{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0001","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-finite-certificate；資料={\"n\":4,\"metric\":[[0,3,4,5],[3,0,5,6],[4,5,0,7],[5,6,7,0]]}。請依 [n,min_positive_distance,universal_witness,complete] 欄序回答。","answer":{"kind":"expression","value":"(4, 3, epsilon=min_positive_distance forces every Cauchy tail to be constant, 是)","explanation_zh":"Wave4根節點核對=(4, 3, epsilon=min_positive_distance forces every Cauchy tail to be constant, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-finite-certificate","seed":202610273001,"parameters":{"n":4,"metric":[[0,3,4,5],[3,0,5,6],[4,5,0,7],[5,6,7,0]],"domain":"Certified_complete_finite_metric_space","operation":"complete-finite-certificate","semantic_template":"qf-mod27-complete-finite-certificate/{\"metric\":[[0,3,4,5],[3,0,5,6],[4,5,0,7],[5,6,7,0]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0002","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-finite-certificate；資料={\"n\":4,\"metric\":[[0,4,7,9],[4,0,7,9],[7,7,0,12],[9,9,12,0]]}。請依 [n,min_positive_distance,universal_witness,complete] 欄序回答。","answer":{"kind":"expression","value":"(4, 4, epsilon=min_positive_distance forces every Cauchy tail to be constant, 是)","explanation_zh":"Wave4根節點核對=(4, 4, epsilon=min_positive_distance forces every Cauchy tail to be constant, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-finite-certificate","seed":202610273002,"parameters":{"n":4,"metric":[[0,4,7,9],[4,0,7,9],[7,7,0,12],[9,9,12,0]],"domain":"Certified_complete_finite_metric_space","operation":"complete-finite-certificate","semantic_template":"qf-mod27-complete-finite-certificate/{\"metric\":[[0,4,7,9],[4,0,7,9],[7,7,0,12],[9,9,12,0]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0003","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-finite-certificate；資料={\"n\":4,\"metric\":[[0,5,10,13],[5,0,9,12],[10,9,0,17],[13,12,17,0]]}。請依 [n,min_positive_distance,universal_witness,complete] 欄序回答。","answer":{"kind":"expression","value":"(4, 5, epsilon=min_positive_distance forces every Cauchy tail to be constant, 是)","explanation_zh":"Wave4根節點核對=(4, 5, epsilon=min_positive_distance forces every Cauchy tail to be constant, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-finite-certificate","seed":202610273003,"parameters":{"n":4,"metric":[[0,5,10,13],[5,0,9,12],[10,9,0,17],[13,12,17,0]],"domain":"Certified_complete_finite_metric_space","operation":"complete-finite-certificate","semantic_template":"qf-mod27-complete-finite-certificate/{\"metric\":[[0,5,10,13],[5,0,9,12],[10,9,0,17],[13,12,17,0]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0004","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-finite-certificate；資料={\"n\":4,\"metric\":[[0,6,13,17],[6,0,11,15],[13,11,0,22],[17,15,22,0]]}。請依 [n,min_positive_distance,universal_witness,complete] 欄序回答。","answer":{"kind":"expression","value":"(4, 6, epsilon=min_positive_distance forces every Cauchy tail to be constant, 是)","explanation_zh":"Wave4根節點核對=(4, 6, epsilon=min_positive_distance forces every Cauchy tail to be constant, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-finite-certificate","seed":202610273004,"parameters":{"n":4,"metric":[[0,6,13,17],[6,0,11,15],[13,11,0,22],[17,15,22,0]],"domain":"Certified_complete_finite_metric_space","operation":"complete-finite-certificate","semantic_template":"qf-mod27-complete-finite-certificate/{\"metric\":[[0,6,13,17],[6,0,11,15],[13,11,0,22],[17,15,22,0]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0005","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-finite-certificate；資料={\"n\":4,\"metric\":[[0,7,9,10],[7,0,6,7],[9,6,0,9],[10,7,9,0]]}。請依 [n,min_positive_distance,universal_witness,complete] 欄序回答。","answer":{"kind":"expression","value":"(4, 6, epsilon=min_positive_distance forces every Cauchy tail to be constant, 是)","explanation_zh":"Wave4根節點核對=(4, 6, epsilon=min_positive_distance forces every Cauchy tail to be constant, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-finite-certificate","seed":202610273005,"parameters":{"n":4,"metric":[[0,7,9,10],[7,0,6,7],[9,6,0,9],[10,7,9,0]],"domain":"Certified_complete_finite_metric_space","operation":"complete-finite-certificate","semantic_template":"qf-mod27-complete-finite-certificate/{\"metric\":[[0,7,9,10],[7,0,6,7],[9,6,0,9],[10,7,9,0]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0006","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-finite-certificate；資料={\"n\":4,\"metric\":[[0,4,7,9],[4,0,9,11],[7,9,0,14],[9,11,14,0]]}。請依 [n,min_positive_distance,universal_witness,complete] 欄序回答。","answer":{"kind":"expression","value":"(4, 4, epsilon=min_positive_distance forces every Cauchy tail to be constant, 是)","explanation_zh":"Wave4根節點核對=(4, 4, epsilon=min_positive_distance forces every Cauchy tail to be constant, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-finite-certificate","seed":202610273006,"parameters":{"n":4,"metric":[[0,4,7,9],[4,0,9,11],[7,9,0,14],[9,11,14,0]],"domain":"Certified_complete_finite_metric_space","operation":"complete-finite-certificate","semantic_template":"qf-mod27-complete-finite-certificate/{\"metric\":[[0,4,7,9],[4,0,9,11],[7,9,0,14],[9,11,14,0]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0007","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-finite-certificate；資料={\"n\":4,\"metric\":[[0,5,10,13],[5,0,11,14],[10,11,0,19],[13,14,19,0]]}。請依 [n,min_positive_distance,universal_witness,complete] 欄序回答。","answer":{"kind":"expression","value":"(4, 5, epsilon=min_positive_distance forces every Cauchy tail to be constant, 是)","explanation_zh":"Wave4根節點核對=(4, 5, epsilon=min_positive_distance forces every Cauchy tail to be constant, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-finite-certificate","seed":202610273007,"parameters":{"n":4,"metric":[[0,5,10,13],[5,0,11,14],[10,11,0,19],[13,14,19,0]],"domain":"Certified_complete_finite_metric_space","operation":"complete-finite-certificate","semantic_template":"qf-mod27-complete-finite-certificate/{\"metric\":[[0,5,10,13],[5,0,11,14],[10,11,0,19],[13,14,19,0]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0008","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-finite-certificate；資料={\"n\":4,\"metric\":[[0,6,6,17],[6,0,6,17],[6,6,0,17],[17,17,17,0]]}。請依 [n,min_positive_distance,universal_witness,complete] 欄序回答。","answer":{"kind":"expression","value":"(4, 6, epsilon=min_positive_distance forces every Cauchy tail to be constant, 是)","explanation_zh":"Wave4根節點核對=(4, 6, epsilon=min_positive_distance forces every Cauchy tail to be constant, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-finite-certificate","seed":202610273008,"parameters":{"n":4,"metric":[[0,6,6,17],[6,0,6,17],[6,6,0,17],[17,17,17,0]],"domain":"Certified_complete_finite_metric_space","operation":"complete-finite-certificate","semantic_template":"qf-mod27-complete-finite-certificate/{\"metric\":[[0,6,6,17],[6,0,6,17],[6,6,0,17],[17,17,17,0]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0009","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-finite-certificate；資料={\"n\":4,\"metric\":[[0,7,9,10],[7,0,8,9],[9,8,0,11],[10,9,11,0]]}。請依 [n,min_positive_distance,universal_witness,complete] 欄序回答。","answer":{"kind":"expression","value":"(4, 7, epsilon=min_positive_distance forces every Cauchy tail to be constant, 是)","explanation_zh":"Wave4根節點核對=(4, 7, epsilon=min_positive_distance forces every Cauchy tail to be constant, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-finite-certificate","seed":202610273009,"parameters":{"n":4,"metric":[[0,7,9,10],[7,0,8,9],[9,8,0,11],[10,9,11,0]],"domain":"Certified_complete_finite_metric_space","operation":"complete-finite-certificate","semantic_template":"qf-mod27-complete-finite-certificate/{\"metric\":[[0,7,9,10],[7,0,8,9],[9,8,0,11],[10,9,11,0]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0010","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-finite-certificate；資料={\"n\":4,\"metric\":[[0,8,12,14],[8,0,10,12],[12,10,0,16],[14,12,16,0]]}。請依 [n,min_positive_distance,universal_witness,complete] 欄序回答。","answer":{"kind":"expression","value":"(4, 8, epsilon=min_positive_distance forces every Cauchy tail to be constant, 是)","explanation_zh":"Wave4根節點核對=(4, 8, epsilon=min_positive_distance forces every Cauchy tail to be constant, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-finite-certificate","seed":202610273010,"parameters":{"n":4,"metric":[[0,8,12,14],[8,0,10,12],[12,10,0,16],[14,12,16,0]],"domain":"Certified_complete_finite_metric_space","operation":"complete-finite-certificate","semantic_template":"qf-mod27-complete-finite-certificate/{\"metric\":[[0,8,12,14],[8,0,10,12],[12,10,0,16],[14,12,16,0]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0011","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-finite-certificate；資料={\"n\":4,\"metric\":[[0,5,10,13],[5,0,13,16],[10,13,0,21],[13,16,21,0]]}。請依 [n,min_positive_distance,universal_witness,complete] 欄序回答。","answer":{"kind":"expression","value":"(4, 5, epsilon=min_positive_distance forces every Cauchy tail to be constant, 是)","explanation_zh":"Wave4根節點核對=(4, 5, epsilon=min_positive_distance forces every Cauchy tail to be constant, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-finite-certificate","seed":202610273011,"parameters":{"n":4,"metric":[[0,5,10,13],[5,0,13,16],[10,13,0,21],[13,16,21,0]],"domain":"Certified_complete_finite_metric_space","operation":"complete-finite-certificate","semantic_template":"qf-mod27-complete-finite-certificate/{\"metric\":[[0,5,10,13],[5,0,13,16],[10,13,0,21],[13,16,21,0]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0012","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-finite-certificate；資料={\"n\":4,\"metric\":[[0,6,6,6],[6,0,8,8],[6,8,0,8],[6,8,8,0]]}。請依 [n,min_positive_distance,universal_witness,complete] 欄序回答。","answer":{"kind":"expression","value":"(4, 6, epsilon=min_positive_distance forces every Cauchy tail to be constant, 是)","explanation_zh":"Wave4根節點核對=(4, 6, epsilon=min_positive_distance forces every Cauchy tail to be constant, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-finite-certificate","seed":202610273012,"parameters":{"n":4,"metric":[[0,6,6,6],[6,0,8,8],[6,8,0,8],[6,8,8,0]],"domain":"Certified_complete_finite_metric_space","operation":"complete-finite-certificate","semantic_template":"qf-mod27-complete-finite-certificate/{\"metric\":[[0,6,6,6],[6,0,8,8],[6,8,0,8],[6,8,8,0]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0013","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-finite-certificate；資料={\"n\":4,\"metric\":[[0,7,9,10],[7,0,10,11],[9,10,0,13],[10,11,13,0]]}。請依 [n,min_positive_distance,universal_witness,complete] 欄序回答。","answer":{"kind":"expression","value":"(4, 7, epsilon=min_positive_distance forces every Cauchy tail to be constant, 是)","explanation_zh":"Wave4根節點核對=(4, 7, epsilon=min_positive_distance forces every Cauchy tail to be constant, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-finite-certificate","seed":202610273013,"parameters":{"n":4,"metric":[[0,7,9,10],[7,0,10,11],[9,10,0,13],[10,11,13,0]],"domain":"Certified_complete_finite_metric_space","operation":"complete-finite-certificate","semantic_template":"qf-mod27-complete-finite-certificate/{\"metric\":[[0,7,9,10],[7,0,10,11],[9,10,0,13],[10,11,13,0]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0014","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-finite-certificate；資料={\"n\":4,\"metric\":[[0,8,12,14],[8,0,12,14],[12,12,0,18],[14,14,18,0]]}。