([1, 1], 10, [0, 0], [9, 0], 0, 9, 9/11, [18/11, 0], 18/11, 是)","explanation_zh":"依賴結構核對=([1, 1], 10, [0, 0], [9, 0], 0, 9, 9/11, [18/11, 0], 18/11, 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-halfspace-combination","seed":202610247059,"parameters":{"normal":[1,1],"bound":10,"left":[0,0],"right":[9,0],"lambda":[9,11],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-halfspace-combination","semantic_template":"qf-mod24-convexset-halfspace-combination/{\"bound\":10,\"lambda\":[9,11],\"left\":[0,0],\"normal\":[1,1],\"right\":[9,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0060","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-halfspace-combination；資料={\"normal\":[1,1],\"bound\":11,\"left\":[0,0],\"right\":[10,0],\"lambda\":[10,12]}。請依 [normal,bound,left,right,left_dot,right_dot,lambda,point,point_dot,inside] 欄序回答。","answer":{"kind":"expression","value":"([1, 1], 11, [0, 0], [10, 0], 0, 10, 5/6, [5/3, 0], 5/3, 是)","explanation_zh":"依賴結構核對=([1, 1], 11, [0, 0], [10, 0], 0, 10, 5/6, [5/3, 0], 5/3, 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-halfspace-combination","seed":202610247060,"parameters":{"normal":[1,1],"bound":11,"left":[0,0],"right":[10,0],"lambda":[10,12],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-halfspace-combination","semantic_template":"qf-mod24-convexset-halfspace-combination/{\"bound\":11,\"lambda\":[10,12],\"left\":[0,0],\"normal\":[1,1],\"right\":[10,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0061","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-halfspace-combination；資料={\"normal\":[1,1],\"bound\":12,\"left\":[0,0],\"right\":[11,0],\"lambda\":[11,13]}。請依 [normal,bound,left,right,left_dot,right_dot,lambda,point,point_dot,inside] 欄序回答。","answer":{"kind":"expression","value":"([1, 1], 12, [0, 0], [11, 0], 0, 11, 11/13, [22/13, 0], 22/13, 是)","explanation_zh":"依賴結構核對=([1, 1], 12, [0, 0], [11, 0], 0, 11, 11/13, [22/13, 0], 22/13, 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-halfspace-combination","seed":202610247061,"parameters":{"normal":[1,1],"bound":12,"left":[0,0],"right":[11,0],"lambda":[11,13],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-halfspace-combination","semantic_template":"qf-mod24-convexset-halfspace-combination/{\"bound\":12,\"lambda\":[11,13],\"left\":[0,0],\"normal\":[1,1],\"right\":[11,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0062","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-halfspace-combination；資料={\"normal\":[1,1],\"bound\":13,\"left\":[0,0],\"right\":[12,0],\"lambda\":[12,14]}。請依 [normal,bound,left,right,left_dot,right_dot,lambda,point,point_dot,inside] 欄序回答。","answer":{"kind":"expression","value":"([1, 1], 13, [0, 0], [12, 0], 0, 12, 6/7, [12/7, 0], 12/7, 是)","explanation_zh":"依賴結構核對=([1, 1], 13, [0, 0], [12, 0], 0, 12, 6/7, [12/7, 0], 12/7, 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-halfspace-combination","seed":202610247062,"parameters":{"normal":[1,1],"bound":13,"left":[0,0],"right":[12,0],"lambda":[12,14],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-halfspace-combination","semantic_template":"qf-mod24-convexset-halfspace-combination/{\"bound\":13,\"lambda\":[12,14],\"left\":[0,0],\"normal\":[1,1],\"right\":[12,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0063","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-halfspace-combination；資料={\"normal\":[1,1],\"bound\":14,\"left\":[0,0],\"right\":[13,0],\"lambda\":[13,15]}。請依 [normal,bound,left,right,left_dot,right_dot,lambda,point,point_dot,inside] 欄序回答。","answer":{"kind":"expression","value":"([1, 1], 14, [0, 0], [13, 0], 0, 13, 13/15, [26/15, 0], 26/15, 是)","explanation_zh":"依賴結構核對=([1, 1], 14, [0, 0], [13, 0], 0, 13, 13/15, [26/15, 0], 26/15, 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-halfspace-combination","seed":202610247063,"parameters":{"normal":[1,1],"bound":14,"left":[0,0],"right":[13,0],"lambda":[13,15],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-halfspace-combination","semantic_template":"qf-mod24-convexset-halfspace-combination/{\"bound\":14,\"lambda\":[13,15],\"left\":[0,0],\"normal\":[1,1],\"right\":[13,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0064","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-halfspace-combination；資料={\"normal\":[1,1],\"bound\":15,\"left\":[0,0],\"right\":[14,0],\"lambda\":[14,16]}。