{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0001","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-named-certificate；資料={\"dimension\":2,\"matrix\":[[2,0],[0,1]]}。請依 [dimension,matrix,binding,completeness,Hilbert,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, [[2, 0], [0, 1]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, 是, no_compactness_or_separability_claim)","explanation_zh":"Wave4第四層核對=(2, [[2, 0], [0, 1]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, 是, no_compactness_or_separability_claim)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-named-certificate","seed":202610302001,"parameters":{"dimension":2,"matrix":[[2,0],[0,1]],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-named-certificate","semantic_template":"qf-mod30-hilbert-named-certificate/{\"dimension\":2,\"matrix\":[[2,0],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0002","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-named-certificate；資料={\"dimension\":2,\"matrix\":[[3,1],[1,2]]}。請依 [dimension,matrix,binding,completeness,Hilbert,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, [[3, 1], [1, 2]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, 是, no_compactness_or_separability_claim)","explanation_zh":"Wave4第四層核對=(2, [[3, 1], [1, 2]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, 是, no_compactness_or_separability_claim)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-named-certificate","seed":202610302002,"parameters":{"dimension":2,"matrix":[[3,1],[1,2]],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-named-certificate","semantic_template":"qf-mod30-hilbert-named-certificate/{\"dimension\":2,\"matrix\":[[3,1],[1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0003","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-named-certificate；資料={\"dimension\":2,\"matrix\":[[4,2],[2,5]]}。請依 [dimension,matrix,binding,completeness,Hilbert,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, [[4, 2], [2, 5]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, 是, no_compactness_or_separability_claim)","explanation_zh":"Wave4第四層核對=(2, [[4, 2], [2, 5]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, 是, no_compactness_or_separability_claim)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-named-certificate","seed":202610302003,"parameters":{"dimension":2,"matrix":[[4,2],[2,5]],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-named-certificate","semantic_template":"qf-mod30-hilbert-named-certificate/{\"dimension\":2,\"matrix\":[[4,2],[2,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0004","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-named-certificate；資料={\"dimension\":2,\"matrix\":[[5,0],[0,1]]}。請依 [dimension,matrix,binding,completeness,Hilbert,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, [[5, 0], [0, 1]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, 是, no_compactness_or_separability_claim)","explanation_zh":"Wave4第四層核對=(2, [[5, 0], [0, 1]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, 是, no_compactness_or_separability_claim)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-named-certificate","seed":202610302004,"parameters":{"dimension":2,"matrix":[[5,0],[0,1]],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-named-certificate","semantic_template":"qf-mod30-hilbert-named-certificate/{\"dimension\":2,\"matrix\":[[5,0],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0005","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-named-certificate；資料={\"dimension\":2,\"matrix\":[[6,1],[1,2]]}。請依 [dimension,matrix,binding,completeness,Hilbert,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, [[6, 1], [1, 2]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, 是, no_compactness_or_separability_claim)","explanation_zh":"Wave4第四層核對=(2, [[6, 1], [1, 2]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, 是, no_compactness_or_separability_claim)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-named-certificate","seed":202610302005,"parameters":{"dimension":2,"matrix":[[6,1],[1,2]],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-named-certificate","semantic_template":"qf-mod30-hilbert-named-certificate/{\"dimension\":2,\"matrix\":[[6,1],[1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0006","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-named-certificate；資料={\"dimension\":2,\"matrix\":[[7,2],[2,5]]}。請依 [dimension,matrix,binding,completeness,Hilbert,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, [[7, 2], [2, 5]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, 是, no_compactness_or_separability_claim)","explanation_zh":"Wave4第四層核對=(2, [[7, 2], [2, 5]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, 是, no_compactness_or_separability_claim)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-named-certificate","seed":202610302006,"parameters":{"dimension":2,"matrix":[[7,2],[2,5]],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-named-certificate","semantic_template":"qf-mod30-hilbert-named-certificate/{\"dimension\":2,\"matrix\":[[7,2],[2,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0007","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-named-certificate；資料={\"dimension\":2,\"matrix\":[[8,0],[0,1]]}。請依 [dimension,matrix,binding,completeness,Hilbert,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, [[8, 0], [0, 1]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, 是, no_compactness_or_separability_claim)","explanation_zh":"Wave4第四層核對=(2, [[8, 0], [0, 1]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, 是, no_compactness_or_separability_claim)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-named-certificate","seed":202610302007,"parameters":{"dimension":2,"matrix":[[8,0],[0,1]],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-named-certificate","semantic_template":"qf-mod30-hilbert-named-certificate/{\"dimension\":2,\"matrix\":[[8,0],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0008","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-named-certificate；資料={\"dimension\":2,\"matrix\":[[9,1],[1,2]]}。請依 [dimension,matrix,binding,completeness,Hilbert,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, [[9, 1], [1, 2]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, 是, no_compactness_or_separability_claim)","explanation_zh":"Wave4第四層核對=(2, [[9, 1], [1, 2]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, 是, no_compactness_or_separability_claim)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-named-certificate","seed":202610302008,"parameters":{"dimension":2,"matrix":[[9,1],[1,2]],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-named-certificate","semantic_template":"qf-mod30-hilbert-named-certificate/{\"dimension\":2,\"matrix\":[[9,1],[1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0009","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-named-certificate；資料={\"dimension\":2,\"matrix\":[[10,2],[2,5]]}。