請依 [n,min_positive_distance,universal_witness,complete] 欄序回答。","answer":{"kind":"expression","value":"(4, 8, epsilon=min_positive_distance forces every Cauchy tail to be constant, 是)","explanation_zh":"Wave4根節點核對=(4, 8, epsilon=min_positive_distance forces every Cauchy tail to be constant, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-finite-certificate","seed":202610273014,"parameters":{"n":4,"metric":[[0,8,12,14],[8,0,12,14],[12,12,0,18],[14,14,18,0]],"domain":"Certified_complete_finite_metric_space","operation":"complete-finite-certificate","semantic_template":"qf-mod27-complete-finite-certificate/{\"metric\":[[0,8,12,14],[8,0,12,14],[12,12,0,18],[14,14,18,0]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0015","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-finite-certificate；資料={\"n\":4,\"metric\":[[0,9,8,18],[9,0,7,17],[8,7,0,16],[18,17,16,0]]}。請依 [n,min_positive_distance,universal_witness,complete] 欄序回答。","answer":{"kind":"expression","value":"(4, 7, epsilon=min_positive_distance forces every Cauchy tail to be constant, 是)","explanation_zh":"Wave4根節點核對=(4, 7, epsilon=min_positive_distance forces every Cauchy tail to be constant, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-finite-certificate","seed":202610273015,"parameters":{"n":4,"metric":[[0,9,8,18],[9,0,7,17],[8,7,0,16],[18,17,16,0]],"domain":"Certified_complete_finite_metric_space","operation":"complete-finite-certificate","semantic_template":"qf-mod27-complete-finite-certificate/{\"metric\":[[0,9,8,18],[9,0,7,17],[8,7,0,16],[18,17,16,0]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0016","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-finite-certificate；資料={\"n\":4,\"metric\":[[0,6,6,6],[6,0,10,10],[6,10,0,10],[6,10,10,0]]}。請依 [n,min_positive_distance,universal_witness,complete] 欄序回答。","answer":{"kind":"expression","value":"(4, 6, epsilon=min_positive_distance forces every Cauchy tail to be constant, 是)","explanation_zh":"Wave4根節點核對=(4, 6, epsilon=min_positive_distance forces every Cauchy tail to be constant, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-finite-certificate","seed":202610273016,"parameters":{"n":4,"metric":[[0,6,6,6],[6,0,10,10],[6,10,0,10],[6,10,10,0]],"domain":"Certified_complete_finite_metric_space","operation":"complete-finite-certificate","semantic_template":"qf-mod27-complete-finite-certificate/{\"metric\":[[0,6,6,6],[6,0,10,10],[6,10,0,10],[6,10,10,0]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0017","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-finite-certificate；資料={\"n\":4,\"metric\":[[0,7,9,10],[7,0,12,13],[9,12,0,15],[10,13,15,0]]}。請依 [n,min_positive_distance,universal_witness,complete] 欄序回答。","answer":{"kind":"expression","value":"(4, 7, epsilon=min_positive_distance forces every Cauchy tail to be constant, 是)","explanation_zh":"Wave4根節點核對=(4, 7, epsilon=min_positive_distance forces every Cauchy tail to be constant, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-finite-certificate","seed":202610273017,"parameters":{"n":4,"metric":[[0,7,9,10],[7,0,12,13],[9,12,0,15],[10,13,15,0]],"domain":"Certified_complete_finite_metric_space","operation":"complete-finite-certificate","semantic_template":"qf-mod27-complete-finite-certificate/{\"metric\":[[0,7,9,10],[7,0,12,13],[9,12,0,15],[10,13,15,0]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0018","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-finite-certificate；資料={\"n\":4,\"metric\":[[0,8,12,14],[8,0,14,16],[12,14,0,20],[14,16,20,0]]}。請依 [n,min_positive_distance,universal_witness,complete] 欄序回答。","answer":{"kind":"expression","value":"(4, 8, epsilon=min_positive_distance forces every Cauchy tail to be constant, 是)","explanation_zh":"Wave4根節點核對=(4, 8, epsilon=min_positive_distance forces every Cauchy tail to be constant, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-finite-certificate","seed":202610273018,"parameters":{"n":4,"metric":[[0,8,12,14],[8,0,14,16],[12,14,0,20],[14,16,20,0]],"domain":"Certified_complete_finite_metric_space","operation":"complete-finite-certificate","semantic_template":"qf-mod27-complete-finite-certificate/{\"metric\":[[0,8,12,14],[8,0,14,16],[12,14,0,20],[14,16,20,0]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0019","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-finite-certificate；資料={\"n\":4,\"metric\":[[0,9,8,18],[9,0,9,19],[8,9,0,18],[18,19,18,0]]}。請依 [n,min_positive_distance,universal_witness,complete] 欄序回答。","answer":{"kind":"expression","value":"(4, 8, epsilon=min_positive_distance forces every Cauchy tail to be constant, 是)","explanation_zh":"Wave4根節點核對=(4, 8, epsilon=min_positive_distance forces every Cauchy tail to be constant, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-finite-certificate","seed":202610273019,"parameters":{"n":4,"metric":[[0,9,8,18],[9,0,9,19],[8,9,0,18],[18,19,18,0]],"domain":"Certified_complete_finite_metric_space","operation":"complete-finite-certificate","semantic_template":"qf-mod27-complete-finite-certificate/{\"metric\":[[0,9,8,18],[9,0,9,19],[8,9,0,18],[18,19,18,0]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0020","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-finite-certificate；資料={\"n\":4,\"metric\":[[0,10,11,11],[10,0,11,11],[11,11,0,12],[11,11,12,0]]}。請依 [n,min_positive_distance,universal_witness,complete] 欄序回答。","answer":{"kind":"expression","value":"(4, 10, epsilon=min_positive_distance forces every Cauchy tail to be constant, 是)","explanation_zh":"Wave4根節點核對=(4, 10, epsilon=min_positive_distance forces every Cauchy tail to be constant, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-finite-certificate","seed":202610273020,"parameters":{"n":4,"metric":[[0,10,11,11],[10,0,11,11],[11,11,0,12],[11,11,12,0]],"domain":"Certified_complete_finite_metric_space","operation":"complete-finite-certificate","semantic_template":"qf-mod27-complete-finite-certificate/{\"metric\":[[0,10,11,11],[10,0,11,11],[11,11,0,12],[11,11,12,0]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0021","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-finite-certificate；資料={\"n\":4,\"metric\":[[0,7,9,10],[7,0,14,15],[9,14,0,17],[10,15,17,0]]}。請依 [n,min_positive_distance,universal_witness,complete] 欄序回答。","answer":{"kind":"expression","value":"(4, 7, epsilon=min_positive_distance forces every Cauchy tail to be constant, 是)","explanation_zh":"Wave4根節點核對=(4, 7, epsilon=min_positive_distance forces every Cauchy tail to be constant, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-finite-certificate","seed":202610273021,"parameters":{"n":4,"metric":[[0,7,9,10],[7,0,14,15],[9,14,0,17],[10,15,17,0]],"domain":"Certified_complete_finite_metric_space","operation":"complete-finite-certificate","semantic_template":"qf-mod27-complete-finite-certificate/{\"metric\":[[0,7,9,10],[7,0,14,15],[9,14,0,17],[10,15,17,0]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0022","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-finite-certificate；資料={\"n\":4,\"metric\":[[0,8,5,14],[8,0,9,18],[5,9,0,15],[14,18,15,0]]}。請依 [n,min_positive_distance,universal_witness,complete] 欄序回答。","answer":{"kind":"expression","value":"(4, 5, epsilon=min_positive_distance forces every Cauchy tail to be constant, 是)","explanation_zh":"Wave4根節點核對=(4, 5, epsilon=min_positive_distance forces every Cauchy tail to be constant, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-finite-certificate","seed":202610273022,"parameters":{"n":4,"metric":[[0,8,5,14],[8,0,9,18],[5,9,0,15],[14,18,15,0]],"domain":"Certified_complete_finite_metric_space","operation":"complete-finite-certificate","semantic_template":"qf-mod27-complete-finite-certificate/{\"metric\":[[0,8,5,14],[8,0,9,18],[5,9,0,15],[14,18,15,0]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0023","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-finite-certificate；資料={\"n\":4,\"metric\":[[0,9,8,7],[9,0,11,10],[8,11,0,9],[7,10,9,0]]}。請依 [n,min_positive_distance,universal_witness,complete] 欄序回答。","answer":{"kind":"expression","value":"(4, 7, epsilon=min_positive_distance forces every Cauchy tail to be constant, 是)","explanation_zh":"Wave4根節點核對=(4, 7, epsilon=min_positive_distance forces every Cauchy tail to be constant, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-finite-certificate","seed":202610273023,"parameters":{"n":4,"metric":[[0,9,8,7],[9,0,11,10],[8,11,0,9],[7,10,9,0]],"domain":"Certified_complete_finite_metric_space","operation":"complete-finite-certificate","semantic_template":"qf-mod27-complete-finite-certificate/{\"metric\":[[0,9,8,7],[9,0,11,10],[8,11,0,9],[7,10,9,0]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0024","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-finite-certificate；資料={\"n\":4,\"metric\":[[0,10,11,11],[10,0,13,13],[11,13,0,14],[11,13,14,0]]}。請依 [n,min_positive_distance,universal_witness,complete] 欄序回答。","answer":{"kind":"expression","value":"(4, 10, epsilon=min_positive_distance forces every Cauchy tail to be constant, 是)","explanation_zh":"Wave4根節點核對=(4, 10, epsilon=min_positive_distance forces every Cauchy tail to be constant, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-finite-certificate","seed":202610273024,"parameters":{"n":4,"metric":[[0,10,11,11],[10,0,13,13],[11,13,0,14],[11,13,14,0]],"domain":"Certified_complete_finite_metric_space","operation":"complete-finite-certificate","semantic_template":"qf-mod27-complete-finite-certificate/{\"metric\":[[0,10,11,11],[10,0,13,13],[11,13,0,14],[11,13,14,0]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0025","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-finite-certificate；資料={\"n\":4,\"metric\":[[0,11,14,15],[11,0,15,16],[14,15,0,19],[15,16,19,0]]}。請依 [n,min_positive_distance,universal_witness,complete] 欄序回答。","answer":{"kind":"expression","value":"(4, 11, epsilon=min_positive_distance forces every Cauchy tail to be constant, 是)","explanation_zh":"Wave4根節點核對=(4, 11, epsilon=min_positive_distance forces every Cauchy tail to be constant, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-finite-certificate","seed":202610273025,"parameters":{"n":4,"metric":[[0,11,14,15],[11,0,15,16],[14,15,0,19],[15,16,19,0]],"domain":"Certified_complete_finite_metric_space","operation":"complete-finite-certificate","semantic_template":"qf-mod27-complete-finite-certificate/{\"metric\":[[0,11,14,15],[11,0,15,16],[14,15,0,19],[15,16,19,0]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0026","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-ultimately-constant-limit；資料={\"n\":4,\"metric\":[[0,3,4,5],[3,0,5,6],[4,5,0,7],[5,6,7,0]],\"prefix\":[0,1,2],\"tail\":3}。請依 [prefix,tail,N,limit,cauchy,converges_in_space] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], 3, 3, 3, 是, 是)","explanation_zh":"Wave4根節點核對=([0, 1, 2], 3, 3, 3, 是, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-ultimately-constant-limit","seed":202610273026,"parameters":{"n":4,"metric":[[0,3,4,5],[3,0,5,6],[4,5,0,7],[5,6,7,0]],"prefix":[0,1,2],"tail":3,"domain":"Certified_complete_finite_metric_space","operation":"complete-ultimately-constant-limit","semantic_template":"qf-mod27-complete-ultimately-constant-limit/{\"metric\":[[0,3,4,5],[3,0,5,6],[4,5,0,7],[5,6,7,0]],\"n\":4,\"prefix\":[0,1,2],\"tail\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0027","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-ultimately-constant-limit；資料={\"n\":4,\"metric\":[[0,4,7,9],[4,0,7,9],[7,7,0,12],[9,9,12,0]],\"prefix\":[1,2,3],\"tail\":0}。