請依 [normal,bound,left,right,left_dot,right_dot,lambda,point,point_dot,inside] 欄序回答。","answer":{"kind":"expression","value":"([1, 1], 15, [0, 0], [14, 0], 0, 14, 7/8, [7/4, 0], 7/4, 是)","explanation_zh":"依賴結構核對=([1, 1], 15, [0, 0], [14, 0], 0, 14, 7/8, [7/4, 0], 7/4, 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-halfspace-combination","seed":202610247064,"parameters":{"normal":[1,1],"bound":15,"left":[0,0],"right":[14,0],"lambda":[14,16],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-halfspace-combination","semantic_template":"qf-mod24-convexset-halfspace-combination/{\"bound\":15,\"lambda\":[14,16],\"left\":[0,0],\"normal\":[1,1],\"right\":[14,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0065","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-halfspace-combination；資料={\"normal\":[1,1],\"bound\":16,\"left\":[0,0],\"right\":[15,0],\"lambda\":[15,17]}。請依 [normal,bound,left,right,left_dot,right_dot,lambda,point,point_dot,inside] 欄序回答。","answer":{"kind":"expression","value":"([1, 1], 16, [0, 0], [15, 0], 0, 15, 15/17, [30/17, 0], 30/17, 是)","explanation_zh":"依賴結構核對=([1, 1], 16, [0, 0], [15, 0], 0, 15, 15/17, [30/17, 0], 30/17, 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-halfspace-combination","seed":202610247065,"parameters":{"normal":[1,1],"bound":16,"left":[0,0],"right":[15,0],"lambda":[15,17],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-halfspace-combination","semantic_template":"qf-mod24-convexset-halfspace-combination/{\"bound\":16,\"lambda\":[15,17],\"left\":[0,0],\"normal\":[1,1],\"right\":[15,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0066","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-halfspace-combination；資料={\"normal\":[1,1],\"bound\":17,\"left\":[0,0],\"right\":[16,0],\"lambda\":[16,18]}。請依 [normal,bound,left,right,left_dot,right_dot,lambda,point,point_dot,inside] 欄序回答。","answer":{"kind":"expression","value":"([1, 1], 17, [0, 0], [16, 0], 0, 16, 8/9, [16/9, 0], 16/9, 是)","explanation_zh":"依賴結構核對=([1, 1], 17, [0, 0], [16, 0], 0, 16, 8/9, [16/9, 0], 16/9, 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-halfspace-combination","seed":202610247066,"parameters":{"normal":[1,1],"bound":17,"left":[0,0],"right":[16,0],"lambda":[16,18],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-halfspace-combination","semantic_template":"qf-mod24-convexset-halfspace-combination/{\"bound\":17,\"lambda\":[16,18],\"left\":[0,0],\"normal\":[1,1],\"right\":[16,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0067","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-halfspace-combination；資料={\"normal\":[1,1],\"bound\":18,\"left\":[0,0],\"right\":[17,0],\"lambda\":[17,19]}。請依 [normal,bound,left,right,left_dot,right_dot,lambda,point,point_dot,inside] 欄序回答。","answer":{"kind":"expression","value":"([1, 1], 18, [0, 0], [17, 0], 0, 17, 17/19, [34/19, 0], 34/19, 是)","explanation_zh":"依賴結構核對=([1, 1], 18, [0, 0], [17, 0], 0, 17, 17/19, [34/19, 0], 34/19, 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-halfspace-combination","seed":202610247067,"parameters":{"normal":[1,1],"bound":18,"left":[0,0],"right":[17,0],"lambda":[17,19],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-halfspace-combination","semantic_template":"qf-mod24-convexset-halfspace-combination/{\"bound\":18,\"lambda\":[17,19],\"left\":[0,0],\"normal\":[1,1],\"right\":[17,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0068","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-halfspace-combination；資料={\"normal\":[1,1],\"bound\":19,\"left\":[0,0],\"right\":[18,0],\"lambda\":[18,20]}。請依 [normal,bound,left,right,left_dot,right_dot,lambda,point,point_dot,inside] 欄序回答。","answer":{"kind":"expression","value":"([1, 1], 19, [0, 0], [18, 0], 0, 18, 9/10, [9/5, 0], 9/5, 是)","explanation_zh":"依賴結構核對=([1, 1], 19, [0, 0], [18, 0], 0, 18, 9/10, [9/5, 0], 9/5, 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-halfspace-combination","seed":202610247068,"parameters":{"normal":[1,1],"bound":19,"left":[0,0],"right":[18,0],"lambda":[18,20],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-halfspace-combination","semantic_template":"qf-mod24-convexset-halfspace-combination/{\"bound\":19,\"lambda\":[18,20],\"left\":[0,0],\"normal\":[1,1],\"right\":[18,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0069","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-halfspace-combination；資料={\"normal\":[1,1],\"bound\":20,\"left\":[0,0],\"right\":[19,0],\"lambda\":[19,21]}。請依 [normal,bound,left,right,left_dot,right_dot,lambda,point,point_dot,inside] 欄序回答。","answer":{"kind":"expression","value":"([1, 1], 20, [0, 0], [19, 0], 0, 19, 19/21, [38/21, 0], 38/21, 是)","explanation_zh":"依賴結構核對=([1, 1], 20, [0, 0], [19, 0], 0, 19, 19/21, [38/21, 0], 38/21, 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-halfspace-combination","seed":202610247069,"parameters":{"normal":[1,1],"bound":20,"left":[0,0],"right":[19,0],"lambda":[19,21],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-halfspace-combination","semantic_template":"qf-mod24-convexset-halfspace-combination/{\"bound\":20,\"lambda\":[19,21],\"left\":[0,0],\"normal\":[1,1],\"right\":[19,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0070","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-halfspace-combination；資料={\"normal\":[1,1],\"bound\":21,\"left\":[0,0],\"right\":[20,0],\"lambda\":[20,22]}。