請依 [dimension,matrix,binding,completeness,Hilbert,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, [[10, 2], [2, 5]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, 是, no_compactness_or_separability_claim)","explanation_zh":"Wave4第四層核對=(2, [[10, 2], [2, 5]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, 是, no_compactness_or_separability_claim)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-named-certificate","seed":202610302009,"parameters":{"dimension":2,"matrix":[[10,2],[2,5]],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-named-certificate","semantic_template":"qf-mod30-hilbert-named-certificate/{\"dimension\":2,\"matrix\":[[10,2],[2,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0010","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-named-certificate；資料={\"dimension\":2,\"matrix\":[[11,0],[0,1]]}。請依 [dimension,matrix,binding,completeness,Hilbert,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, [[11, 0], [0, 1]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, 是, no_compactness_or_separability_claim)","explanation_zh":"Wave4第四層核對=(2, [[11, 0], [0, 1]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, 是, no_compactness_or_separability_claim)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-named-certificate","seed":202610302010,"parameters":{"dimension":2,"matrix":[[11,0],[0,1]],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-named-certificate","semantic_template":"qf-mod30-hilbert-named-certificate/{\"dimension\":2,\"matrix\":[[11,0],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0011","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-named-certificate；資料={\"dimension\":2,\"matrix\":[[12,1],[1,2]]}。請依 [dimension,matrix,binding,completeness,Hilbert,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, [[12, 1], [1, 2]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, 是, no_compactness_or_separability_claim)","explanation_zh":"Wave4第四層核對=(2, [[12, 1], [1, 2]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, 是, no_compactness_or_separability_claim)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-named-certificate","seed":202610302011,"parameters":{"dimension":2,"matrix":[[12,1],[1,2]],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-named-certificate","semantic_template":"qf-mod30-hilbert-named-certificate/{\"dimension\":2,\"matrix\":[[12,1],[1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0012","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-named-certificate；資料={\"dimension\":2,\"matrix\":[[13,2],[2,5]]}。請依 [dimension,matrix,binding,completeness,Hilbert,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, [[13, 2], [2, 5]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, 是, no_compactness_or_separability_claim)","explanation_zh":"Wave4第四層核對=(2, [[13, 2], [2, 5]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, 是, no_compactness_or_separability_claim)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-named-certificate","seed":202610302012,"parameters":{"dimension":2,"matrix":[[13,2],[2,5]],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-named-certificate","semantic_template":"qf-mod30-hilbert-named-certificate/{\"dimension\":2,\"matrix\":[[13,2],[2,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0013","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-named-certificate；資料={\"dimension\":2,\"matrix\":[[14,0],[0,1]]}。請依 [dimension,matrix,binding,completeness,Hilbert,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, [[14, 0], [0, 1]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, 是, no_compactness_or_separability_claim)","explanation_zh":"Wave4第四層核對=(2, [[14, 0], [0, 1]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, 是, no_compactness_or_separability_claim)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-named-certificate","seed":202610302013,"parameters":{"dimension":2,"matrix":[[14,0],[0,1]],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-named-certificate","semantic_template":"qf-mod30-hilbert-named-certificate/{\"dimension\":2,\"matrix\":[[14,0],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0014","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-named-certificate；資料={\"dimension\":2,\"matrix\":[[15,1],[1,2]]}。請依 [dimension,matrix,binding,completeness,Hilbert,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, [[15, 1], [1, 2]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, 是, no_compactness_or_separability_claim)","explanation_zh":"Wave4第四層核對=(2, [[15, 1], [1, 2]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, 是, no_compactness_or_separability_claim)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-named-certificate","seed":202610302014,"parameters":{"dimension":2,"matrix":[[15,1],[1,2]],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-named-certificate","semantic_template":"qf-mod30-hilbert-named-certificate/{\"dimension\":2,\"matrix\":[[15,1],[1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0015","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-named-certificate；資料={\"dimension\":2,\"matrix\":[[16,2],[2,5]]}。請依 [dimension,matrix,binding,completeness,Hilbert,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, [[16, 2], [2, 5]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, 是, no_compactness_or_separability_claim)","explanation_zh":"Wave4第四層核對=(2, [[16, 2], [2, 5]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, 是, no_compactness_or_separability_claim)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-named-certificate","seed":202610302015,"parameters":{"dimension":2,"matrix":[[16,2],[2,5]],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-named-certificate","semantic_template":"qf-mod30-hilbert-named-certificate/{\"dimension\":2,\"matrix\":[[16,2],[2,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0016","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-named-certificate；資料={\"dimension\":2,\"matrix\":[[17,0],[0,1]]}。請依 [dimension,matrix,binding,completeness,Hilbert,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, [[17, 0], [0, 1]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, 是, no_compactness_or_separability_claim)","explanation_zh":"Wave4第四層核對=(2, [[17, 0], [0, 1]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, 是, no_compactness_or_separability_claim)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-named-certificate","seed":202610302016,"parameters":{"dimension":2,"matrix":[[17,0],[0,1]],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-named-certificate","semantic_template":"qf-mod30-hilbert-named-certificate/{\"dimension\":2,\"matrix\":[[17,0],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0017","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-named-certificate；資料={\"dimension\":2,\"matrix\":[[18,1],[1,2]]}。