請依 [prefix,tail,N,limit,cauchy,converges_in_space] 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 3], 0, 3, 0, 是, 是)","explanation_zh":"Wave4根節點核對=([1, 2, 3], 0, 3, 0, 是, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-ultimately-constant-limit","seed":202610273027,"parameters":{"n":4,"metric":[[0,4,7,9],[4,0,7,9],[7,7,0,12],[9,9,12,0]],"prefix":[1,2,3],"tail":0,"domain":"Certified_complete_finite_metric_space","operation":"complete-ultimately-constant-limit","semantic_template":"qf-mod27-complete-ultimately-constant-limit/{\"metric\":[[0,4,7,9],[4,0,7,9],[7,7,0,12],[9,9,12,0]],\"n\":4,\"prefix\":[1,2,3],\"tail\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0028","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-ultimately-constant-limit；資料={\"n\":4,\"metric\":[[0,5,10,13],[5,0,9,12],[10,9,0,17],[13,12,17,0]],\"prefix\":[2,3,0],\"tail\":1}。請依 [prefix,tail,N,limit,cauchy,converges_in_space] 欄序回答。","answer":{"kind":"expression","value":"([2, 3, 0], 1, 3, 1, 是, 是)","explanation_zh":"Wave4根節點核對=([2, 3, 0], 1, 3, 1, 是, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-ultimately-constant-limit","seed":202610273028,"parameters":{"n":4,"metric":[[0,5,10,13],[5,0,9,12],[10,9,0,17],[13,12,17,0]],"prefix":[2,3,0],"tail":1,"domain":"Certified_complete_finite_metric_space","operation":"complete-ultimately-constant-limit","semantic_template":"qf-mod27-complete-ultimately-constant-limit/{\"metric\":[[0,5,10,13],[5,0,9,12],[10,9,0,17],[13,12,17,0]],\"n\":4,\"prefix\":[2,3,0],\"tail\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0029","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-ultimately-constant-limit；資料={\"n\":4,\"metric\":[[0,6,13,17],[6,0,11,15],[13,11,0,22],[17,15,22,0]],\"prefix\":[3,0,1],\"tail\":2}。請依 [prefix,tail,N,limit,cauchy,converges_in_space] 欄序回答。","answer":{"kind":"expression","value":"([3, 0, 1], 2, 3, 2, 是, 是)","explanation_zh":"Wave4根節點核對=([3, 0, 1], 2, 3, 2, 是, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-ultimately-constant-limit","seed":202610273029,"parameters":{"n":4,"metric":[[0,6,13,17],[6,0,11,15],[13,11,0,22],[17,15,22,0]],"prefix":[3,0,1],"tail":2,"domain":"Certified_complete_finite_metric_space","operation":"complete-ultimately-constant-limit","semantic_template":"qf-mod27-complete-ultimately-constant-limit/{\"metric\":[[0,6,13,17],[6,0,11,15],[13,11,0,22],[17,15,22,0]],\"n\":4,\"prefix\":[3,0,1],\"tail\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0030","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-ultimately-constant-limit；資料={\"n\":4,\"metric\":[[0,7,9,10],[7,0,6,7],[9,6,0,9],[10,7,9,0]],\"prefix\":[0,1,2],\"tail\":3}。請依 [prefix,tail,N,limit,cauchy,converges_in_space] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], 3, 3, 3, 是, 是)","explanation_zh":"Wave4根節點核對=([0, 1, 2], 3, 3, 3, 是, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-ultimately-constant-limit","seed":202610273030,"parameters":{"n":4,"metric":[[0,7,9,10],[7,0,6,7],[9,6,0,9],[10,7,9,0]],"prefix":[0,1,2],"tail":3,"domain":"Certified_complete_finite_metric_space","operation":"complete-ultimately-constant-limit","semantic_template":"qf-mod27-complete-ultimately-constant-limit/{\"metric\":[[0,7,9,10],[7,0,6,7],[9,6,0,9],[10,7,9,0]],\"n\":4,\"prefix\":[0,1,2],\"tail\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0031","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-ultimately-constant-limit；資料={\"n\":4,\"metric\":[[0,4,7,9],[4,0,9,11],[7,9,0,14],[9,11,14,0]],\"prefix\":[1,2,3],\"tail\":0}。請依 [prefix,tail,N,limit,cauchy,converges_in_space] 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 3], 0, 3, 0, 是, 是)","explanation_zh":"Wave4根節點核對=([1, 2, 3], 0, 3, 0, 是, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-ultimately-constant-limit","seed":202610273031,"parameters":{"n":4,"metric":[[0,4,7,9],[4,0,9,11],[7,9,0,14],[9,11,14,0]],"prefix":[1,2,3],"tail":0,"domain":"Certified_complete_finite_metric_space","operation":"complete-ultimately-constant-limit","semantic_template":"qf-mod27-complete-ultimately-constant-limit/{\"metric\":[[0,4,7,9],[4,0,9,11],[7,9,0,14],[9,11,14,0]],\"n\":4,\"prefix\":[1,2,3],\"tail\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0032","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-ultimately-constant-limit；資料={\"n\":4,\"metric\":[[0,5,10,13],[5,0,11,14],[10,11,0,19],[13,14,19,0]],\"prefix\":[2,3,0],\"tail\":1}。請依 [prefix,tail,N,limit,cauchy,converges_in_space] 欄序回答。","answer":{"kind":"expression","value":"([2, 3, 0], 1, 3, 1, 是, 是)","explanation_zh":"Wave4根節點核對=([2, 3, 0], 1, 3, 1, 是, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-ultimately-constant-limit","seed":202610273032,"parameters":{"n":4,"metric":[[0,5,10,13],[5,0,11,14],[10,11,0,19],[13,14,19,0]],"prefix":[2,3,0],"tail":1,"domain":"Certified_complete_finite_metric_space","operation":"complete-ultimately-constant-limit","semantic_template":"qf-mod27-complete-ultimately-constant-limit/{\"metric\":[[0,5,10,13],[5,0,11,14],[10,11,0,19],[13,14,19,0]],\"n\":4,\"prefix\":[2,3,0],\"tail\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0033","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-ultimately-constant-limit；資料={\"n\":4,\"metric\":[[0,6,6,17],[6,0,6,17],[6,6,0,17],[17,17,17,0]],\"prefix\":[3,0,1],\"tail\":2}。請依 [prefix,tail,N,limit,cauchy,converges_in_space] 欄序回答。","answer":{"kind":"expression","value":"([3, 0, 1], 2, 3, 2, 是, 是)","explanation_zh":"Wave4根節點核對=([3, 0, 1], 2, 3, 2, 是, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-ultimately-constant-limit","seed":202610273033,"parameters":{"n":4,"metric":[[0,6,6,17],[6,0,6,17],[6,6,0,17],[17,17,17,0]],"prefix":[3,0,1],"tail":2,"domain":"Certified_complete_finite_metric_space","operation":"complete-ultimately-constant-limit","semantic_template":"qf-mod27-complete-ultimately-constant-limit/{\"metric\":[[0,6,6,17],[6,0,6,17],[6,6,0,17],[17,17,17,0]],\"n\":4,\"prefix\":[3,0,1],\"tail\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0034","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-ultimately-constant-limit；資料={\"n\":4,\"metric\":[[0,7,9,10],[7,0,8,9],[9,8,0,11],[10,9,11,0]],\"prefix\":[0,1,2],\"tail\":3}。請依 [prefix,tail,N,limit,cauchy,converges_in_space] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], 3, 3, 3, 是, 是)","explanation_zh":"Wave4根節點核對=([0, 1, 2], 3, 3, 3, 是, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-ultimately-constant-limit","seed":202610273034,"parameters":{"n":4,"metric":[[0,7,9,10],[7,0,8,9],[9,8,0,11],[10,9,11,0]],"prefix":[0,1,2],"tail":3,"domain":"Certified_complete_finite_metric_space","operation":"complete-ultimately-constant-limit","semantic_template":"qf-mod27-complete-ultimately-constant-limit/{\"metric\":[[0,7,9,10],[7,0,8,9],[9,8,0,11],[10,9,11,0]],\"n\":4,\"prefix\":[0,1,2],\"tail\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0035","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-ultimately-constant-limit；資料={\"n\":4,\"metric\":[[0,8,12,14],[8,0,10,12],[12,10,0,16],[14,12,16,0]],\"prefix\":[1,2,3],\"tail\":0}。請依 [prefix,tail,N,limit,cauchy,converges_in_space] 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 3], 0, 3, 0, 是, 是)","explanation_zh":"Wave4根節點核對=([1, 2, 3], 0, 3, 0, 是, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-ultimately-constant-limit","seed":202610273035,"parameters":{"n":4,"metric":[[0,8,12,14],[8,0,10,12],[12,10,0,16],[14,12,16,0]],"prefix":[1,2,3],"tail":0,"domain":"Certified_complete_finite_metric_space","operation":"complete-ultimately-constant-limit","semantic_template":"qf-mod27-complete-ultimately-constant-limit/{\"metric\":[[0,8,12,14],[8,0,10,12],[12,10,0,16],[14,12,16,0]],\"n\":4,\"prefix\":[1,2,3],\"tail\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0036","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-ultimately-constant-limit；資料={\"n\":4,\"metric\":[[0,5,10,13],[5,0,13,16],[10,13,0,21],[13,16,21,0]],\"prefix\":[2,3,0],\"tail\":1}。請依 [prefix,tail,N,limit,cauchy,converges_in_space] 欄序回答。","answer":{"kind":"expression","value":"([2, 3, 0], 1, 3, 1, 是, 是)","explanation_zh":"Wave4根節點核對=([2, 3, 0], 1, 3, 1, 是, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-ultimately-constant-limit","seed":202610273036,"parameters":{"n":4,"metric":[[0,5,10,13],[5,0,13,16],[10,13,0,21],[13,16,21,0]],"prefix":[2,3,0],"tail":1,"domain":"Certified_complete_finite_metric_space","operation":"complete-ultimately-constant-limit","semantic_template":"qf-mod27-complete-ultimately-constant-limit/{\"metric\":[[0,5,10,13],[5,0,13,16],[10,13,0,21],[13,16,21,0]],\"n\":4,\"prefix\":[2,3,0],\"tail\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0037","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-ultimately-constant-limit；資料={\"n\":4,\"metric\":[[0,6,6,6],[6,0,8,8],[6,8,0,8],[6,8,8,0]],\"prefix\":[3,0,1],\"tail\":2}。請依 [prefix,tail,N,limit,cauchy,converges_in_space] 欄序回答。","answer":{"kind":"expression","value":"([3, 0, 1], 2, 3, 2, 是, 是)","explanation_zh":"Wave4根節點核對=([3, 0, 1], 2, 3, 2, 是, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-ultimately-constant-limit","seed":202610273037,"parameters":{"n":4,"metric":[[0,6,6,6],[6,0,8,8],[6,8,0,8],[6,8,8,0]],"prefix":[3,0,1],"tail":2,"domain":"Certified_complete_finite_metric_space","operation":"complete-ultimately-constant-limit","semantic_template":"qf-mod27-complete-ultimately-constant-limit/{\"metric\":[[0,6,6,6],[6,0,8,8],[6,8,0,8],[6,8,8,0]],\"n\":4,\"prefix\":[3,0,1],\"tail\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0038","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-ultimately-constant-limit；資料={\"n\":4,\"metric\":[[0,7,9,10],[7,0,10,11],[9,10,0,13],[10,11,13,0]],\"prefix\":[0,1,2],\"tail\":3}。請依 [prefix,tail,N,limit,cauchy,converges_in_space] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], 3, 3, 3, 是, 是)","explanation_zh":"Wave4根節點核對=([0, 1, 2], 3, 3, 3, 是, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-ultimately-constant-limit","seed":202610273038,"parameters":{"n":4,"metric":[[0,7,9,10],[7,0,10,11],[9,10,0,13],[10,11,13,0]],"prefix":[0,1,2],"tail":3,"domain":"Certified_complete_finite_metric_space","operation":"complete-ultimately-constant-limit","semantic_template":"qf-mod27-complete-ultimately-constant-limit/{\"metric\":[[0,7,9,10],[7,0,10,11],[9,10,0,13],[10,11,13,0]],\"n\":4,\"prefix\":[0,1,2],\"tail\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0039","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-ultimately-constant-limit；資料={\"n\":4,\"metric\":[[0,8,12,14],[8,0,12,14],[12,12,0,18],[14,14,18,0]],\"prefix\":[1,2,3],\"tail\":0}。請依 [prefix,tail,N,limit,cauchy,converges_in_space] 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 3], 0, 3, 0, 是, 是)","explanation_zh":"Wave4根節點核對=([1, 2, 3], 0, 3, 0, 是, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-ultimately-constant-limit","seed":202610273039,"parameters":{"n":4,"metric":[[0,8,12,14],[8,0,12,14],[12,12,0,18],[14,14,18,0]],"prefix":[1,2,3],"tail":0,"domain":"Certified_complete_finite_metric_space","operation":"complete-ultimately-constant-limit","semantic_template":"qf-mod27-complete-ultimately-constant-limit/{\"metric\":[[0,8,12,14],[8,0,12,14],[12,12,0,18],[14,14,18,0]],\"n\":4,\"prefix\":[1,2,3],\"tail\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0040","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-ultimately-constant-limit；資料={\"n\":4,\"metric\":[[0,9,8,18],[9,0,7,17],[8,7,0,16],[18,17,16,0]],\"prefix\":[2,3,0],\"tail\":1}。