請依 [normal,bound,left,right,left_dot,right_dot,lambda,point,point_dot,inside] 欄序回答。","answer":{"kind":"expression","value":"([1, 1], 21, [0, 0], [20, 0], 0, 20, 10/11, [20/11, 0], 20/11, 是)","explanation_zh":"依賴結構核對=([1, 1], 21, [0, 0], [20, 0], 0, 20, 10/11, [20/11, 0], 20/11, 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-halfspace-combination","seed":202610247070,"parameters":{"normal":[1,1],"bound":21,"left":[0,0],"right":[20,0],"lambda":[20,22],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-halfspace-combination","semantic_template":"qf-mod24-convexset-halfspace-combination/{\"bound\":21,\"lambda\":[20,22],\"left\":[0,0],\"normal\":[1,1],\"right\":[20,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0071","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-halfspace-combination；資料={\"normal\":[1,1],\"bound\":22,\"left\":[0,0],\"right\":[21,0],\"lambda\":[21,23]}。請依 [normal,bound,left,right,left_dot,right_dot,lambda,point,point_dot,inside] 欄序回答。","answer":{"kind":"expression","value":"([1, 1], 22, [0, 0], [21, 0], 0, 21, 21/23, [42/23, 0], 42/23, 是)","explanation_zh":"依賴結構核對=([1, 1], 22, [0, 0], [21, 0], 0, 21, 21/23, [42/23, 0], 42/23, 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-halfspace-combination","seed":202610247071,"parameters":{"normal":[1,1],"bound":22,"left":[0,0],"right":[21,0],"lambda":[21,23],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-halfspace-combination","semantic_template":"qf-mod24-convexset-halfspace-combination/{\"bound\":22,\"lambda\":[21,23],\"left\":[0,0],\"normal\":[1,1],\"right\":[21,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0072","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-halfspace-combination；資料={\"normal\":[1,1],\"bound\":23,\"left\":[0,0],\"right\":[22,0],\"lambda\":[22,24]}。請依 [normal,bound,left,right,left_dot,right_dot,lambda,point,point_dot,inside] 欄序回答。","answer":{"kind":"expression","value":"([1, 1], 23, [0, 0], [22, 0], 0, 22, 11/12, [11/6, 0], 11/6, 是)","explanation_zh":"依賴結構核對=([1, 1], 23, [0, 0], [22, 0], 0, 22, 11/12, [11/6, 0], 11/6, 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-halfspace-combination","seed":202610247072,"parameters":{"normal":[1,1],"bound":23,"left":[0,0],"right":[22,0],"lambda":[22,24],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-halfspace-combination","semantic_template":"qf-mod24-convexset-halfspace-combination/{\"bound\":23,\"lambda\":[22,24],\"left\":[0,0],\"normal\":[1,1],\"right\":[22,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0073","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-halfspace-combination；資料={\"normal\":[1,1],\"bound\":24,\"left\":[0,0],\"right\":[23,0],\"lambda\":[23,25]}。請依 [normal,bound,left,right,left_dot,right_dot,lambda,point,point_dot,inside] 欄序回答。","answer":{"kind":"expression","value":"([1, 1], 24, [0, 0], [23, 0], 0, 23, 23/25, [46/25, 0], 46/25, 是)","explanation_zh":"依賴結構核對=([1, 1], 24, [0, 0], [23, 0], 0, 23, 23/25, [46/25, 0], 46/25, 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-halfspace-combination","seed":202610247073,"parameters":{"normal":[1,1],"bound":24,"left":[0,0],"right":[23,0],"lambda":[23,25],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-halfspace-combination","semantic_template":"qf-mod24-convexset-halfspace-combination/{\"bound\":24,\"lambda\":[23,25],\"left\":[0,0],\"normal\":[1,1],\"right\":[23,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0074","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-halfspace-combination；資料={\"normal\":[1,1],\"bound\":25,\"left\":[0,0],\"right\":[24,0],\"lambda\":[24,26]}。請依 [normal,bound,left,right,left_dot,right_dot,lambda,point,point_dot,inside] 欄序回答。","answer":{"kind":"expression","value":"([1, 1], 25, [0, 0], [24, 0], 0, 24, 12/13, [24/13, 0], 24/13, 是)","explanation_zh":"依賴結構核對=([1, 1], 25, [0, 0], [24, 0], 0, 24, 12/13, [24/13, 0], 24/13, 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-halfspace-combination","seed":202610247074,"parameters":{"normal":[1,1],"bound":25,"left":[0,0],"right":[24,0],"lambda":[24,26],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-halfspace-combination","semantic_template":"qf-mod24-convexset-halfspace-combination/{\"bound\":25,\"lambda\":[24,26],\"left\":[0,0],\"normal\":[1,1],\"right\":[24,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0075","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-halfspace-combination；資料={\"normal\":[1,1],\"bound\":26,\"left\":[0,0],\"right\":[25,0],\"lambda\":[25,27]}。