請依 [dimension,matrix,binding,completeness,Hilbert,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, [[18, 1], [1, 2]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, 是, no_compactness_or_separability_claim)","explanation_zh":"Wave4第四層核對=(2, [[18, 1], [1, 2]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, 是, no_compactness_or_separability_claim)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-named-certificate","seed":202610302017,"parameters":{"dimension":2,"matrix":[[18,1],[1,2]],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-named-certificate","semantic_template":"qf-mod30-hilbert-named-certificate/{\"dimension\":2,\"matrix\":[[18,1],[1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0018","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-named-certificate；資料={\"dimension\":2,\"matrix\":[[19,2],[2,5]]}。請依 [dimension,matrix,binding,completeness,Hilbert,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, [[19, 2], [2, 5]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, 是, no_compactness_or_separability_claim)","explanation_zh":"Wave4第四層核對=(2, [[19, 2], [2, 5]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, 是, no_compactness_or_separability_claim)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-named-certificate","seed":202610302018,"parameters":{"dimension":2,"matrix":[[19,2],[2,5]],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-named-certificate","semantic_template":"qf-mod30-hilbert-named-certificate/{\"dimension\":2,\"matrix\":[[19,2],[2,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0019","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-named-certificate；資料={\"dimension\":2,\"matrix\":[[20,0],[0,1]]}。請依 [dimension,matrix,binding,completeness,Hilbert,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, [[20, 0], [0, 1]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, 是, no_compactness_or_separability_claim)","explanation_zh":"Wave4第四層核對=(2, [[20, 0], [0, 1]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, 是, no_compactness_or_separability_claim)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-named-certificate","seed":202610302019,"parameters":{"dimension":2,"matrix":[[20,0],[0,1]],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-named-certificate","semantic_template":"qf-mod30-hilbert-named-certificate/{\"dimension\":2,\"matrix\":[[20,0],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0020","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-named-certificate；資料={\"dimension\":2,\"matrix\":[[21,1],[1,2]]}。請依 [dimension,matrix,binding,completeness,Hilbert,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, [[21, 1], [1, 2]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, 是, no_compactness_or_separability_claim)","explanation_zh":"Wave4第四層核對=(2, [[21, 1], [1, 2]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, 是, no_compactness_or_separability_claim)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-named-certificate","seed":202610302020,"parameters":{"dimension":2,"matrix":[[21,1],[1,2]],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-named-certificate","semantic_template":"qf-mod30-hilbert-named-certificate/{\"dimension\":2,\"matrix\":[[21,1],[1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0021","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-named-certificate；資料={\"dimension\":2,\"matrix\":[[22,2],[2,5]]}。請依 [dimension,matrix,binding,completeness,Hilbert,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, [[22, 2], [2, 5]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, 是, no_compactness_or_separability_claim)","explanation_zh":"Wave4第四層核對=(2, [[22, 2], [2, 5]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, 是, no_compactness_or_separability_claim)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-named-certificate","seed":202610302021,"parameters":{"dimension":2,"matrix":[[22,2],[2,5]],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-named-certificate","semantic_template":"qf-mod30-hilbert-named-certificate/{\"dimension\":2,\"matrix\":[[22,2],[2,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0022","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-named-certificate；資料={\"dimension\":2,\"matrix\":[[23,0],[0,1]]}。請依 [dimension,matrix,binding,completeness,Hilbert,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, [[23, 0], [0, 1]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, 是, no_compactness_or_separability_claim)","explanation_zh":"Wave4第四層核對=(2, [[23, 0], [0, 1]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, 是, no_compactness_or_separability_claim)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-named-certificate","seed":202610302022,"parameters":{"dimension":2,"matrix":[[23,0],[0,1]],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-named-certificate","semantic_template":"qf-mod30-hilbert-named-certificate/{\"dimension\":2,\"matrix\":[[23,0],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0023","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-named-certificate；資料={\"dimension\":2,\"matrix\":[[24,1],[1,2]]}。請依 [dimension,matrix,binding,completeness,Hilbert,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, [[24, 1], [1, 2]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, 是, no_compactness_or_separability_claim)","explanation_zh":"Wave4第四層核對=(2, [[24, 1], [1, 2]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, 是, no_compactness_or_separability_claim)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-named-certificate","seed":202610302023,"parameters":{"dimension":2,"matrix":[[24,1],[1,2]],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-named-certificate","semantic_template":"qf-mod30-hilbert-named-certificate/{\"dimension\":2,\"matrix\":[[24,1],[1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0024","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-named-certificate；資料={\"dimension\":2,\"matrix\":[[25,2],[2,5]]}。請依 [dimension,matrix,binding,completeness,Hilbert,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, [[25, 2], [2, 5]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, 是, no_compactness_or_separability_claim)","explanation_zh":"Wave4第四層核對=(2, [[25, 2], [2, 5]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, 是, no_compactness_or_separability_claim)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-named-certificate","seed":202610302024,"parameters":{"dimension":2,"matrix":[[25,2],[2,5]],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-named-certificate","semantic_template":"qf-mod30-hilbert-named-certificate/{\"dimension\":2,\"matrix\":[[25,2],[2,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0025","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-named-certificate；資料={\"dimension\":2,\"matrix\":[[26,0],[0,1]]}。