請依 [prefix,tail,N,limit,cauchy,converges_in_space] 欄序回答。","answer":{"kind":"expression","value":"([2, 3, 0], 1, 3, 1, 是, 是)","explanation_zh":"Wave4根節點核對=([2, 3, 0], 1, 3, 1, 是, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-ultimately-constant-limit","seed":202610273040,"parameters":{"n":4,"metric":[[0,9,8,18],[9,0,7,17],[8,7,0,16],[18,17,16,0]],"prefix":[2,3,0],"tail":1,"domain":"Certified_complete_finite_metric_space","operation":"complete-ultimately-constant-limit","semantic_template":"qf-mod27-complete-ultimately-constant-limit/{\"metric\":[[0,9,8,18],[9,0,7,17],[8,7,0,16],[18,17,16,0]],\"n\":4,\"prefix\":[2,3,0],\"tail\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0041","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-ultimately-constant-limit；資料={\"n\":4,\"metric\":[[0,6,6,6],[6,0,10,10],[6,10,0,10],[6,10,10,0]],\"prefix\":[3,0,1],\"tail\":2}。請依 [prefix,tail,N,limit,cauchy,converges_in_space] 欄序回答。","answer":{"kind":"expression","value":"([3, 0, 1], 2, 3, 2, 是, 是)","explanation_zh":"Wave4根節點核對=([3, 0, 1], 2, 3, 2, 是, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-ultimately-constant-limit","seed":202610273041,"parameters":{"n":4,"metric":[[0,6,6,6],[6,0,10,10],[6,10,0,10],[6,10,10,0]],"prefix":[3,0,1],"tail":2,"domain":"Certified_complete_finite_metric_space","operation":"complete-ultimately-constant-limit","semantic_template":"qf-mod27-complete-ultimately-constant-limit/{\"metric\":[[0,6,6,6],[6,0,10,10],[6,10,0,10],[6,10,10,0]],\"n\":4,\"prefix\":[3,0,1],\"tail\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0042","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-ultimately-constant-limit；資料={\"n\":4,\"metric\":[[0,7,9,10],[7,0,12,13],[9,12,0,15],[10,13,15,0]],\"prefix\":[0,1,2],\"tail\":3}。請依 [prefix,tail,N,limit,cauchy,converges_in_space] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], 3, 3, 3, 是, 是)","explanation_zh":"Wave4根節點核對=([0, 1, 2], 3, 3, 3, 是, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-ultimately-constant-limit","seed":202610273042,"parameters":{"n":4,"metric":[[0,7,9,10],[7,0,12,13],[9,12,0,15],[10,13,15,0]],"prefix":[0,1,2],"tail":3,"domain":"Certified_complete_finite_metric_space","operation":"complete-ultimately-constant-limit","semantic_template":"qf-mod27-complete-ultimately-constant-limit/{\"metric\":[[0,7,9,10],[7,0,12,13],[9,12,0,15],[10,13,15,0]],\"n\":4,\"prefix\":[0,1,2],\"tail\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0043","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-ultimately-constant-limit；資料={\"n\":4,\"metric\":[[0,8,12,14],[8,0,14,16],[12,14,0,20],[14,16,20,0]],\"prefix\":[1,2,3],\"tail\":0}。請依 [prefix,tail,N,limit,cauchy,converges_in_space] 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 3], 0, 3, 0, 是, 是)","explanation_zh":"Wave4根節點核對=([1, 2, 3], 0, 3, 0, 是, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-ultimately-constant-limit","seed":202610273043,"parameters":{"n":4,"metric":[[0,8,12,14],[8,0,14,16],[12,14,0,20],[14,16,20,0]],"prefix":[1,2,3],"tail":0,"domain":"Certified_complete_finite_metric_space","operation":"complete-ultimately-constant-limit","semantic_template":"qf-mod27-complete-ultimately-constant-limit/{\"metric\":[[0,8,12,14],[8,0,14,16],[12,14,0,20],[14,16,20,0]],\"n\":4,\"prefix\":[1,2,3],\"tail\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0044","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-ultimately-constant-limit；資料={\"n\":4,\"metric\":[[0,9,8,18],[9,0,9,19],[8,9,0,18],[18,19,18,0]],\"prefix\":[2,3,0],\"tail\":1}。請依 [prefix,tail,N,limit,cauchy,converges_in_space] 欄序回答。","answer":{"kind":"expression","value":"([2, 3, 0], 1, 3, 1, 是, 是)","explanation_zh":"Wave4根節點核對=([2, 3, 0], 1, 3, 1, 是, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-ultimately-constant-limit","seed":202610273044,"parameters":{"n":4,"metric":[[0,9,8,18],[9,0,9,19],[8,9,0,18],[18,19,18,0]],"prefix":[2,3,0],"tail":1,"domain":"Certified_complete_finite_metric_space","operation":"complete-ultimately-constant-limit","semantic_template":"qf-mod27-complete-ultimately-constant-limit/{\"metric\":[[0,9,8,18],[9,0,9,19],[8,9,0,18],[18,19,18,0]],\"n\":4,\"prefix\":[2,3,0],\"tail\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0045","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-ultimately-constant-limit；資料={\"n\":4,\"metric\":[[0,10,11,11],[10,0,11,11],[11,11,0,12],[11,11,12,0]],\"prefix\":[3,0,1],\"tail\":2}。請依 [prefix,tail,N,limit,cauchy,converges_in_space] 欄序回答。","answer":{"kind":"expression","value":"([3, 0, 1], 2, 3, 2, 是, 是)","explanation_zh":"Wave4根節點核對=([3, 0, 1], 2, 3, 2, 是, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-ultimately-constant-limit","seed":202610273045,"parameters":{"n":4,"metric":[[0,10,11,11],[10,0,11,11],[11,11,0,12],[11,11,12,0]],"prefix":[3,0,1],"tail":2,"domain":"Certified_complete_finite_metric_space","operation":"complete-ultimately-constant-limit","semantic_template":"qf-mod27-complete-ultimately-constant-limit/{\"metric\":[[0,10,11,11],[10,0,11,11],[11,11,0,12],[11,11,12,0]],\"n\":4,\"prefix\":[3,0,1],\"tail\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0046","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-ultimately-constant-limit；資料={\"n\":4,\"metric\":[[0,7,9,10],[7,0,14,15],[9,14,0,17],[10,15,17,0]],\"prefix\":[0,1,2],\"tail\":3}。請依 [prefix,tail,N,limit,cauchy,converges_in_space] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], 3, 3, 3, 是, 是)","explanation_zh":"Wave4根節點核對=([0, 1, 2], 3, 3, 3, 是, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-ultimately-constant-limit","seed":202610273046,"parameters":{"n":4,"metric":[[0,7,9,10],[7,0,14,15],[9,14,0,17],[10,15,17,0]],"prefix":[0,1,2],"tail":3,"domain":"Certified_complete_finite_metric_space","operation":"complete-ultimately-constant-limit","semantic_template":"qf-mod27-complete-ultimately-constant-limit/{\"metric\":[[0,7,9,10],[7,0,14,15],[9,14,0,17],[10,15,17,0]],\"n\":4,\"prefix\":[0,1,2],\"tail\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0047","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-ultimately-constant-limit；資料={\"n\":4,\"metric\":[[0,8,5,14],[8,0,9,18],[5,9,0,15],[14,18,15,0]],\"prefix\":[1,2,3],\"tail\":0}。請依 [prefix,tail,N,limit,cauchy,converges_in_space] 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 3], 0, 3, 0, 是, 是)","explanation_zh":"Wave4根節點核對=([1, 2, 3], 0, 3, 0, 是, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-ultimately-constant-limit","seed":202610273047,"parameters":{"n":4,"metric":[[0,8,5,14],[8,0,9,18],[5,9,0,15],[14,18,15,0]],"prefix":[1,2,3],"tail":0,"domain":"Certified_complete_finite_metric_space","operation":"complete-ultimately-constant-limit","semantic_template":"qf-mod27-complete-ultimately-constant-limit/{\"metric\":[[0,8,5,14],[8,0,9,18],[5,9,0,15],[14,18,15,0]],\"n\":4,\"prefix\":[1,2,3],\"tail\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0048","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-ultimately-constant-limit；資料={\"n\":4,\"metric\":[[0,9,8,7],[9,0,11,10],[8,11,0,9],[7,10,9,0]],\"prefix\":[2,3,0],\"tail\":1}。請依 [prefix,tail,N,limit,cauchy,converges_in_space] 欄序回答。","answer":{"kind":"expression","value":"([2, 3, 0], 1, 3, 1, 是, 是)","explanation_zh":"Wave4根節點核對=([2, 3, 0], 1, 3, 1, 是, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-ultimately-constant-limit","seed":202610273048,"parameters":{"n":4,"metric":[[0,9,8,7],[9,0,11,10],[8,11,0,9],[7,10,9,0]],"prefix":[2,3,0],"tail":1,"domain":"Certified_complete_finite_metric_space","operation":"complete-ultimately-constant-limit","semantic_template":"qf-mod27-complete-ultimately-constant-limit/{\"metric\":[[0,9,8,7],[9,0,11,10],[8,11,0,9],[7,10,9,0]],\"n\":4,\"prefix\":[2,3,0],\"tail\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0049","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-ultimately-constant-limit；資料={\"n\":4,\"metric\":[[0,10,11,11],[10,0,13,13],[11,13,0,14],[11,13,14,0]],\"prefix\":[3,0,1],\"tail\":2}。請依 [prefix,tail,N,limit,cauchy,converges_in_space] 欄序回答。","answer":{"kind":"expression","value":"([3, 0, 1], 2, 3, 2, 是, 是)","explanation_zh":"Wave4根節點核對=([3, 0, 1], 2, 3, 2, 是, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-ultimately-constant-limit","seed":202610273049,"parameters":{"n":4,"metric":[[0,10,11,11],[10,0,13,13],[11,13,0,14],[11,13,14,0]],"prefix":[3,0,1],"tail":2,"domain":"Certified_complete_finite_metric_space","operation":"complete-ultimately-constant-limit","semantic_template":"qf-mod27-complete-ultimately-constant-limit/{\"metric\":[[0,10,11,11],[10,0,13,13],[11,13,0,14],[11,13,14,0]],\"n\":4,\"prefix\":[3,0,1],\"tail\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0050","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-ultimately-constant-limit；資料={\"n\":4,\"metric\":[[0,11,14,15],[11,0,15,16],[14,15,0,19],[15,16,19,0]],\"prefix\":[0,1,2],\"tail\":3}。請依 [prefix,tail,N,limit,cauchy,converges_in_space] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], 3, 3, 3, 是, 是)","explanation_zh":"Wave4根節點核對=([0, 1, 2], 3, 3, 3, 是, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-ultimately-constant-limit","seed":202610273050,"parameters":{"n":4,"metric":[[0,11,14,15],[11,0,15,16],[14,15,0,19],[15,16,19,0]],"prefix":[0,1,2],"tail":3,"domain":"Certified_complete_finite_metric_space","operation":"complete-ultimately-constant-limit","semantic_template":"qf-mod27-complete-ultimately-constant-limit/{\"metric\":[[0,11,14,15],[11,0,15,16],[14,15,0,19],[15,16,19,0]],\"n\":4,\"prefix\":[0,1,2],\"tail\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0051","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-alternating-noncauchy；資料={\"n\":4,\"metric\":[[0,3,4,5],[3,0,5,6],[4,5,0,7],[5,6,7,0]],\"left\":0,\"right\":1,\"N\":1}。請依 [left,right,N,next_index,epsilon,distance,counterexample,cauchy] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 1, 2, 3, 3, 是, 否)","explanation_zh":"Wave4根節點核對=(0, 1, 1, 2, 3, 3, 是, 否)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-alternating-noncauchy","seed":202610273051,"parameters":{"n":4,"metric":[[0,3,4,5],[3,0,5,6],[4,5,0,7],[5,6,7,0]],"left":0,"right":1,"N":1,"domain":"Certified_complete_finite_metric_space","operation":"complete-alternating-noncauchy","semantic_template":"qf-mod27-complete-alternating-noncauchy/{\"N\":1,\"left\":0,\"metric\":[[0,3,4,5],[3,0,5,6],[4,5,0,7],[5,6,7,0]],\"n\":4,\"right\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0052","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-alternating-noncauchy；資料={\"n\":4,\"metric\":[[0,4,7,9],[4,0,7,9],[7,7,0,12],[9,9,12,0]],\"left\":1,\"right\":2,\"N\":3}。請依 [left,right,N,next_index,epsilon,distance,counterexample,cauchy] 欄序回答。","answer":{"kind":"expression","value":"(1, 2, 3, 4, 7, 7, 是, 否)","explanation_zh":"Wave4根節點核對=(1, 2, 3, 4, 7, 7, 是, 否)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-alternating-noncauchy","seed":202610273052,"parameters":{"n":4,"metric":[[0,4,7,9],[4,0,7,9],[7,7,0,12],[9,9,12,0]],"left":1,"right":2,"N":3,"domain":"Certified_complete_finite_metric_space","operation":"complete-alternating-noncauchy","semantic_template":"qf-mod27-complete-alternating-noncauchy/{\"N\":3,\"left\":1,\"metric\":[[0,4,7,9],[4,0,7,9],[7,7,0,12],[9,9,12,0]],\"n\":4,\"right\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0053","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-alternating-noncauchy；資料={\"n\":4,\"metric\":[[0,5,10,13],[5,0,9,12],[10,9,0,17],[13,12,17,0]],\"left\":2,\"right\":3,\"N\":5}。