請依 [normal,bound,left,right,left_dot,right_dot,lambda,point,point_dot,inside] 欄序回答。","answer":{"kind":"expression","value":"([1, 1], 26, [0, 0], [25, 0], 0, 25, 25/27, [50/27, 0], 50/27, 是)","explanation_zh":"依賴結構核對=([1, 1], 26, [0, 0], [25, 0], 0, 25, 25/27, [50/27, 0], 50/27, 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-halfspace-combination","seed":202610247075,"parameters":{"normal":[1,1],"bound":26,"left":[0,0],"right":[25,0],"lambda":[25,27],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-halfspace-combination","semantic_template":"qf-mod24-convexset-halfspace-combination/{\"bound\":26,\"lambda\":[25,27],\"left\":[0,0],\"normal\":[1,1],\"right\":[25,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0076","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-finite-midpoint-counterexample；資料={\"points\":[[0,0],[2,0],[0,2]],\"left\":[2,0],\"right\":[0,2]}。請依 [points,left,right,midpoint,midpoint_present,nonconvex_witness] 欄序回答。","answer":{"kind":"expression","value":"([[0, 0], [2, 0], [0, 2]], [2, 0], [0, 2], [1, 1], 否, 是)","explanation_zh":"依賴結構核對=([[0, 0], [2, 0], [0, 2]], [2, 0], [0, 2], [1, 1], 否, 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-finite-midpoint-counterexample","seed":202610247076,"parameters":{"points":[[0,0],[2,0],[0,2]],"left":[2,0],"right":[0,2],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-finite-midpoint-counterexample","semantic_template":"qf-mod24-convexset-finite-midpoint-counterexample/{\"left\":[2,0],\"points\":[[0,0],[0,2],[2,0]],\"right\":[0,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0077","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-finite-midpoint-counterexample；資料={\"points\":[[0,0],[4,0],[0,4]],\"left\":[4,0],\"right\":[0,4]}。請依 [points,left,right,midpoint,midpoint_present,nonconvex_witness] 欄序回答。","answer":{"kind":"expression","value":"([[0, 0], [4, 0], [0, 4]], [4, 0], [0, 4], [2, 2], 否, 是)","explanation_zh":"依賴結構核對=([[0, 0], [4, 0], [0, 4]], [4, 0], [0, 4], [2, 2], 否, 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-finite-midpoint-counterexample","seed":202610247077,"parameters":{"points":[[0,0],[4,0],[0,4]],"left":[4,0],"right":[0,4],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-finite-midpoint-counterexample","semantic_template":"qf-mod24-convexset-finite-midpoint-counterexample/{\"left\":[4,0],\"points\":[[0,0],[0,4],[4,0]],\"right\":[0,4]}"}},"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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0078","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-finite-midpoint-counterexample；資料={\"points\":[[0,0],[6,0],[0,6]],\"left\":[6,0],\"right\":[0,6]}。請依 [points,left,right,midpoint,midpoint_present,nonconvex_witness] 欄序回答。","answer":{"kind":"expression","value":"([[0, 0], [6, 0], [0, 6]], [6, 0], [0, 6], [3, 3], 否, 是)","explanation_zh":"依賴結構核對=([[0, 0], [6, 0], [0, 6]], [6, 0], [0, 6], [3, 3], 否, 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-finite-midpoint-counterexample","seed":202610247078,"parameters":{"points":[[0,0],[6,0],[0,6]],"left":[6,0],"right":[0,6],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-finite-midpoint-counterexample","semantic_template":"qf-mod24-convexset-finite-midpoint-counterexample/{\"left\":[6,0],\"points\":[[0,0],[0,6],[6,0]],\"right\":[0,6]}"}},"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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0079","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-finite-midpoint-counterexample；資料={\"points\":[[0,0],[8,0],[0,8]],\"left\":[8,0],\"right\":[0,8]}。請依 [points,left,right,midpoint,midpoint_present,nonconvex_witness] 欄序回答。","answer":{"kind":"expression","value":"([[0, 0], [8, 0], [0, 8]], [8, 0], [0, 8], [4, 4], 否, 是)","explanation_zh":"依賴結構核對=([[0, 0], [8, 0], [0, 8]], [8, 0], [0, 8], [4, 4], 否, 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-finite-midpoint-counterexample","seed":202610247079,"parameters":{"points":[[0,0],[8,0],[0,8]],"left":[8,0],"right":[0,8],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-finite-midpoint-counterexample","semantic_template":"qf-mod24-convexset-finite-midpoint-counterexample/{\"left\":[8,0],\"points\":[[0,0],[0,8],[8,0]],\"right\":[0,8]}"}},"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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0080","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-finite-midpoint-counterexample；資料={\"points\":[[0,0],[10,0],[0,10]],\"left\":[10,0],\"right\":[0,10]}。