請依 [dimension,matrix,binding,completeness,Hilbert,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, [[26, 0], [0, 1]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, 是, no_compactness_or_separability_claim)","explanation_zh":"Wave4第四層核對=(2, [[26, 0], [0, 1]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, 是, no_compactness_or_separability_claim)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-named-certificate","seed":202610302025,"parameters":{"dimension":2,"matrix":[[26,0],[0,1]],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-named-certificate","semantic_template":"qf-mod30-hilbert-named-certificate/{\"dimension\":2,\"matrix\":[[26,0],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0026","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-component-checklist；資料={\"dimension\":2,\"matrix\":[[2,0],[0,1]],\"norm_source\":\"other\",\"metric_source\":\"same_norm\",\"completeness_source\":\"finite_dimensional_theorem\"}。請依 [dimension,matrix,norm_source,metric_source,completeness_source,Hilbert] 欄序回答。","answer":{"kind":"expression","value":"(2, [[2, 0], [0, 1]], other, same_norm, finite_dimensional_theorem, 否)","explanation_zh":"Wave4第四層核對=(2, [[2, 0], [0, 1]], other, same_norm, finite_dimensional_theorem, 否)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-component-checklist","seed":202610302026,"parameters":{"dimension":2,"matrix":[[2,0],[0,1]],"norm_source":"other","metric_source":"same_norm","completeness_source":"finite_dimensional_theorem","domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-component-checklist","semantic_template":"qf-mod30-hilbert-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":2,\"matrix\":[[2,0],[0,1]],\"metric_source\":\"same_norm\",\"norm_source\":\"other\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0027","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-component-checklist；資料={\"dimension\":2,\"matrix\":[[3,1],[1,2]],\"norm_source\":\"same_inner_product\",\"metric_source\":\"same_norm\",\"completeness_source\":\"finite_dimensional_theorem\"}。請依 [dimension,matrix,norm_source,metric_source,completeness_source,Hilbert] 欄序回答。","answer":{"kind":"expression","value":"(2, [[3, 1], [1, 2]], same_inner_product, same_norm, finite_dimensional_theorem, 是)","explanation_zh":"Wave4第四層核對=(2, [[3, 1], [1, 2]], same_inner_product, same_norm, finite_dimensional_theorem, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-component-checklist","seed":202610302027,"parameters":{"dimension":2,"matrix":[[3,1],[1,2]],"norm_source":"same_inner_product","metric_source":"same_norm","completeness_source":"finite_dimensional_theorem","domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-component-checklist","semantic_template":"qf-mod30-hilbert-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":2,\"matrix\":[[3,1],[1,2]],\"metric_source\":\"same_norm\",\"norm_source\":\"same_inner_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0028","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-component-checklist；資料={\"dimension\":2,\"matrix\":[[4,2],[2,5]],\"norm_source\":\"same_inner_product\",\"metric_source\":\"same_norm\",\"completeness_source\":\"finite_dimensional_theorem\"}。請依 [dimension,matrix,norm_source,metric_source,completeness_source,Hilbert] 欄序回答。","answer":{"kind":"expression","value":"(2, [[4, 2], [2, 5]], same_inner_product, same_norm, finite_dimensional_theorem, 是)","explanation_zh":"Wave4第四層核對=(2, [[4, 2], [2, 5]], same_inner_product, same_norm, finite_dimensional_theorem, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-component-checklist","seed":202610302028,"parameters":{"dimension":2,"matrix":[[4,2],[2,5]],"norm_source":"same_inner_product","metric_source":"same_norm","completeness_source":"finite_dimensional_theorem","domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-component-checklist","semantic_template":"qf-mod30-hilbert-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":2,\"matrix\":[[4,2],[2,5]],\"metric_source\":\"same_norm\",\"norm_source\":\"same_inner_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0029","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-component-checklist；資料={\"dimension\":2,\"matrix\":[[5,0],[0,1]],\"norm_source\":\"same_inner_product\",\"metric_source\":\"same_norm\",\"completeness_source\":\"finite_dimensional_theorem\"}。請依 [dimension,matrix,norm_source,metric_source,completeness_source,Hilbert] 欄序回答。","answer":{"kind":"expression","value":"(2, [[5, 0], [0, 1]], same_inner_product, same_norm, finite_dimensional_theorem, 是)","explanation_zh":"Wave4第四層核對=(2, [[5, 0], [0, 1]], same_inner_product, same_norm, finite_dimensional_theorem, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-component-checklist","seed":202610302029,"parameters":{"dimension":2,"matrix":[[5,0],[0,1]],"norm_source":"same_inner_product","metric_source":"same_norm","completeness_source":"finite_dimensional_theorem","domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-component-checklist","semantic_template":"qf-mod30-hilbert-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":2,\"matrix\":[[5,0],[0,1]],\"metric_source\":\"same_norm\",\"norm_source\":\"same_inner_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0030","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-component-checklist；資料={\"dimension\":2,\"matrix\":[[6,1],[1,2]],\"norm_source\":\"same_inner_product\",\"metric_source\":\"same_norm\",\"completeness_source\":\"finite_dimensional_theorem\"}。請依 [dimension,matrix,norm_source,metric_source,completeness_source,Hilbert] 欄序回答。","answer":{"kind":"expression","value":"(2, [[6, 1], [1, 2]], same_inner_product, same_norm, finite_dimensional_theorem, 是)","explanation_zh":"Wave4第四層核對=(2, [[6, 1], [1, 2]], same_inner_product, same_norm, finite_dimensional_theorem, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-component-checklist","seed":202610302030,"parameters":{"dimension":2,"matrix":[[6,1],[1,2]],"norm_source":"same_inner_product","metric_source":"same_norm","completeness_source":"finite_dimensional_theorem","domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-component-checklist","semantic_template":"qf-mod30-hilbert-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":2,\"matrix\":[[6,1],[1,2]],\"metric_source\":\"same_norm\",\"norm_source\":\"same_inner_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0031","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-component-checklist；資料={\"dimension\":2,\"matrix\":[[7,2],[2,5]],\"norm_source\":\"other\",\"metric_source\":\"same_norm\",\"completeness_source\":\"finite_dimensional_theorem\"}。