請依 [left,right,N,next_index,epsilon,distance,counterexample,cauchy] 欄序回答。","answer":{"kind":"expression","value":"(2, 3, 5, 6, 17, 17, 是, 否)","explanation_zh":"Wave4根節點核對=(2, 3, 5, 6, 17, 17, 是, 否)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-alternating-noncauchy","seed":202610273053,"parameters":{"n":4,"metric":[[0,5,10,13],[5,0,9,12],[10,9,0,17],[13,12,17,0]],"left":2,"right":3,"N":5,"domain":"Certified_complete_finite_metric_space","operation":"complete-alternating-noncauchy","semantic_template":"qf-mod27-complete-alternating-noncauchy/{\"N\":5,\"left\":2,\"metric\":[[0,5,10,13],[5,0,9,12],[10,9,0,17],[13,12,17,0]],\"n\":4,\"right\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0054","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-alternating-noncauchy；資料={\"n\":4,\"metric\":[[0,6,13,17],[6,0,11,15],[13,11,0,22],[17,15,22,0]],\"left\":3,\"right\":0,\"N\":7}。請依 [left,right,N,next_index,epsilon,distance,counterexample,cauchy] 欄序回答。","answer":{"kind":"expression","value":"(3, 0, 7, 8, 17, 17, 是, 否)","explanation_zh":"Wave4根節點核對=(3, 0, 7, 8, 17, 17, 是, 否)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-alternating-noncauchy","seed":202610273054,"parameters":{"n":4,"metric":[[0,6,13,17],[6,0,11,15],[13,11,0,22],[17,15,22,0]],"left":3,"right":0,"N":7,"domain":"Certified_complete_finite_metric_space","operation":"complete-alternating-noncauchy","semantic_template":"qf-mod27-complete-alternating-noncauchy/{\"N\":7,\"left\":3,\"metric\":[[0,6,13,17],[6,0,11,15],[13,11,0,22],[17,15,22,0]],\"n\":4,\"right\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0055","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-alternating-noncauchy；資料={\"n\":4,\"metric\":[[0,7,9,10],[7,0,6,7],[9,6,0,9],[10,7,9,0]],\"left\":0,\"right\":1,\"N\":9}。請依 [left,right,N,next_index,epsilon,distance,counterexample,cauchy] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 9, 10, 7, 7, 是, 否)","explanation_zh":"Wave4根節點核對=(0, 1, 9, 10, 7, 7, 是, 否)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-alternating-noncauchy","seed":202610273055,"parameters":{"n":4,"metric":[[0,7,9,10],[7,0,6,7],[9,6,0,9],[10,7,9,0]],"left":0,"right":1,"N":9,"domain":"Certified_complete_finite_metric_space","operation":"complete-alternating-noncauchy","semantic_template":"qf-mod27-complete-alternating-noncauchy/{\"N\":9,\"left\":0,\"metric\":[[0,7,9,10],[7,0,6,7],[9,6,0,9],[10,7,9,0]],\"n\":4,\"right\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0056","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-alternating-noncauchy；資料={\"n\":4,\"metric\":[[0,4,7,9],[4,0,9,11],[7,9,0,14],[9,11,14,0]],\"left\":1,\"right\":2,\"N\":11}。請依 [left,right,N,next_index,epsilon,distance,counterexample,cauchy] 欄序回答。","answer":{"kind":"expression","value":"(1, 2, 11, 12, 9, 9, 是, 否)","explanation_zh":"Wave4根節點核對=(1, 2, 11, 12, 9, 9, 是, 否)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-alternating-noncauchy","seed":202610273056,"parameters":{"n":4,"metric":[[0,4,7,9],[4,0,9,11],[7,9,0,14],[9,11,14,0]],"left":1,"right":2,"N":11,"domain":"Certified_complete_finite_metric_space","operation":"complete-alternating-noncauchy","semantic_template":"qf-mod27-complete-alternating-noncauchy/{\"N\":11,\"left\":1,\"metric\":[[0,4,7,9],[4,0,9,11],[7,9,0,14],[9,11,14,0]],\"n\":4,\"right\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0057","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-alternating-noncauchy；資料={\"n\":4,\"metric\":[[0,5,10,13],[5,0,11,14],[10,11,0,19],[13,14,19,0]],\"left\":2,\"right\":3,\"N\":13}。請依 [left,right,N,next_index,epsilon,distance,counterexample,cauchy] 欄序回答。","answer":{"kind":"expression","value":"(2, 3, 13, 14, 19, 19, 是, 否)","explanation_zh":"Wave4根節點核對=(2, 3, 13, 14, 19, 19, 是, 否)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-alternating-noncauchy","seed":202610273057,"parameters":{"n":4,"metric":[[0,5,10,13],[5,0,11,14],[10,11,0,19],[13,14,19,0]],"left":2,"right":3,"N":13,"domain":"Certified_complete_finite_metric_space","operation":"complete-alternating-noncauchy","semantic_template":"qf-mod27-complete-alternating-noncauchy/{\"N\":13,\"left\":2,\"metric\":[[0,5,10,13],[5,0,11,14],[10,11,0,19],[13,14,19,0]],\"n\":4,\"right\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0058","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-alternating-noncauchy；資料={\"n\":4,\"metric\":[[0,6,6,17],[6,0,6,17],[6,6,0,17],[17,17,17,0]],\"left\":3,\"right\":0,\"N\":15}。請依 [left,right,N,next_index,epsilon,distance,counterexample,cauchy] 欄序回答。","answer":{"kind":"expression","value":"(3, 0, 15, 16, 17, 17, 是, 否)","explanation_zh":"Wave4根節點核對=(3, 0, 15, 16, 17, 17, 是, 否)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-alternating-noncauchy","seed":202610273058,"parameters":{"n":4,"metric":[[0,6,6,17],[6,0,6,17],[6,6,0,17],[17,17,17,0]],"left":3,"right":0,"N":15,"domain":"Certified_complete_finite_metric_space","operation":"complete-alternating-noncauchy","semantic_template":"qf-mod27-complete-alternating-noncauchy/{\"N\":15,\"left\":3,\"metric\":[[0,6,6,17],[6,0,6,17],[6,6,0,17],[17,17,17,0]],\"n\":4,\"right\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0059","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-alternating-noncauchy；資料={\"n\":4,\"metric\":[[0,7,9,10],[7,0,8,9],[9,8,0,11],[10,9,11,0]],\"left\":0,\"right\":1,\"N\":17}。請依 [left,right,N,next_index,epsilon,distance,counterexample,cauchy] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 17, 18, 7, 7, 是, 否)","explanation_zh":"Wave4根節點核對=(0, 1, 17, 18, 7, 7, 是, 否)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-alternating-noncauchy","seed":202610273059,"parameters":{"n":4,"metric":[[0,7,9,10],[7,0,8,9],[9,8,0,11],[10,9,11,0]],"left":0,"right":1,"N":17,"domain":"Certified_complete_finite_metric_space","operation":"complete-alternating-noncauchy","semantic_template":"qf-mod27-complete-alternating-noncauchy/{\"N\":17,\"left\":0,\"metric\":[[0,7,9,10],[7,0,8,9],[9,8,0,11],[10,9,11,0]],\"n\":4,\"right\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0060","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-alternating-noncauchy；資料={\"n\":4,\"metric\":[[0,8,12,14],[8,0,10,12],[12,10,0,16],[14,12,16,0]],\"left\":1,\"right\":2,\"N\":19}。請依 [left,right,N,next_index,epsilon,distance,counterexample,cauchy] 欄序回答。","answer":{"kind":"expression","value":"(1, 2, 19, 20, 10, 10, 是, 否)","explanation_zh":"Wave4根節點核對=(1, 2, 19, 20, 10, 10, 是, 否)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-alternating-noncauchy","seed":202610273060,"parameters":{"n":4,"metric":[[0,8,12,14],[8,0,10,12],[12,10,0,16],[14,12,16,0]],"left":1,"right":2,"N":19,"domain":"Certified_complete_finite_metric_space","operation":"complete-alternating-noncauchy","semantic_template":"qf-mod27-complete-alternating-noncauchy/{\"N\":19,\"left\":1,\"metric\":[[0,8,12,14],[8,0,10,12],[12,10,0,16],[14,12,16,0]],\"n\":4,\"right\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0061","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-alternating-noncauchy；資料={\"n\":4,\"metric\":[[0,5,10,13],[5,0,13,16],[10,13,0,21],[13,16,21,0]],\"left\":2,\"right\":3,\"N\":21}。請依 [left,right,N,next_index,epsilon,distance,counterexample,cauchy] 欄序回答。","answer":{"kind":"expression","value":"(2, 3, 21, 22, 21, 21, 是, 否)","explanation_zh":"Wave4根節點核對=(2, 3, 21, 22, 21, 21, 是, 否)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-alternating-noncauchy","seed":202610273061,"parameters":{"n":4,"metric":[[0,5,10,13],[5,0,13,16],[10,13,0,21],[13,16,21,0]],"left":2,"right":3,"N":21,"domain":"Certified_complete_finite_metric_space","operation":"complete-alternating-noncauchy","semantic_template":"qf-mod27-complete-alternating-noncauchy/{\"N\":21,\"left\":2,\"metric\":[[0,5,10,13],[5,0,13,16],[10,13,0,21],[13,16,21,0]],\"n\":4,\"right\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0062","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-alternating-noncauchy；資料={\"n\":4,\"metric\":[[0,6,6,6],[6,0,8,8],[6,8,0,8],[6,8,8,0]],\"left\":3,\"right\":0,\"N\":23}。請依 [left,right,N,next_index,epsilon,distance,counterexample,cauchy] 欄序回答。","answer":{"kind":"expression","value":"(3, 0, 23, 24, 6, 6, 是, 否)","explanation_zh":"Wave4根節點核對=(3, 0, 23, 24, 6, 6, 是, 否)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-alternating-noncauchy","seed":202610273062,"parameters":{"n":4,"metric":[[0,6,6,6],[6,0,8,8],[6,8,0,8],[6,8,8,0]],"left":3,"right":0,"N":23,"domain":"Certified_complete_finite_metric_space","operation":"complete-alternating-noncauchy","semantic_template":"qf-mod27-complete-alternating-noncauchy/{\"N\":23,\"left\":3,\"metric\":[[0,6,6,6],[6,0,8,8],[6,8,0,8],[6,8,8,0]],\"n\":4,\"right\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0063","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-alternating-noncauchy；資料={\"n\":4,\"metric\":[[0,7,9,10],[7,0,10,11],[9,10,0,13],[10,11,13,0]],\"left\":0,\"right\":1,\"N\":25}。請依 [left,right,N,next_index,epsilon,distance,counterexample,cauchy] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 25, 26, 7, 7, 是, 否)","explanation_zh":"Wave4根節點核對=(0, 1, 25, 26, 7, 7, 是, 否)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-alternating-noncauchy","seed":202610273063,"parameters":{"n":4,"metric":[[0,7,9,10],[7,0,10,11],[9,10,0,13],[10,11,13,0]],"left":0,"right":1,"N":25,"domain":"Certified_complete_finite_metric_space","operation":"complete-alternating-noncauchy","semantic_template":"qf-mod27-complete-alternating-noncauchy/{\"N\":25,\"left\":0,\"metric\":[[0,7,9,10],[7,0,10,11],[9,10,0,13],[10,11,13,0]],\"n\":4,\"right\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0064","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-alternating-noncauchy；資料={\"n\":4,\"metric\":[[0,8,12,14],[8,0,12,14],[12,12,0,18],[14,14,18,0]],\"left\":1,\"right\":2,\"N\":27}。請依 [left,right,N,next_index,epsilon,distance,counterexample,cauchy] 欄序回答。","answer":{"kind":"expression","value":"(1, 2, 27, 28, 12, 12, 是, 否)","explanation_zh":"Wave4根節點核對=(1, 2, 27, 28, 12, 12, 是, 否)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-alternating-noncauchy","seed":202610273064,"parameters":{"n":4,"metric":[[0,8,12,14],[8,0,12,14],[12,12,0,18],[14,14,18,0]],"left":1,"right":2,"N":27,"domain":"Certified_complete_finite_metric_space","operation":"complete-alternating-noncauchy","semantic_template":"qf-mod27-complete-alternating-noncauchy/{\"N\":27,\"left\":1,\"metric\":[[0,8,12,14],[8,0,12,14],[12,12,0,18],[14,14,18,0]],\"n\":4,\"right\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0065","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-alternating-noncauchy；資料={\"n\":4,\"metric\":[[0,9,8,18],[9,0,7,17],[8,7,0,16],[18,17,16,0]],\"left\":2,\"right\":3,\"N\":29}。