請依 [points,left,right,midpoint,midpoint_present,nonconvex_witness] 欄序回答。","answer":{"kind":"expression","value":"([[0, 0], [10, 0], [0, 10]], [10, 0], [0, 10], [5, 5], 否, 是)","explanation_zh":"依賴結構核對=([[0, 0], [10, 0], [0, 10]], [10, 0], [0, 10], [5, 5], 否, 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-finite-midpoint-counterexample","seed":202610247080,"parameters":{"points":[[0,0],[10,0],[0,10]],"left":[10,0],"right":[0,10],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-finite-midpoint-counterexample","semantic_template":"qf-mod24-convexset-finite-midpoint-counterexample/{\"left\":[10,0],\"points\":[[0,0],[0,10],[10,0]],\"right\":[0,10]}"}},"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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0081","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-finite-midpoint-counterexample；資料={\"points\":[[0,0],[12,0],[0,12]],\"left\":[12,0],\"right\":[0,12]}。請依 [points,left,right,midpoint,midpoint_present,nonconvex_witness] 欄序回答。","answer":{"kind":"expression","value":"([[0, 0], [12, 0], [0, 12]], [12, 0], [0, 12], [6, 6], 否, 是)","explanation_zh":"依賴結構核對=([[0, 0], [12, 0], [0, 12]], [12, 0], [0, 12], [6, 6], 否, 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-finite-midpoint-counterexample","seed":202610247081,"parameters":{"points":[[0,0],[12,0],[0,12]],"left":[12,0],"right":[0,12],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-finite-midpoint-counterexample","semantic_template":"qf-mod24-convexset-finite-midpoint-counterexample/{\"left\":[12,0],\"points\":[[0,0],[0,12],[12,0]],\"right\":[0,12]}"}},"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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0082","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-finite-midpoint-counterexample；資料={\"points\":[[0,0],[14,0],[0,14]],\"left\":[14,0],\"right\":[0,14]}。請依 [points,left,right,midpoint,midpoint_present,nonconvex_witness] 欄序回答。","answer":{"kind":"expression","value":"([[0, 0], [14, 0], [0, 14]], [14, 0], [0, 14], [7, 7], 否, 是)","explanation_zh":"依賴結構核對=([[0, 0], [14, 0], [0, 14]], [14, 0], [0, 14], [7, 7], 否, 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-finite-midpoint-counterexample","seed":202610247082,"parameters":{"points":[[0,0],[14,0],[0,14]],"left":[14,0],"right":[0,14],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-finite-midpoint-counterexample","semantic_template":"qf-mod24-convexset-finite-midpoint-counterexample/{\"left\":[14,0],\"points\":[[0,0],[0,14],[14,0]],\"right\":[0,14]}"}},"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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0083","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-finite-midpoint-counterexample；資料={\"points\":[[0,0],[16,0],[0,16]],\"left\":[16,0],\"right\":[0,16]}。請依 [points,left,right,midpoint,midpoint_present,nonconvex_witness] 欄序回答。","answer":{"kind":"expression","value":"([[0, 0], [16, 0], [0, 16]], [16, 0], [0, 16], [8, 8], 否, 是)","explanation_zh":"依賴結構核對=([[0, 0], [16, 0], [0, 16]], [16, 0], [0, 16], [8, 8], 否, 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-finite-midpoint-counterexample","seed":202610247083,"parameters":{"points":[[0,0],[16,0],[0,16]],"left":[16,0],"right":[0,16],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-finite-midpoint-counterexample","semantic_template":"qf-mod24-convexset-finite-midpoint-counterexample/{\"left\":[16,0],\"points\":[[0,0],[0,16],[16,0]],\"right\":[0,16]}"}},"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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0084","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-finite-midpoint-counterexample；資料={\"points\":[[0,0],[18,0],[0,18]],\"left\":[18,0],\"right\":[0,18]}。請依 [points,left,right,midpoint,midpoint_present,nonconvex_witness] 欄序回答。","answer":{"kind":"expression","value":"([[0, 0], [18, 0], [0, 18]], [18, 0], [0, 18], [9, 9], 否, 是)","explanation_zh":"依賴結構核對=([[0, 0], [18, 0], [0, 18]], [18, 0], [0, 18], [9, 9], 否, 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-finite-midpoint-counterexample","seed":202610247084,"parameters":{"points":[[0,0],[18,0],[0,18]],"left":[18,0],"right":[0,18],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-finite-midpoint-counterexample","semantic_template":"qf-mod24-convexset-finite-midpoint-counterexample/{\"left\":[18,0],\"points\":[[0,0],[0,18],[18,0]],\"right\":[0,18]}"}},"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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0085","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-finite-midpoint-counterexample；資料={\"points\":[[0,0],[20,0],[0,20]],\"left\":[20,0],\"right\":[0,20]}。