請依 [dimension,matrix,norm_source,metric_source,completeness_source,Hilbert] 欄序回答。","answer":{"kind":"expression","value":"(2, [[7, 2], [2, 5]], other, same_norm, finite_dimensional_theorem, 否)","explanation_zh":"Wave4第四層核對=(2, [[7, 2], [2, 5]], other, same_norm, finite_dimensional_theorem, 否)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-component-checklist","seed":202610302031,"parameters":{"dimension":2,"matrix":[[7,2],[2,5]],"norm_source":"other","metric_source":"same_norm","completeness_source":"finite_dimensional_theorem","domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-component-checklist","semantic_template":"qf-mod30-hilbert-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":2,\"matrix\":[[7,2],[2,5]],\"metric_source\":\"same_norm\",\"norm_source\":\"other\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0032","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-component-checklist；資料={\"dimension\":2,\"matrix\":[[8,0],[0,1]],\"norm_source\":\"same_inner_product\",\"metric_source\":\"same_norm\",\"completeness_source\":\"finite_dimensional_theorem\"}。請依 [dimension,matrix,norm_source,metric_source,completeness_source,Hilbert] 欄序回答。","answer":{"kind":"expression","value":"(2, [[8, 0], [0, 1]], same_inner_product, same_norm, finite_dimensional_theorem, 是)","explanation_zh":"Wave4第四層核對=(2, [[8, 0], [0, 1]], same_inner_product, same_norm, finite_dimensional_theorem, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-component-checklist","seed":202610302032,"parameters":{"dimension":2,"matrix":[[8,0],[0,1]],"norm_source":"same_inner_product","metric_source":"same_norm","completeness_source":"finite_dimensional_theorem","domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-component-checklist","semantic_template":"qf-mod30-hilbert-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":2,\"matrix\":[[8,0],[0,1]],\"metric_source\":\"same_norm\",\"norm_source\":\"same_inner_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0033","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-component-checklist；資料={\"dimension\":2,\"matrix\":[[9,1],[1,2]],\"norm_source\":\"same_inner_product\",\"metric_source\":\"same_norm\",\"completeness_source\":\"finite_dimensional_theorem\"}。請依 [dimension,matrix,norm_source,metric_source,completeness_source,Hilbert] 欄序回答。","answer":{"kind":"expression","value":"(2, [[9, 1], [1, 2]], same_inner_product, same_norm, finite_dimensional_theorem, 是)","explanation_zh":"Wave4第四層核對=(2, [[9, 1], [1, 2]], same_inner_product, same_norm, finite_dimensional_theorem, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-component-checklist","seed":202610302033,"parameters":{"dimension":2,"matrix":[[9,1],[1,2]],"norm_source":"same_inner_product","metric_source":"same_norm","completeness_source":"finite_dimensional_theorem","domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-component-checklist","semantic_template":"qf-mod30-hilbert-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":2,\"matrix\":[[9,1],[1,2]],\"metric_source\":\"same_norm\",\"norm_source\":\"same_inner_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0034","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-component-checklist；資料={\"dimension\":2,\"matrix\":[[10,2],[2,5]],\"norm_source\":\"same_inner_product\",\"metric_source\":\"same_norm\",\"completeness_source\":\"finite_dimensional_theorem\"}。請依 [dimension,matrix,norm_source,metric_source,completeness_source,Hilbert] 欄序回答。","answer":{"kind":"expression","value":"(2, [[10, 2], [2, 5]], same_inner_product, same_norm, finite_dimensional_theorem, 是)","explanation_zh":"Wave4第四層核對=(2, [[10, 2], [2, 5]], same_inner_product, same_norm, finite_dimensional_theorem, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-component-checklist","seed":202610302034,"parameters":{"dimension":2,"matrix":[[10,2],[2,5]],"norm_source":"same_inner_product","metric_source":"same_norm","completeness_source":"finite_dimensional_theorem","domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-component-checklist","semantic_template":"qf-mod30-hilbert-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":2,\"matrix\":[[10,2],[2,5]],\"metric_source\":\"same_norm\",\"norm_source\":\"same_inner_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0035","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-component-checklist；資料={\"dimension\":2,\"matrix\":[[11,0],[0,1]],\"norm_source\":\"same_inner_product\",\"metric_source\":\"same_norm\",\"completeness_source\":\"finite_dimensional_theorem\"}。請依 [dimension,matrix,norm_source,metric_source,completeness_source,Hilbert] 欄序回答。","answer":{"kind":"expression","value":"(2, [[11, 0], [0, 1]], same_inner_product, same_norm, finite_dimensional_theorem, 是)","explanation_zh":"Wave4第四層核對=(2, [[11, 0], [0, 1]], same_inner_product, same_norm, finite_dimensional_theorem, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-component-checklist","seed":202610302035,"parameters":{"dimension":2,"matrix":[[11,0],[0,1]],"norm_source":"same_inner_product","metric_source":"same_norm","completeness_source":"finite_dimensional_theorem","domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-component-checklist","semantic_template":"qf-mod30-hilbert-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":2,\"matrix\":[[11,0],[0,1]],\"metric_source\":\"same_norm\",\"norm_source\":\"same_inner_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0036","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-component-checklist；資料={\"dimension\":2,\"matrix\":[[12,1],[1,2]],\"norm_source\":\"other\",\"metric_source\":\"same_norm\",\"completeness_source\":\"finite_dimensional_theorem\"}。請依 [dimension,matrix,norm_source,metric_source,completeness_source,Hilbert] 欄序回答。","answer":{"kind":"expression","value":"(2, [[12, 1], [1, 2]], other, same_norm, finite_dimensional_theorem, 否)","explanation_zh":"Wave4第四層核對=(2, [[12, 1], [1, 2]], other, same_norm, finite_dimensional_theorem, 否)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-component-checklist","seed":202610302036,"parameters":{"dimension":2,"matrix":[[12,1],[1,2]],"norm_source":"other","metric_source":"same_norm","completeness_source":"finite_dimensional_theorem","domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-component-checklist","semantic_template":"qf-mod30-hilbert-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":2,\"matrix\":[[12,1],[1,2]],\"metric_source\":\"same_norm\",\"norm_source\":\"other\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0037","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-component-checklist；資料={\"dimension\":2,\"matrix\":[[13,2],[2,5]],\"norm_source\":\"same_inner_product\",\"metric_source\":\"same_norm\",\"completeness_source\":\"finite_dimensional_theorem\"}。