請依 [left,right,N,next_index,epsilon,distance,counterexample,cauchy] 欄序回答。","answer":{"kind":"expression","value":"(2, 3, 29, 30, 16, 16, 是, 否)","explanation_zh":"Wave4根節點核對=(2, 3, 29, 30, 16, 16, 是, 否)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-alternating-noncauchy","seed":202610273065,"parameters":{"n":4,"metric":[[0,9,8,18],[9,0,7,17],[8,7,0,16],[18,17,16,0]],"left":2,"right":3,"N":29,"domain":"Certified_complete_finite_metric_space","operation":"complete-alternating-noncauchy","semantic_template":"qf-mod27-complete-alternating-noncauchy/{\"N\":29,\"left\":2,\"metric\":[[0,9,8,18],[9,0,7,17],[8,7,0,16],[18,17,16,0]],\"n\":4,\"right\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0066","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-alternating-noncauchy；資料={\"n\":4,\"metric\":[[0,6,6,6],[6,0,10,10],[6,10,0,10],[6,10,10,0]],\"left\":3,\"right\":0,\"N\":31}。請依 [left,right,N,next_index,epsilon,distance,counterexample,cauchy] 欄序回答。","answer":{"kind":"expression","value":"(3, 0, 31, 32, 6, 6, 是, 否)","explanation_zh":"Wave4根節點核對=(3, 0, 31, 32, 6, 6, 是, 否)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-alternating-noncauchy","seed":202610273066,"parameters":{"n":4,"metric":[[0,6,6,6],[6,0,10,10],[6,10,0,10],[6,10,10,0]],"left":3,"right":0,"N":31,"domain":"Certified_complete_finite_metric_space","operation":"complete-alternating-noncauchy","semantic_template":"qf-mod27-complete-alternating-noncauchy/{\"N\":31,\"left\":3,\"metric\":[[0,6,6,6],[6,0,10,10],[6,10,0,10],[6,10,10,0]],\"n\":4,\"right\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0067","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-alternating-noncauchy；資料={\"n\":4,\"metric\":[[0,7,9,10],[7,0,12,13],[9,12,0,15],[10,13,15,0]],\"left\":0,\"right\":1,\"N\":33}。請依 [left,right,N,next_index,epsilon,distance,counterexample,cauchy] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 33, 34, 7, 7, 是, 否)","explanation_zh":"Wave4根節點核對=(0, 1, 33, 34, 7, 7, 是, 否)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-alternating-noncauchy","seed":202610273067,"parameters":{"n":4,"metric":[[0,7,9,10],[7,0,12,13],[9,12,0,15],[10,13,15,0]],"left":0,"right":1,"N":33,"domain":"Certified_complete_finite_metric_space","operation":"complete-alternating-noncauchy","semantic_template":"qf-mod27-complete-alternating-noncauchy/{\"N\":33,\"left\":0,\"metric\":[[0,7,9,10],[7,0,12,13],[9,12,0,15],[10,13,15,0]],\"n\":4,\"right\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0068","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-alternating-noncauchy；資料={\"n\":4,\"metric\":[[0,8,12,14],[8,0,14,16],[12,14,0,20],[14,16,20,0]],\"left\":1,\"right\":2,\"N\":35}。請依 [left,right,N,next_index,epsilon,distance,counterexample,cauchy] 欄序回答。","answer":{"kind":"expression","value":"(1, 2, 35, 36, 14, 14, 是, 否)","explanation_zh":"Wave4根節點核對=(1, 2, 35, 36, 14, 14, 是, 否)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-alternating-noncauchy","seed":202610273068,"parameters":{"n":4,"metric":[[0,8,12,14],[8,0,14,16],[12,14,0,20],[14,16,20,0]],"left":1,"right":2,"N":35,"domain":"Certified_complete_finite_metric_space","operation":"complete-alternating-noncauchy","semantic_template":"qf-mod27-complete-alternating-noncauchy/{\"N\":35,\"left\":1,\"metric\":[[0,8,12,14],[8,0,14,16],[12,14,0,20],[14,16,20,0]],\"n\":4,\"right\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0069","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-alternating-noncauchy；資料={\"n\":4,\"metric\":[[0,9,8,18],[9,0,9,19],[8,9,0,18],[18,19,18,0]],\"left\":2,\"right\":3,\"N\":37}。請依 [left,right,N,next_index,epsilon,distance,counterexample,cauchy] 欄序回答。","answer":{"kind":"expression","value":"(2, 3, 37, 38, 18, 18, 是, 否)","explanation_zh":"Wave4根節點核對=(2, 3, 37, 38, 18, 18, 是, 否)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-alternating-noncauchy","seed":202610273069,"parameters":{"n":4,"metric":[[0,9,8,18],[9,0,9,19],[8,9,0,18],[18,19,18,0]],"left":2,"right":3,"N":37,"domain":"Certified_complete_finite_metric_space","operation":"complete-alternating-noncauchy","semantic_template":"qf-mod27-complete-alternating-noncauchy/{\"N\":37,\"left\":2,\"metric\":[[0,9,8,18],[9,0,9,19],[8,9,0,18],[18,19,18,0]],\"n\":4,\"right\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0070","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-alternating-noncauchy；資料={\"n\":4,\"metric\":[[0,10,11,11],[10,0,11,11],[11,11,0,12],[11,11,12,0]],\"left\":3,\"right\":0,\"N\":39}。請依 [left,right,N,next_index,epsilon,distance,counterexample,cauchy] 欄序回答。","answer":{"kind":"expression","value":"(3, 0, 39, 40, 11, 11, 是, 否)","explanation_zh":"Wave4根節點核對=(3, 0, 39, 40, 11, 11, 是, 否)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-alternating-noncauchy","seed":202610273070,"parameters":{"n":4,"metric":[[0,10,11,11],[10,0,11,11],[11,11,0,12],[11,11,12,0]],"left":3,"right":0,"N":39,"domain":"Certified_complete_finite_metric_space","operation":"complete-alternating-noncauchy","semantic_template":"qf-mod27-complete-alternating-noncauchy/{\"N\":39,\"left\":3,\"metric\":[[0,10,11,11],[10,0,11,11],[11,11,0,12],[11,11,12,0]],\"n\":4,\"right\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0071","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-alternating-noncauchy；資料={\"n\":4,\"metric\":[[0,7,9,10],[7,0,14,15],[9,14,0,17],[10,15,17,0]],\"left\":0,\"right\":1,\"N\":41}。請依 [left,right,N,next_index,epsilon,distance,counterexample,cauchy] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 41, 42, 7, 7, 是, 否)","explanation_zh":"Wave4根節點核對=(0, 1, 41, 42, 7, 7, 是, 否)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-alternating-noncauchy","seed":202610273071,"parameters":{"n":4,"metric":[[0,7,9,10],[7,0,14,15],[9,14,0,17],[10,15,17,0]],"left":0,"right":1,"N":41,"domain":"Certified_complete_finite_metric_space","operation":"complete-alternating-noncauchy","semantic_template":"qf-mod27-complete-alternating-noncauchy/{\"N\":41,\"left\":0,\"metric\":[[0,7,9,10],[7,0,14,15],[9,14,0,17],[10,15,17,0]],\"n\":4,\"right\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0072","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-alternating-noncauchy；資料={\"n\":4,\"metric\":[[0,8,5,14],[8,0,9,18],[5,9,0,15],[14,18,15,0]],\"left\":1,\"right\":2,\"N\":43}。請依 [left,right,N,next_index,epsilon,distance,counterexample,cauchy] 欄序回答。","answer":{"kind":"expression","value":"(1, 2, 43, 44, 9, 9, 是, 否)","explanation_zh":"Wave4根節點核對=(1, 2, 43, 44, 9, 9, 是, 否)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-alternating-noncauchy","seed":202610273072,"parameters":{"n":4,"metric":[[0,8,5,14],[8,0,9,18],[5,9,0,15],[14,18,15,0]],"left":1,"right":2,"N":43,"domain":"Certified_complete_finite_metric_space","operation":"complete-alternating-noncauchy","semantic_template":"qf-mod27-complete-alternating-noncauchy/{\"N\":43,\"left\":1,\"metric\":[[0,8,5,14],[8,0,9,18],[5,9,0,15],[14,18,15,0]],\"n\":4,\"right\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0073","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-alternating-noncauchy；資料={\"n\":4,\"metric\":[[0,9,8,7],[9,0,11,10],[8,11,0,9],[7,10,9,0]],\"left\":2,\"right\":3,\"N\":45}。請依 [left,right,N,next_index,epsilon,distance,counterexample,cauchy] 欄序回答。","answer":{"kind":"expression","value":"(2, 3, 45, 46, 9, 9, 是, 否)","explanation_zh":"Wave4根節點核對=(2, 3, 45, 46, 9, 9, 是, 否)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-alternating-noncauchy","seed":202610273073,"parameters":{"n":4,"metric":[[0,9,8,7],[9,0,11,10],[8,11,0,9],[7,10,9,0]],"left":2,"right":3,"N":45,"domain":"Certified_complete_finite_metric_space","operation":"complete-alternating-noncauchy","semantic_template":"qf-mod27-complete-alternating-noncauchy/{\"N\":45,\"left\":2,\"metric\":[[0,9,8,7],[9,0,11,10],[8,11,0,9],[7,10,9,0]],\"n\":4,\"right\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0074","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-alternating-noncauchy；資料={\"n\":4,\"metric\":[[0,10,11,11],[10,0,13,13],[11,13,0,14],[11,13,14,0]],\"left\":3,\"right\":0,\"N\":47}。請依 [left,right,N,next_index,epsilon,distance,counterexample,cauchy] 欄序回答。","answer":{"kind":"expression","value":"(3, 0, 47, 48, 11, 11, 是, 否)","explanation_zh":"Wave4根節點核對=(3, 0, 47, 48, 11, 11, 是, 否)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-alternating-noncauchy","seed":202610273074,"parameters":{"n":4,"metric":[[0,10,11,11],[10,0,13,13],[11,13,0,14],[11,13,14,0]],"left":3,"right":0,"N":47,"domain":"Certified_complete_finite_metric_space","operation":"complete-alternating-noncauchy","semantic_template":"qf-mod27-complete-alternating-noncauchy/{\"N\":47,\"left\":3,\"metric\":[[0,10,11,11],[10,0,13,13],[11,13,0,14],[11,13,14,0]],\"n\":4,\"right\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0075","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-alternating-noncauchy；資料={\"n\":4,\"metric\":[[0,11,14,15],[11,0,15,16],[14,15,0,19],[15,16,19,0]],\"left\":0,\"right\":1,\"N\":49}。請依 [left,right,N,next_index,epsilon,distance,counterexample,cauchy] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 49, 50, 11, 11, 是, 否)","explanation_zh":"Wave4根節點核對=(0, 1, 49, 50, 11, 11, 是, 否)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-alternating-noncauchy","seed":202610273075,"parameters":{"n":4,"metric":[[0,11,14,15],[11,0,15,16],[14,15,0,19],[15,16,19,0]],"left":0,"right":1,"N":49,"domain":"Certified_complete_finite_metric_space","operation":"complete-alternating-noncauchy","semantic_template":"qf-mod27-complete-alternating-noncauchy/{\"N\":49,\"left\":0,\"metric\":[[0,11,14,15],[11,0,15,16],[14,15,0,19],[15,16,19,0]],\"n\":4,\"right\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0076","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-evidence-mode-audit；資料={\"n\":4,\"metric\":[[0,3,4,5],[3,0,5,6],[4,5,0,7],[5,6,7,0]],\"evidence_mode\":\"sampled_sequences\"}。請依 [n,min_positive_distance,evidence_mode,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(4, 3, sampled_sequences, 否, sampled_sequences_insufficient)","explanation_zh":"Wave4根節點核對=(4, 3, sampled_sequences, 否, sampled_sequences_insufficient)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-evidence-mode-audit","seed":202610273076,"parameters":{"n":4,"metric":[[0,3,4,5],[3,0,5,6],[4,5,0,7],[5,6,7,0]],"evidence_mode":"sampled_sequences","domain":"Certified_complete_finite_metric_space","operation":"complete-evidence-mode-audit","semantic_template":"qf-mod27-complete-evidence-mode-audit/{\"evidence_mode\":\"sampled_sequences\",\"metric\":[[0,3,4,5],[3,0,5,6],[4,5,0,7],[5,6,7,0]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0077","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-evidence-mode-audit；資料={\"n\":4,\"metric\":[[0,4,7,9],[4,0,7,9],[7,7,0,12],[9,9,12,0]],\"evidence_mode\":\"finite_minimum_separation\"}。請依 [n,min_positive_distance,evidence_mode,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(4, 4, finite_minimum_separation, 是, universal_finite_certificate)","explanation_zh":"Wave4根節點核對=(4, 4, finite_minimum_separation, 是, universal_finite_certificate)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-evidence-mode-audit","seed":202610273077,"parameters":{"n":4,"metric":[[0,4,7,9],[4,0,7,9],[7,7,0,12],[9,9,12,0]],"evidence_mode":"finite_minimum_separation","domain":"Certified_complete_finite_metric_space","operation":"complete-evidence-mode-audit","semantic_template":"qf-mod27-complete-evidence-mode-audit/{\"evidence_mode\":\"finite_minimum_separation\",\"metric\":[[0,4,7,9],[4,0,7,9],[7,7,0,12],[9,9,12,0]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0078","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-evidence-mode-audit；資料={\"n\":4,\"metric\":[[0,5,10,13],[5,0,9,12],[10,9,0,17],[13,12,17,0]],\"evidence_mode\":\"finite_minimum_separation\"}。