請依 [points,left,right,midpoint,midpoint_present,nonconvex_witness] 欄序回答。","answer":{"kind":"expression","value":"([[0, 0], [20, 0], [0, 20]], [20, 0], [0, 20], [10, 10], 否, 是)","explanation_zh":"依賴結構核對=([[0, 0], [20, 0], [0, 20]], [20, 0], [0, 20], [10, 10], 否, 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-finite-midpoint-counterexample","seed":202610247085,"parameters":{"points":[[0,0],[20,0],[0,20]],"left":[20,0],"right":[0,20],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-finite-midpoint-counterexample","semantic_template":"qf-mod24-convexset-finite-midpoint-counterexample/{\"left\":[20,0],\"points\":[[0,0],[0,20],[20,0]],\"right\":[0,20]}"}},"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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0086","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-finite-midpoint-counterexample；資料={\"points\":[[0,0],[22,0],[0,22]],\"left\":[22,0],\"right\":[0,22]}。請依 [points,left,right,midpoint,midpoint_present,nonconvex_witness] 欄序回答。","answer":{"kind":"expression","value":"([[0, 0], [22, 0], [0, 22]], [22, 0], [0, 22], [11, 11], 否, 是)","explanation_zh":"依賴結構核對=([[0, 0], [22, 0], [0, 22]], [22, 0], [0, 22], [11, 11], 否, 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-finite-midpoint-counterexample","seed":202610247086,"parameters":{"points":[[0,0],[22,0],[0,22]],"left":[22,0],"right":[0,22],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-finite-midpoint-counterexample","semantic_template":"qf-mod24-convexset-finite-midpoint-counterexample/{\"left\":[22,0],\"points\":[[0,0],[0,22],[22,0]],\"right\":[0,22]}"}},"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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0087","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-finite-midpoint-counterexample；資料={\"points\":[[0,0],[24,0],[0,24]],\"left\":[24,0],\"right\":[0,24]}。請依 [points,left,right,midpoint,midpoint_present,nonconvex_witness] 欄序回答。","answer":{"kind":"expression","value":"([[0, 0], [24, 0], [0, 24]], [24, 0], [0, 24], [12, 12], 否, 是)","explanation_zh":"依賴結構核對=([[0, 0], [24, 0], [0, 24]], [24, 0], [0, 24], [12, 12], 否, 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-finite-midpoint-counterexample","seed":202610247087,"parameters":{"points":[[0,0],[24,0],[0,24]],"left":[24,0],"right":[0,24],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-finite-midpoint-counterexample","semantic_template":"qf-mod24-convexset-finite-midpoint-counterexample/{\"left\":[24,0],\"points\":[[0,0],[0,24],[24,0]],\"right\":[0,24]}"}},"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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0088","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-finite-midpoint-counterexample；資料={\"points\":[[0,0],[26,0],[0,26]],\"left\":[26,0],\"right\":[0,26]}。請依 [points,left,right,midpoint,midpoint_present,nonconvex_witness] 欄序回答。","answer":{"kind":"expression","value":"([[0, 0], [26, 0], [0, 26]], [26, 0], [0, 26], [13, 13], 否, 是)","explanation_zh":"依賴結構核對=([[0, 0], [26, 0], [0, 26]], [26, 0], [0, 26], [13, 13], 否, 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-finite-midpoint-counterexample","seed":202610247088,"parameters":{"points":[[0,0],[26,0],[0,26]],"left":[26,0],"right":[0,26],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-finite-midpoint-counterexample","semantic_template":"qf-mod24-convexset-finite-midpoint-counterexample/{\"left\":[26,0],\"points\":[[0,0],[0,26],[26,0]],\"right\":[0,26]}"}},"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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0089","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-finite-midpoint-counterexample；資料={\"points\":[[0,0],[28,0],[0,28]],\"left\":[28,0],\"right\":[0,28]}。請依 [points,left,right,midpoint,midpoint_present,nonconvex_witness] 欄序回答。","answer":{"kind":"expression","value":"([[0, 0], [28, 0], [0, 28]], [28, 0], [0, 28], [14, 14], 否, 是)","explanation_zh":"依賴結構核對=([[0, 0], [28, 0], [0, 28]], [28, 0], [0, 28], [14, 14], 否, 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-finite-midpoint-counterexample","seed":202610247089,"parameters":{"points":[[0,0],[28,0],[0,28]],"left":[28,0],"right":[0,28],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-finite-midpoint-counterexample","semantic_template":"qf-mod24-convexset-finite-midpoint-counterexample/{\"left\":[28,0],\"points\":[[0,0],[0,28],[28,0]],\"right\":[0,28]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0090","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-finite-midpoint-counterexample；資料={\"points\":[[0,0],[30,0],[0,30]],\"left\":[30,0],\"right\":[0,30]}。請依 [points,left,right,midpoint,midpoint_present,nonconvex_witness] 欄序回答。","answer":{"kind":"expression","value":"([[0, 0], [30, 0], [0, 30]], [30, 0], [0, 30], [15, 15], 否, 是)","explanation_zh":"依賴結構核對=([[0, 0], [30, 0], [0, 30]], [30, 0], [0, 30], [15, 15], 否, 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-finite-midpoint-counterexample","seed":202610247090,"parameters":{"points":[[0,0],[30,0],[0,30]],"left":[30,0],"right":[0,30],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-finite-midpoint-counterexample","semantic_template":"qf-mod24-convexset-finite-midpoint-counterexample/{\"left\":[30,0],\"points\":[[0,0],[0,30],[30,0]],\"right\":[0,30]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0091","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-finite-midpoint-counterexample；資料={\"points\":[[0,0],[32,0],[0,32]],\"left\":[32,0],\"right\":[0,32]}。