請依 [dimension,matrix,norm_source,metric_source,completeness_source,Hilbert] 欄序回答。","answer":{"kind":"expression","value":"(2, [[13, 2], [2, 5]], same_inner_product, same_norm, finite_dimensional_theorem, 是)","explanation_zh":"Wave4第四層核對=(2, [[13, 2], [2, 5]], same_inner_product, same_norm, finite_dimensional_theorem, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-component-checklist","seed":202610302037,"parameters":{"dimension":2,"matrix":[[13,2],[2,5]],"norm_source":"same_inner_product","metric_source":"same_norm","completeness_source":"finite_dimensional_theorem","domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-component-checklist","semantic_template":"qf-mod30-hilbert-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":2,\"matrix\":[[13,2],[2,5]],\"metric_source\":\"same_norm\",\"norm_source\":\"same_inner_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0038","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-component-checklist；資料={\"dimension\":2,\"matrix\":[[14,0],[0,1]],\"norm_source\":\"same_inner_product\",\"metric_source\":\"same_norm\",\"completeness_source\":\"finite_dimensional_theorem\"}。請依 [dimension,matrix,norm_source,metric_source,completeness_source,Hilbert] 欄序回答。","answer":{"kind":"expression","value":"(2, [[14, 0], [0, 1]], same_inner_product, same_norm, finite_dimensional_theorem, 是)","explanation_zh":"Wave4第四層核對=(2, [[14, 0], [0, 1]], same_inner_product, same_norm, finite_dimensional_theorem, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-component-checklist","seed":202610302038,"parameters":{"dimension":2,"matrix":[[14,0],[0,1]],"norm_source":"same_inner_product","metric_source":"same_norm","completeness_source":"finite_dimensional_theorem","domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-component-checklist","semantic_template":"qf-mod30-hilbert-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":2,\"matrix\":[[14,0],[0,1]],\"metric_source\":\"same_norm\",\"norm_source\":\"same_inner_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0039","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-component-checklist；資料={\"dimension\":2,\"matrix\":[[15,1],[1,2]],\"norm_source\":\"same_inner_product\",\"metric_source\":\"same_norm\",\"completeness_source\":\"finite_dimensional_theorem\"}。請依 [dimension,matrix,norm_source,metric_source,completeness_source,Hilbert] 欄序回答。","answer":{"kind":"expression","value":"(2, [[15, 1], [1, 2]], same_inner_product, same_norm, finite_dimensional_theorem, 是)","explanation_zh":"Wave4第四層核對=(2, [[15, 1], [1, 2]], same_inner_product, same_norm, finite_dimensional_theorem, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-component-checklist","seed":202610302039,"parameters":{"dimension":2,"matrix":[[15,1],[1,2]],"norm_source":"same_inner_product","metric_source":"same_norm","completeness_source":"finite_dimensional_theorem","domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-component-checklist","semantic_template":"qf-mod30-hilbert-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":2,\"matrix\":[[15,1],[1,2]],\"metric_source\":\"same_norm\",\"norm_source\":\"same_inner_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0040","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-component-checklist；資料={\"dimension\":2,\"matrix\":[[16,2],[2,5]],\"norm_source\":\"same_inner_product\",\"metric_source\":\"same_norm\",\"completeness_source\":\"finite_dimensional_theorem\"}。請依 [dimension,matrix,norm_source,metric_source,completeness_source,Hilbert] 欄序回答。","answer":{"kind":"expression","value":"(2, [[16, 2], [2, 5]], same_inner_product, same_norm, finite_dimensional_theorem, 是)","explanation_zh":"Wave4第四層核對=(2, [[16, 2], [2, 5]], same_inner_product, same_norm, finite_dimensional_theorem, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-component-checklist","seed":202610302040,"parameters":{"dimension":2,"matrix":[[16,2],[2,5]],"norm_source":"same_inner_product","metric_source":"same_norm","completeness_source":"finite_dimensional_theorem","domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-component-checklist","semantic_template":"qf-mod30-hilbert-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":2,\"matrix\":[[16,2],[2,5]],\"metric_source\":\"same_norm\",\"norm_source\":\"same_inner_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0041","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-component-checklist；資料={\"dimension\":2,\"matrix\":[[17,0],[0,1]],\"norm_source\":\"other\",\"metric_source\":\"same_norm\",\"completeness_source\":\"finite_dimensional_theorem\"}。請依 [dimension,matrix,norm_source,metric_source,completeness_source,Hilbert] 欄序回答。","answer":{"kind":"expression","value":"(2, [[17, 0], [0, 1]], other, same_norm, finite_dimensional_theorem, 否)","explanation_zh":"Wave4第四層核對=(2, [[17, 0], [0, 1]], other, same_norm, finite_dimensional_theorem, 否)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-component-checklist","seed":202610302041,"parameters":{"dimension":2,"matrix":[[17,0],[0,1]],"norm_source":"other","metric_source":"same_norm","completeness_source":"finite_dimensional_theorem","domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-component-checklist","semantic_template":"qf-mod30-hilbert-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":2,\"matrix\":[[17,0],[0,1]],\"metric_source\":\"same_norm\",\"norm_source\":\"other\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0042","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-component-checklist；資料={\"dimension\":2,\"matrix\":[[18,1],[1,2]],\"norm_source\":\"same_inner_product\",\"metric_source\":\"same_norm\",\"completeness_source\":\"finite_dimensional_theorem\"}。請依 [dimension,matrix,norm_source,metric_source,completeness_source,Hilbert] 欄序回答。","answer":{"kind":"expression","value":"(2, [[18, 1], [1, 2]], same_inner_product, same_norm, finite_dimensional_theorem, 是)","explanation_zh":"Wave4第四層核對=(2, [[18, 1], [1, 2]], same_inner_product, same_norm, finite_dimensional_theorem, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-component-checklist","seed":202610302042,"parameters":{"dimension":2,"matrix":[[18,1],[1,2]],"norm_source":"same_inner_product","metric_source":"same_norm","completeness_source":"finite_dimensional_theorem","domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-component-checklist","semantic_template":"qf-mod30-hilbert-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":2,\"matrix\":[[18,1],[1,2]],\"metric_source\":\"same_norm\",\"norm_source\":\"same_inner_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0043","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-component-checklist；資料={\"dimension\":2,\"matrix\":[[19,2],[2,5]],\"norm_source\":\"same_inner_product\",\"metric_source\":\"same_norm\",\"completeness_source\":\"finite_dimensional_theorem\"}。