請依 [n,min_positive_distance,evidence_mode,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(4, 5, finite_minimum_separation, 是, universal_finite_certificate)","explanation_zh":"Wave4根節點核對=(4, 5, finite_minimum_separation, 是, universal_finite_certificate)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-evidence-mode-audit","seed":202610273078,"parameters":{"n":4,"metric":[[0,5,10,13],[5,0,9,12],[10,9,0,17],[13,12,17,0]],"evidence_mode":"finite_minimum_separation","domain":"Certified_complete_finite_metric_space","operation":"complete-evidence-mode-audit","semantic_template":"qf-mod27-complete-evidence-mode-audit/{\"evidence_mode\":\"finite_minimum_separation\",\"metric\":[[0,5,10,13],[5,0,9,12],[10,9,0,17],[13,12,17,0]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0079","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-evidence-mode-audit；資料={\"n\":4,\"metric\":[[0,6,13,17],[6,0,11,15],[13,11,0,22],[17,15,22,0]],\"evidence_mode\":\"finite_minimum_separation\"}。請依 [n,min_positive_distance,evidence_mode,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(4, 6, finite_minimum_separation, 是, universal_finite_certificate)","explanation_zh":"Wave4根節點核對=(4, 6, finite_minimum_separation, 是, universal_finite_certificate)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-evidence-mode-audit","seed":202610273079,"parameters":{"n":4,"metric":[[0,6,13,17],[6,0,11,15],[13,11,0,22],[17,15,22,0]],"evidence_mode":"finite_minimum_separation","domain":"Certified_complete_finite_metric_space","operation":"complete-evidence-mode-audit","semantic_template":"qf-mod27-complete-evidence-mode-audit/{\"evidence_mode\":\"finite_minimum_separation\",\"metric\":[[0,6,13,17],[6,0,11,15],[13,11,0,22],[17,15,22,0]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0080","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-evidence-mode-audit；資料={\"n\":4,\"metric\":[[0,7,9,10],[7,0,6,7],[9,6,0,9],[10,7,9,0]],\"evidence_mode\":\"sampled_sequences\"}。請依 [n,min_positive_distance,evidence_mode,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(4, 6, sampled_sequences, 否, sampled_sequences_insufficient)","explanation_zh":"Wave4根節點核對=(4, 6, sampled_sequences, 否, sampled_sequences_insufficient)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-evidence-mode-audit","seed":202610273080,"parameters":{"n":4,"metric":[[0,7,9,10],[7,0,6,7],[9,6,0,9],[10,7,9,0]],"evidence_mode":"sampled_sequences","domain":"Certified_complete_finite_metric_space","operation":"complete-evidence-mode-audit","semantic_template":"qf-mod27-complete-evidence-mode-audit/{\"evidence_mode\":\"sampled_sequences\",\"metric\":[[0,7,9,10],[7,0,6,7],[9,6,0,9],[10,7,9,0]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0081","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-evidence-mode-audit；資料={\"n\":4,\"metric\":[[0,4,7,9],[4,0,9,11],[7,9,0,14],[9,11,14,0]],\"evidence_mode\":\"finite_minimum_separation\"}。請依 [n,min_positive_distance,evidence_mode,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(4, 4, finite_minimum_separation, 是, universal_finite_certificate)","explanation_zh":"Wave4根節點核對=(4, 4, finite_minimum_separation, 是, universal_finite_certificate)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-evidence-mode-audit","seed":202610273081,"parameters":{"n":4,"metric":[[0,4,7,9],[4,0,9,11],[7,9,0,14],[9,11,14,0]],"evidence_mode":"finite_minimum_separation","domain":"Certified_complete_finite_metric_space","operation":"complete-evidence-mode-audit","semantic_template":"qf-mod27-complete-evidence-mode-audit/{\"evidence_mode\":\"finite_minimum_separation\",\"metric\":[[0,4,7,9],[4,0,9,11],[7,9,0,14],[9,11,14,0]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0082","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-evidence-mode-audit；資料={\"n\":4,\"metric\":[[0,5,10,13],[5,0,11,14],[10,11,0,19],[13,14,19,0]],\"evidence_mode\":\"finite_minimum_separation\"}。請依 [n,min_positive_distance,evidence_mode,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(4, 5, finite_minimum_separation, 是, universal_finite_certificate)","explanation_zh":"Wave4根節點核對=(4, 5, finite_minimum_separation, 是, universal_finite_certificate)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-evidence-mode-audit","seed":202610273082,"parameters":{"n":4,"metric":[[0,5,10,13],[5,0,11,14],[10,11,0,19],[13,14,19,0]],"evidence_mode":"finite_minimum_separation","domain":"Certified_complete_finite_metric_space","operation":"complete-evidence-mode-audit","semantic_template":"qf-mod27-complete-evidence-mode-audit/{\"evidence_mode\":\"finite_minimum_separation\",\"metric\":[[0,5,10,13],[5,0,11,14],[10,11,0,19],[13,14,19,0]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0083","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-evidence-mode-audit；資料={\"n\":4,\"metric\":[[0,6,6,17],[6,0,6,17],[6,6,0,17],[17,17,17,0]],\"evidence_mode\":\"finite_minimum_separation\"}。請依 [n,min_positive_distance,evidence_mode,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(4, 6, finite_minimum_separation, 是, universal_finite_certificate)","explanation_zh":"Wave4根節點核對=(4, 6, finite_minimum_separation, 是, universal_finite_certificate)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-evidence-mode-audit","seed":202610273083,"parameters":{"n":4,"metric":[[0,6,6,17],[6,0,6,17],[6,6,0,17],[17,17,17,0]],"evidence_mode":"finite_minimum_separation","domain":"Certified_complete_finite_metric_space","operation":"complete-evidence-mode-audit","semantic_template":"qf-mod27-complete-evidence-mode-audit/{\"evidence_mode\":\"finite_minimum_separation\",\"metric\":[[0,6,6,17],[6,0,6,17],[6,6,0,17],[17,17,17,0]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0084","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-evidence-mode-audit；資料={\"n\":4,\"metric\":[[0,7,9,10],[7,0,8,9],[9,8,0,11],[10,9,11,0]],\"evidence_mode\":\"sampled_sequences\"}。請依 [n,min_positive_distance,evidence_mode,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(4, 7, sampled_sequences, 否, sampled_sequences_insufficient)","explanation_zh":"Wave4根節點核對=(4, 7, sampled_sequences, 否, sampled_sequences_insufficient)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-evidence-mode-audit","seed":202610273084,"parameters":{"n":4,"metric":[[0,7,9,10],[7,0,8,9],[9,8,0,11],[10,9,11,0]],"evidence_mode":"sampled_sequences","domain":"Certified_complete_finite_metric_space","operation":"complete-evidence-mode-audit","semantic_template":"qf-mod27-complete-evidence-mode-audit/{\"evidence_mode\":\"sampled_sequences\",\"metric\":[[0,7,9,10],[7,0,8,9],[9,8,0,11],[10,9,11,0]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0085","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-evidence-mode-audit；資料={\"n\":4,\"metric\":[[0,8,12,14],[8,0,10,12],[12,10,0,16],[14,12,16,0]],\"evidence_mode\":\"finite_minimum_separation\"}。請依 [n,min_positive_distance,evidence_mode,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(4, 8, finite_minimum_separation, 是, universal_finite_certificate)","explanation_zh":"Wave4根節點核對=(4, 8, finite_minimum_separation, 是, universal_finite_certificate)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-evidence-mode-audit","seed":202610273085,"parameters":{"n":4,"metric":[[0,8,12,14],[8,0,10,12],[12,10,0,16],[14,12,16,0]],"evidence_mode":"finite_minimum_separation","domain":"Certified_complete_finite_metric_space","operation":"complete-evidence-mode-audit","semantic_template":"qf-mod27-complete-evidence-mode-audit/{\"evidence_mode\":\"finite_minimum_separation\",\"metric\":[[0,8,12,14],[8,0,10,12],[12,10,0,16],[14,12,16,0]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0086","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-evidence-mode-audit；資料={\"n\":4,\"metric\":[[0,5,10,13],[5,0,13,16],[10,13,0,21],[13,16,21,0]],\"evidence_mode\":\"finite_minimum_separation\"}。請依 [n,min_positive_distance,evidence_mode,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(4, 5, finite_minimum_separation, 是, universal_finite_certificate)","explanation_zh":"Wave4根節點核對=(4, 5, finite_minimum_separation, 是, universal_finite_certificate)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-evidence-mode-audit","seed":202610273086,"parameters":{"n":4,"metric":[[0,5,10,13],[5,0,13,16],[10,13,0,21],[13,16,21,0]],"evidence_mode":"finite_minimum_separation","domain":"Certified_complete_finite_metric_space","operation":"complete-evidence-mode-audit","semantic_template":"qf-mod27-complete-evidence-mode-audit/{\"evidence_mode\":\"finite_minimum_separation\",\"metric\":[[0,5,10,13],[5,0,13,16],[10,13,0,21],[13,16,21,0]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0087","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-evidence-mode-audit；資料={\"n\":4,\"metric\":[[0,6,6,6],[6,0,8,8],[6,8,0,8],[6,8,8,0]],\"evidence_mode\":\"finite_minimum_separation\"}。請依 [n,min_positive_distance,evidence_mode,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(4, 6, finite_minimum_separation, 是, universal_finite_certificate)","explanation_zh":"Wave4根節點核對=(4, 6, finite_minimum_separation, 是, universal_finite_certificate)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-evidence-mode-audit","seed":202610273087,"parameters":{"n":4,"metric":[[0,6,6,6],[6,0,8,8],[6,8,0,8],[6,8,8,0]],"evidence_mode":"finite_minimum_separation","domain":"Certified_complete_finite_metric_space","operation":"complete-evidence-mode-audit","semantic_template":"qf-mod27-complete-evidence-mode-audit/{\"evidence_mode\":\"finite_minimum_separation\",\"metric\":[[0,6,6,6],[6,0,8,8],[6,8,0,8],[6,8,8,0]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0088","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-evidence-mode-audit；資料={\"n\":4,\"metric\":[[0,7,9,10],[7,0,10,11],[9,10,0,13],[10,11,13,0]],\"evidence_mode\":\"sampled_sequences\"}。請依 [n,min_positive_distance,evidence_mode,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(4, 7, sampled_sequences, 否, sampled_sequences_insufficient)","explanation_zh":"Wave4根節點核對=(4, 7, sampled_sequences, 否, sampled_sequences_insufficient)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-evidence-mode-audit","seed":202610273088,"parameters":{"n":4,"metric":[[0,7,9,10],[7,0,10,11],[9,10,0,13],[10,11,13,0]],"evidence_mode":"sampled_sequences","domain":"Certified_complete_finite_metric_space","operation":"complete-evidence-mode-audit","semantic_template":"qf-mod27-complete-evidence-mode-audit/{\"evidence_mode\":\"sampled_sequences\",\"metric\":[[0,7,9,10],[7,0,10,11],[9,10,0,13],[10,11,13,0]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0089","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-evidence-mode-audit；資料={\"n\":4,\"metric\":[[0,8,12,14],[8,0,12,14],[12,12,0,18],[14,14,18,0]],\"evidence_mode\":\"finite_minimum_separation\"}。