請依 [points,left,right,midpoint,midpoint_present,nonconvex_witness] 欄序回答。","answer":{"kind":"expression","value":"([[0, 0], [32, 0], [0, 32]], [32, 0], [0, 32], [16, 16], 否, 是)","explanation_zh":"依賴結構核對=([[0, 0], [32, 0], [0, 32]], [32, 0], [0, 32], [16, 16], 否, 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-finite-midpoint-counterexample","seed":202610247091,"parameters":{"points":[[0,0],[32,0],[0,32]],"left":[32,0],"right":[0,32],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-finite-midpoint-counterexample","semantic_template":"qf-mod24-convexset-finite-midpoint-counterexample/{\"left\":[32,0],\"points\":[[0,0],[0,32],[32,0]],\"right\":[0,32]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0092","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-finite-midpoint-counterexample；資料={\"points\":[[0,0],[34,0],[0,34]],\"left\":[34,0],\"right\":[0,34]}。請依 [points,left,right,midpoint,midpoint_present,nonconvex_witness] 欄序回答。","answer":{"kind":"expression","value":"([[0, 0], [34, 0], [0, 34]], [34, 0], [0, 34], [17, 17], 否, 是)","explanation_zh":"依賴結構核對=([[0, 0], [34, 0], [0, 34]], [34, 0], [0, 34], [17, 17], 否, 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-finite-midpoint-counterexample","seed":202610247092,"parameters":{"points":[[0,0],[34,0],[0,34]],"left":[34,0],"right":[0,34],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-finite-midpoint-counterexample","semantic_template":"qf-mod24-convexset-finite-midpoint-counterexample/{\"left\":[34,0],\"points\":[[0,0],[0,34],[34,0]],\"right\":[0,34]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0093","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-finite-midpoint-counterexample；資料={\"points\":[[0,0],[36,0],[0,36]],\"left\":[36,0],\"right\":[0,36]}。請依 [points,left,right,midpoint,midpoint_present,nonconvex_witness] 欄序回答。","answer":{"kind":"expression","value":"([[0, 0], [36, 0], [0, 36]], [36, 0], [0, 36], [18, 18], 否, 是)","explanation_zh":"依賴結構核對=([[0, 0], [36, 0], [0, 36]], [36, 0], [0, 36], [18, 18], 否, 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-finite-midpoint-counterexample","seed":202610247093,"parameters":{"points":[[0,0],[36,0],[0,36]],"left":[36,0],"right":[0,36],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-finite-midpoint-counterexample","semantic_template":"qf-mod24-convexset-finite-midpoint-counterexample/{\"left\":[36,0],\"points\":[[0,0],[0,36],[36,0]],\"right\":[0,36]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0094","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-finite-midpoint-counterexample；資料={\"points\":[[0,0],[38,0],[0,38]],\"left\":[38,0],\"right\":[0,38]}。請依 [points,left,right,midpoint,midpoint_present,nonconvex_witness] 欄序回答。","answer":{"kind":"expression","value":"([[0, 0], [38, 0], [0, 38]], [38, 0], [0, 38], [19, 19], 否, 是)","explanation_zh":"依賴結構核對=([[0, 0], [38, 0], [0, 38]], [38, 0], [0, 38], [19, 19], 否, 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-finite-midpoint-counterexample","seed":202610247094,"parameters":{"points":[[0,0],[38,0],[0,38]],"left":[38,0],"right":[0,38],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-finite-midpoint-counterexample","semantic_template":"qf-mod24-convexset-finite-midpoint-counterexample/{\"left\":[38,0],\"points\":[[0,0],[0,38],[38,0]],\"right\":[0,38]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0095","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-finite-midpoint-counterexample；資料={\"points\":[[0,0],[40,0],[0,40]],\"left\":[40,0],\"right\":[0,40]}。請依 [points,left,right,midpoint,midpoint_present,nonconvex_witness] 欄序回答。","answer":{"kind":"expression","value":"([[0, 0], [40, 0], [0, 40]], [40, 0], [0, 40], [20, 20], 否, 是)","explanation_zh":"依賴結構核對=([[0, 0], [40, 0], [0, 40]], [40, 0], [0, 40], [20, 20], 否, 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-finite-midpoint-counterexample","seed":202610247095,"parameters":{"points":[[0,0],[40,0],[0,40]],"left":[40,0],"right":[0,40],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-finite-midpoint-counterexample","semantic_template":"qf-mod24-convexset-finite-midpoint-counterexample/{\"left\":[40,0],\"points\":[[0,0],[0,40],[40,0]],\"right\":[0,40]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0096","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-finite-midpoint-counterexample；資料={\"points\":[[0,0],[42,0],[0,42]],\"left\":[42,0],\"right\":[0,42]}。