請依 [dimension,matrix,norm_source,metric_source,completeness_source,Hilbert] 欄序回答。","answer":{"kind":"expression","value":"(2, [[19, 2], [2, 5]], same_inner_product, same_norm, finite_dimensional_theorem, 是)","explanation_zh":"Wave4第四層核對=(2, [[19, 2], [2, 5]], same_inner_product, same_norm, finite_dimensional_theorem, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-component-checklist","seed":202610302043,"parameters":{"dimension":2,"matrix":[[19,2],[2,5]],"norm_source":"same_inner_product","metric_source":"same_norm","completeness_source":"finite_dimensional_theorem","domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-component-checklist","semantic_template":"qf-mod30-hilbert-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":2,\"matrix\":[[19,2],[2,5]],\"metric_source\":\"same_norm\",\"norm_source\":\"same_inner_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0044","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-component-checklist；資料={\"dimension\":2,\"matrix\":[[20,0],[0,1]],\"norm_source\":\"same_inner_product\",\"metric_source\":\"same_norm\",\"completeness_source\":\"finite_dimensional_theorem\"}。請依 [dimension,matrix,norm_source,metric_source,completeness_source,Hilbert] 欄序回答。","answer":{"kind":"expression","value":"(2, [[20, 0], [0, 1]], same_inner_product, same_norm, finite_dimensional_theorem, 是)","explanation_zh":"Wave4第四層核對=(2, [[20, 0], [0, 1]], same_inner_product, same_norm, finite_dimensional_theorem, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-component-checklist","seed":202610302044,"parameters":{"dimension":2,"matrix":[[20,0],[0,1]],"norm_source":"same_inner_product","metric_source":"same_norm","completeness_source":"finite_dimensional_theorem","domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-component-checklist","semantic_template":"qf-mod30-hilbert-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":2,\"matrix\":[[20,0],[0,1]],\"metric_source\":\"same_norm\",\"norm_source\":\"same_inner_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0045","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-component-checklist；資料={\"dimension\":2,\"matrix\":[[21,1],[1,2]],\"norm_source\":\"same_inner_product\",\"metric_source\":\"same_norm\",\"completeness_source\":\"finite_dimensional_theorem\"}。請依 [dimension,matrix,norm_source,metric_source,completeness_source,Hilbert] 欄序回答。","answer":{"kind":"expression","value":"(2, [[21, 1], [1, 2]], same_inner_product, same_norm, finite_dimensional_theorem, 是)","explanation_zh":"Wave4第四層核對=(2, [[21, 1], [1, 2]], same_inner_product, same_norm, finite_dimensional_theorem, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-component-checklist","seed":202610302045,"parameters":{"dimension":2,"matrix":[[21,1],[1,2]],"norm_source":"same_inner_product","metric_source":"same_norm","completeness_source":"finite_dimensional_theorem","domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-component-checklist","semantic_template":"qf-mod30-hilbert-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":2,\"matrix\":[[21,1],[1,2]],\"metric_source\":\"same_norm\",\"norm_source\":\"same_inner_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0046","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-component-checklist；資料={\"dimension\":2,\"matrix\":[[22,2],[2,5]],\"norm_source\":\"other\",\"metric_source\":\"same_norm\",\"completeness_source\":\"finite_dimensional_theorem\"}。請依 [dimension,matrix,norm_source,metric_source,completeness_source,Hilbert] 欄序回答。","answer":{"kind":"expression","value":"(2, [[22, 2], [2, 5]], other, same_norm, finite_dimensional_theorem, 否)","explanation_zh":"Wave4第四層核對=(2, [[22, 2], [2, 5]], other, same_norm, finite_dimensional_theorem, 否)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-component-checklist","seed":202610302046,"parameters":{"dimension":2,"matrix":[[22,2],[2,5]],"norm_source":"other","metric_source":"same_norm","completeness_source":"finite_dimensional_theorem","domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-component-checklist","semantic_template":"qf-mod30-hilbert-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":2,\"matrix\":[[22,2],[2,5]],\"metric_source\":\"same_norm\",\"norm_source\":\"other\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0047","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-component-checklist；資料={\"dimension\":2,\"matrix\":[[23,0],[0,1]],\"norm_source\":\"same_inner_product\",\"metric_source\":\"same_norm\",\"completeness_source\":\"finite_dimensional_theorem\"}。請依 [dimension,matrix,norm_source,metric_source,completeness_source,Hilbert] 欄序回答。","answer":{"kind":"expression","value":"(2, [[23, 0], [0, 1]], same_inner_product, same_norm, finite_dimensional_theorem, 是)","explanation_zh":"Wave4第四層核對=(2, [[23, 0], [0, 1]], same_inner_product, same_norm, finite_dimensional_theorem, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-component-checklist","seed":202610302047,"parameters":{"dimension":2,"matrix":[[23,0],[0,1]],"norm_source":"same_inner_product","metric_source":"same_norm","completeness_source":"finite_dimensional_theorem","domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-component-checklist","semantic_template":"qf-mod30-hilbert-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":2,\"matrix\":[[23,0],[0,1]],\"metric_source\":\"same_norm\",\"norm_source\":\"same_inner_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0048","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-component-checklist；資料={\"dimension\":2,\"matrix\":[[24,1],[1,2]],\"norm_source\":\"same_inner_product\",\"metric_source\":\"same_norm\",\"completeness_source\":\"finite_dimensional_theorem\"}。請依 [dimension,matrix,norm_source,metric_source,completeness_source,Hilbert] 欄序回答。","answer":{"kind":"expression","value":"(2, [[24, 1], [1, 2]], same_inner_product, same_norm, finite_dimensional_theorem, 是)","explanation_zh":"Wave4第四層核對=(2, [[24, 1], [1, 2]], same_inner_product, same_norm, finite_dimensional_theorem, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-component-checklist","seed":202610302048,"parameters":{"dimension":2,"matrix":[[24,1],[1,2]],"norm_source":"same_inner_product","metric_source":"same_norm","completeness_source":"finite_dimensional_theorem","domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-component-checklist","semantic_template":"qf-mod30-hilbert-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":2,\"matrix\":[[24,1],[1,2]],\"metric_source\":\"same_norm\",\"norm_source\":\"same_inner_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0049","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-component-checklist；資料={\"dimension\":2,\"matrix\":[[25,2],[2,5]],\"norm_source\":\"same_inner_product\",\"metric_source\":\"same_norm\",\"completeness_source\":\"finite_dimensional_theorem\"}。