請依 [n,min_positive_distance,evidence_mode,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(4, 8, finite_minimum_separation, 是, universal_finite_certificate)","explanation_zh":"Wave4根節點核對=(4, 8, finite_minimum_separation, 是, universal_finite_certificate)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-evidence-mode-audit","seed":202610273089,"parameters":{"n":4,"metric":[[0,8,12,14],[8,0,12,14],[12,12,0,18],[14,14,18,0]],"evidence_mode":"finite_minimum_separation","domain":"Certified_complete_finite_metric_space","operation":"complete-evidence-mode-audit","semantic_template":"qf-mod27-complete-evidence-mode-audit/{\"evidence_mode\":\"finite_minimum_separation\",\"metric\":[[0,8,12,14],[8,0,12,14],[12,12,0,18],[14,14,18,0]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0090","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-evidence-mode-audit；資料={\"n\":4,\"metric\":[[0,9,8,18],[9,0,7,17],[8,7,0,16],[18,17,16,0]],\"evidence_mode\":\"finite_minimum_separation\"}。請依 [n,min_positive_distance,evidence_mode,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(4, 7, finite_minimum_separation, 是, universal_finite_certificate)","explanation_zh":"Wave4根節點核對=(4, 7, finite_minimum_separation, 是, universal_finite_certificate)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-evidence-mode-audit","seed":202610273090,"parameters":{"n":4,"metric":[[0,9,8,18],[9,0,7,17],[8,7,0,16],[18,17,16,0]],"evidence_mode":"finite_minimum_separation","domain":"Certified_complete_finite_metric_space","operation":"complete-evidence-mode-audit","semantic_template":"qf-mod27-complete-evidence-mode-audit/{\"evidence_mode\":\"finite_minimum_separation\",\"metric\":[[0,9,8,18],[9,0,7,17],[8,7,0,16],[18,17,16,0]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0091","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-evidence-mode-audit；資料={\"n\":4,\"metric\":[[0,6,6,6],[6,0,10,10],[6,10,0,10],[6,10,10,0]],\"evidence_mode\":\"finite_minimum_separation\"}。請依 [n,min_positive_distance,evidence_mode,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(4, 6, finite_minimum_separation, 是, universal_finite_certificate)","explanation_zh":"Wave4根節點核對=(4, 6, finite_minimum_separation, 是, universal_finite_certificate)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-evidence-mode-audit","seed":202610273091,"parameters":{"n":4,"metric":[[0,6,6,6],[6,0,10,10],[6,10,0,10],[6,10,10,0]],"evidence_mode":"finite_minimum_separation","domain":"Certified_complete_finite_metric_space","operation":"complete-evidence-mode-audit","semantic_template":"qf-mod27-complete-evidence-mode-audit/{\"evidence_mode\":\"finite_minimum_separation\",\"metric\":[[0,6,6,6],[6,0,10,10],[6,10,0,10],[6,10,10,0]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0092","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-evidence-mode-audit；資料={\"n\":4,\"metric\":[[0,7,9,10],[7,0,12,13],[9,12,0,15],[10,13,15,0]],\"evidence_mode\":\"sampled_sequences\"}。請依 [n,min_positive_distance,evidence_mode,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(4, 7, sampled_sequences, 否, sampled_sequences_insufficient)","explanation_zh":"Wave4根節點核對=(4, 7, sampled_sequences, 否, sampled_sequences_insufficient)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-evidence-mode-audit","seed":202610273092,"parameters":{"n":4,"metric":[[0,7,9,10],[7,0,12,13],[9,12,0,15],[10,13,15,0]],"evidence_mode":"sampled_sequences","domain":"Certified_complete_finite_metric_space","operation":"complete-evidence-mode-audit","semantic_template":"qf-mod27-complete-evidence-mode-audit/{\"evidence_mode\":\"sampled_sequences\",\"metric\":[[0,7,9,10],[7,0,12,13],[9,12,0,15],[10,13,15,0]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0093","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-evidence-mode-audit；資料={\"n\":4,\"metric\":[[0,8,12,14],[8,0,14,16],[12,14,0,20],[14,16,20,0]],\"evidence_mode\":\"finite_minimum_separation\"}。請依 [n,min_positive_distance,evidence_mode,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(4, 8, finite_minimum_separation, 是, universal_finite_certificate)","explanation_zh":"Wave4根節點核對=(4, 8, finite_minimum_separation, 是, universal_finite_certificate)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-evidence-mode-audit","seed":202610273093,"parameters":{"n":4,"metric":[[0,8,12,14],[8,0,14,16],[12,14,0,20],[14,16,20,0]],"evidence_mode":"finite_minimum_separation","domain":"Certified_complete_finite_metric_space","operation":"complete-evidence-mode-audit","semantic_template":"qf-mod27-complete-evidence-mode-audit/{\"evidence_mode\":\"finite_minimum_separation\",\"metric\":[[0,8,12,14],[8,0,14,16],[12,14,0,20],[14,16,20,0]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0094","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-evidence-mode-audit；資料={\"n\":4,\"metric\":[[0,9,8,18],[9,0,9,19],[8,9,0,18],[18,19,18,0]],\"evidence_mode\":\"finite_minimum_separation\"}。請依 [n,min_positive_distance,evidence_mode,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(4, 8, finite_minimum_separation, 是, universal_finite_certificate)","explanation_zh":"Wave4根節點核對=(4, 8, finite_minimum_separation, 是, universal_finite_certificate)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-evidence-mode-audit","seed":202610273094,"parameters":{"n":4,"metric":[[0,9,8,18],[9,0,9,19],[8,9,0,18],[18,19,18,0]],"evidence_mode":"finite_minimum_separation","domain":"Certified_complete_finite_metric_space","operation":"complete-evidence-mode-audit","semantic_template":"qf-mod27-complete-evidence-mode-audit/{\"evidence_mode\":\"finite_minimum_separation\",\"metric\":[[0,9,8,18],[9,0,9,19],[8,9,0,18],[18,19,18,0]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0095","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-evidence-mode-audit；資料={\"n\":4,\"metric\":[[0,10,11,11],[10,0,11,11],[11,11,0,12],[11,11,12,0]],\"evidence_mode\":\"finite_minimum_separation\"}。請依 [n,min_positive_distance,evidence_mode,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(4, 10, finite_minimum_separation, 是, universal_finite_certificate)","explanation_zh":"Wave4根節點核對=(4, 10, finite_minimum_separation, 是, universal_finite_certificate)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-evidence-mode-audit","seed":202610273095,"parameters":{"n":4,"metric":[[0,10,11,11],[10,0,11,11],[11,11,0,12],[11,11,12,0]],"evidence_mode":"finite_minimum_separation","domain":"Certified_complete_finite_metric_space","operation":"complete-evidence-mode-audit","semantic_template":"qf-mod27-complete-evidence-mode-audit/{\"evidence_mode\":\"finite_minimum_separation\",\"metric\":[[0,10,11,11],[10,0,11,11],[11,11,0,12],[11,11,12,0]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0096","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-evidence-mode-audit；資料={\"n\":4,\"metric\":[[0,7,9,10],[7,0,14,15],[9,14,0,17],[10,15,17,0]],\"evidence_mode\":\"sampled_sequences\"}。請依 [n,min_positive_distance,evidence_mode,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(4, 7, sampled_sequences, 否, sampled_sequences_insufficient)","explanation_zh":"Wave4根節點核對=(4, 7, sampled_sequences, 否, sampled_sequences_insufficient)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-evidence-mode-audit","seed":202610273096,"parameters":{"n":4,"metric":[[0,7,9,10],[7,0,14,15],[9,14,0,17],[10,15,17,0]],"evidence_mode":"sampled_sequences","domain":"Certified_complete_finite_metric_space","operation":"complete-evidence-mode-audit","semantic_template":"qf-mod27-complete-evidence-mode-audit/{\"evidence_mode\":\"sampled_sequences\",\"metric\":[[0,7,9,10],[7,0,14,15],[9,14,0,17],[10,15,17,0]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0097","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-evidence-mode-audit；資料={\"n\":4,\"metric\":[[0,8,5,14],[8,0,9,18],[5,9,0,15],[14,18,15,0]],\"evidence_mode\":\"finite_minimum_separation\"}。請依 [n,min_positive_distance,evidence_mode,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(4, 5, finite_minimum_separation, 是, universal_finite_certificate)","explanation_zh":"Wave4根節點核對=(4, 5, finite_minimum_separation, 是, universal_finite_certificate)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-evidence-mode-audit","seed":202610273097,"parameters":{"n":4,"metric":[[0,8,5,14],[8,0,9,18],[5,9,0,15],[14,18,15,0]],"evidence_mode":"finite_minimum_separation","domain":"Certified_complete_finite_metric_space","operation":"complete-evidence-mode-audit","semantic_template":"qf-mod27-complete-evidence-mode-audit/{\"evidence_mode\":\"finite_minimum_separation\",\"metric\":[[0,8,5,14],[8,0,9,18],[5,9,0,15],[14,18,15,0]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0098","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-evidence-mode-audit；資料={\"n\":4,\"metric\":[[0,9,8,7],[9,0,11,10],[8,11,0,9],[7,10,9,0]],\"evidence_mode\":\"finite_minimum_separation\"}。請依 [n,min_positive_distance,evidence_mode,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(4, 7, finite_minimum_separation, 是, universal_finite_certificate)","explanation_zh":"Wave4根節點核對=(4, 7, finite_minimum_separation, 是, universal_finite_certificate)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-evidence-mode-audit","seed":202610273098,"parameters":{"n":4,"metric":[[0,9,8,7],[9,0,11,10],[8,11,0,9],[7,10,9,0]],"evidence_mode":"finite_minimum_separation","domain":"Certified_complete_finite_metric_space","operation":"complete-evidence-mode-audit","semantic_template":"qf-mod27-complete-evidence-mode-audit/{\"evidence_mode\":\"finite_minimum_separation\",\"metric\":[[0,9,8,7],[9,0,11,10],[8,11,0,9],[7,10,9,0]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0099","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-evidence-mode-audit；資料={\"n\":4,\"metric\":[[0,10,11,11],[10,0,13,13],[11,13,0,14],[11,13,14,0]],\"evidence_mode\":\"finite_minimum_separation\"}。請依 [n,min_positive_distance,evidence_mode,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(4, 10, finite_minimum_separation, 是, universal_finite_certificate)","explanation_zh":"Wave4根節點核對=(4, 10, finite_minimum_separation, 是, universal_finite_certificate)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-evidence-mode-audit","seed":202610273099,"parameters":{"n":4,"metric":[[0,10,11,11],[10,0,13,13],[11,13,0,14],[11,13,14,0]],"evidence_mode":"finite_minimum_separation","domain":"Certified_complete_finite_metric_space","operation":"complete-evidence-mode-audit","semantic_template":"qf-mod27-complete-evidence-mode-audit/{\"evidence_mode\":\"finite_minimum_separation\",\"metric\":[[0,10,11,11],[10,0,13,13],[11,13,0,14],[11,13,14,0]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r03-complete-metric-space-0100","batch_id":"batch-20261014-advance-20261027-r03-complete-metric-space-foundations","language":"zh-Hant","source_mko_ids":["mko-complete-metric-space"],"title_zh":"完備度量空間","stem_zh":"完備性量化空間內每個Cauchy序列。操作=complete-evidence-mode-audit；資料={\"n\":4,\"metric\":[[0,11,14,15],[11,0,15,16],[14,15,0,19],[15,16,19,0]],\"evidence_mode\":\"sampled_sequences\"}。請依 [n,min_positive_distance,evidence_mode,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(4, 11, sampled_sequences, 否, sampled_sequences_insufficient)","explanation_zh":"Wave4根節點核對=(4, 11, sampled_sequences, 否, sampled_sequences_insufficient)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["完備度量空間"]},"variant":{"family_id":"qf-mod27-complete-evidence-mode-audit","seed":202610273100,"parameters":{"n":4,"metric":[[0,11,14,15],[11,0,15,16],[14,15,0,19],[15,16,19,0]],"evidence_mode":"sampled_sequences","domain":"Certified_complete_finite_metric_space","operation":"complete-evidence-mode-audit","semantic_template":"qf-mod27-complete-evidence-mode-audit/{\"evidence_mode\":\"sampled_sequences\",\"metric\":[[0,11,14,15],[11,0,15,16],[14,15,0,19],[15,16,19,0]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}