請依 [points,left,right,midpoint,midpoint_present,nonconvex_witness] 欄序回答。","answer":{"kind":"expression","value":"([[0, 0], [42, 0], [0, 42]], [42, 0], [0, 42], [21, 21], 否, 是)","explanation_zh":"依賴結構核對=([[0, 0], [42, 0], [0, 42]], [42, 0], [0, 42], [21, 21], 否, 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-finite-midpoint-counterexample","seed":202610247096,"parameters":{"points":[[0,0],[42,0],[0,42]],"left":[42,0],"right":[0,42],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-finite-midpoint-counterexample","semantic_template":"qf-mod24-convexset-finite-midpoint-counterexample/{\"left\":[42,0],\"points\":[[0,0],[0,42],[42,0]],\"right\":[0,42]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0097","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-finite-midpoint-counterexample；資料={\"points\":[[0,0],[44,0],[0,44]],\"left\":[44,0],\"right\":[0,44]}。請依 [points,left,right,midpoint,midpoint_present,nonconvex_witness] 欄序回答。","answer":{"kind":"expression","value":"([[0, 0], [44, 0], [0, 44]], [44, 0], [0, 44], [22, 22], 否, 是)","explanation_zh":"依賴結構核對=([[0, 0], [44, 0], [0, 44]], [44, 0], [0, 44], [22, 22], 否, 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-finite-midpoint-counterexample","seed":202610247097,"parameters":{"points":[[0,0],[44,0],[0,44]],"left":[44,0],"right":[0,44],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-finite-midpoint-counterexample","semantic_template":"qf-mod24-convexset-finite-midpoint-counterexample/{\"left\":[44,0],\"points\":[[0,0],[0,44],[44,0]],\"right\":[0,44]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0098","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-finite-midpoint-counterexample；資料={\"points\":[[0,0],[46,0],[0,46]],\"left\":[46,0],\"right\":[0,46]}。請依 [points,left,right,midpoint,midpoint_present,nonconvex_witness] 欄序回答。","answer":{"kind":"expression","value":"([[0, 0], [46, 0], [0, 46]], [46, 0], [0, 46], [23, 23], 否, 是)","explanation_zh":"依賴結構核對=([[0, 0], [46, 0], [0, 46]], [46, 0], [0, 46], [23, 23], 否, 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-finite-midpoint-counterexample","seed":202610247098,"parameters":{"points":[[0,0],[46,0],[0,46]],"left":[46,0],"right":[0,46],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-finite-midpoint-counterexample","semantic_template":"qf-mod24-convexset-finite-midpoint-counterexample/{\"left\":[46,0],\"points\":[[0,0],[0,46],[46,0]],\"right\":[0,46]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0099","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-finite-midpoint-counterexample；資料={\"points\":[[0,0],[48,0],[0,48]],\"left\":[48,0],\"right\":[0,48]}。請依 [points,left,right,midpoint,midpoint_present,nonconvex_witness] 欄序回答。","answer":{"kind":"expression","value":"([[0, 0], [48, 0], [0, 48]], [48, 0], [0, 48], [24, 24], 否, 是)","explanation_zh":"依賴結構核對=([[0, 0], [48, 0], [0, 48]], [48, 0], [0, 48], [24, 24], 否, 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-finite-midpoint-counterexample","seed":202610247099,"parameters":{"points":[[0,0],[48,0],[0,48]],"left":[48,0],"right":[0,48],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-finite-midpoint-counterexample","semantic_template":"qf-mod24-convexset-finite-midpoint-counterexample/{\"left\":[48,0],\"points\":[[0,0],[0,48],[48,0]],\"right\":[0,48]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、Markov長期性、PDE適定性、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-20261024-r07-convex-set-0100","batch_id":"batch-20261013-advance-20261024-r07-convex-set-foundations","language":"zh-Hant","source_mko_ids":["mko-convex-set"],"title_zh":"凸集合證書與反例","stem_zh":"量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-finite-midpoint-counterexample；資料={\"points\":[[0,0],[50,0],[0,50]],\"left\":[50,0],\"right\":[0,50]}。請依 [points,left,right,midpoint,midpoint_present,nonconvex_witness] 欄序回答。","answer":{"kind":"expression","value":"([[0, 0], [50, 0], [0, 50]], [50, 0], [0, 50], [25, 25], 否, 是)","explanation_zh":"依賴結構核對=([[0, 0], [50, 0], [0, 50]], [50, 0], [0, 50], [25, 25], 否, 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；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["凸集合證書與反例"]},"variant":{"family_id":"qf-mod24-convexset-finite-midpoint-counterexample","seed":202610247100,"parameters":{"points":[[0,0],[50,0],[0,50]],"left":[50,0],"right":[0,50],"domain":"Exact_algebraic_convex_set_certificates","operation":"convexset-finite-midpoint-counterexample","semantic_template":"qf-mod24-convexset-finite-midpoint-counterexample/{\"left\":[50,0],\"points\":[[0,0],[0,50],[50,0]],\"right\":[0,50]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_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、Markov長期性、PDE適定性、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"}}