請依 [dimension,matrix,norm_source,metric_source,completeness_source,Hilbert] 欄序回答。","answer":{"kind":"expression","value":"(2, [[25, 2], [2, 5]], same_inner_product, same_norm, finite_dimensional_theorem, 是)","explanation_zh":"Wave4第四層核對=(2, [[25, 2], [2, 5]], same_inner_product, same_norm, finite_dimensional_theorem, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-component-checklist","seed":202610302049,"parameters":{"dimension":2,"matrix":[[25,2],[2,5]],"norm_source":"same_inner_product","metric_source":"same_norm","completeness_source":"finite_dimensional_theorem","domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-component-checklist","semantic_template":"qf-mod30-hilbert-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":2,\"matrix\":[[25,2],[2,5]],\"metric_source\":\"same_norm\",\"norm_source\":\"same_inner_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0050","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-component-checklist；資料={\"dimension\":2,\"matrix\":[[26,0],[0,1]],\"norm_source\":\"same_inner_product\",\"metric_source\":\"same_norm\",\"completeness_source\":\"finite_dimensional_theorem\"}。請依 [dimension,matrix,norm_source,metric_source,completeness_source,Hilbert] 欄序回答。","answer":{"kind":"expression","value":"(2, [[26, 0], [0, 1]], same_inner_product, same_norm, finite_dimensional_theorem, 是)","explanation_zh":"Wave4第四層核對=(2, [[26, 0], [0, 1]], same_inner_product, same_norm, finite_dimensional_theorem, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-component-checklist","seed":202610302050,"parameters":{"dimension":2,"matrix":[[26,0],[0,1]],"norm_source":"same_inner_product","metric_source":"same_norm","completeness_source":"finite_dimensional_theorem","domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-component-checklist","semantic_template":"qf-mod30-hilbert-component-checklist/{\"completeness_source\":\"finite_dimensional_theorem\",\"dimension\":2,\"matrix\":[[26,0],[0,1]],\"metric_source\":\"same_norm\",\"norm_source\":\"same_inner_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0051","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-distance；資料={\"matrix\":[[2,0],[0,1]],\"left\":[-12,-2],\"right\":[12,-3]}。請依 [left,right,difference,distance_square,distance] 欄序回答。","answer":{"kind":"expression","value":"([-12, -2], [12, -3], [-24, 1], 1153, sqrt(1153))","explanation_zh":"Wave4第四層核對=([-12, -2], [12, -3], [-24, 1], 1153, sqrt(1153))；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-distance","seed":202610302051,"parameters":{"matrix":[[2,0],[0,1]],"left":[-12,-2],"right":[12,-3],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-distance","semantic_template":"qf-mod30-hilbert-distance/{\"left\":[-12,-2],\"matrix\":[[2,0],[0,1]],\"right\":[12,-3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0052","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-distance；資料={\"matrix\":[[3,1],[1,2]],\"left\":[-11,-1],\"right\":[11,-2]}。請依 [left,right,difference,distance_square,distance] 欄序回答。","answer":{"kind":"expression","value":"([-11, -1], [11, -2], [-22, 1], 1410, sqrt(1410))","explanation_zh":"Wave4第四層核對=([-11, -1], [11, -2], [-22, 1], 1410, sqrt(1410))；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-distance","seed":202610302052,"parameters":{"matrix":[[3,1],[1,2]],"left":[-11,-1],"right":[11,-2],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-distance","semantic_template":"qf-mod30-hilbert-distance/{\"left\":[-11,-1],\"matrix\":[[3,1],[1,2]],\"right\":[11,-2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0053","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-distance；資料={\"matrix\":[[4,2],[2,5]],\"left\":[-10,0],\"right\":[10,-1]}。請依 [left,right,difference,distance_square,distance] 欄序回答。","answer":{"kind":"expression","value":"([-10, 0], [10, -1], [-20, 1], 1525, sqrt(1525))","explanation_zh":"Wave4第四層核對=([-10, 0], [10, -1], [-20, 1], 1525, sqrt(1525))；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-distance","seed":202610302053,"parameters":{"matrix":[[4,2],[2,5]],"left":[-10,0],"right":[10,-1],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-distance","semantic_template":"qf-mod30-hilbert-distance/{\"left\":[-10,0],\"matrix\":[[4,2],[2,5]],\"right\":[10,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0054","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-distance；資料={\"matrix\":[[5,0],[0,1]],\"left\":[-9,1],\"right\":[9,0]}。請依 [left,right,difference,distance_square,distance] 欄序回答。","answer":{"kind":"expression","value":"([-9, 1], [9, 0], [-18, 1], 1621, sqrt(1621))","explanation_zh":"Wave4第四層核對=([-9, 1], [9, 0], [-18, 1], 1621, sqrt(1621))；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-distance","seed":202610302054,"parameters":{"matrix":[[5,0],[0,1]],"left":[-9,1],"right":[9,0],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-distance","semantic_template":"qf-mod30-hilbert-distance/{\"left\":[-9,1],\"matrix\":[[5,0],[0,1]],\"right\":[9,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0055","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-distance；資料={\"matrix\":[[6,1],[1,2]],\"left\":[-8,2],\"right\":[8,1]}。請依 [left,right,difference,distance_square,distance] 欄序回答。","answer":{"kind":"expression","value":"([-8, 2], [8, 1], [-16, 1], 1506, sqrt(1506))","explanation_zh":"Wave4第四層核對=([-8, 2], [8, 1], [-16, 1], 1506, sqrt(1506))；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-distance","seed":202610302055,"parameters":{"matrix":[[6,1],[1,2]],"left":[-8,2],"right":[8,1],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-distance","semantic_template":"qf-mod30-hilbert-distance/{\"left\":[-8,2],\"matrix\":[[6,1],[1,2]],\"right\":[8,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0056","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-distance；資料={\"matrix\":[[7,2],[2,5]],\"left\":[-7,-2],\"right\":[7,2]}。請依 [left,right,difference,distance_square,distance] 欄序回答。","answer":{"kind":"expression","value":"([-7, -2], [7, 2], [-14, -4], 1676, sqrt(1676))","explanation_zh":"Wave4第四層核對=([-7, -2], [7, 2], [-14, -4], 1676, sqrt(1676))；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-distance","seed":202610302056,"parameters":{"matrix":[[7,2],[2,5]],"left":[-7,-2],"right":[7,2],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-distance","semantic_template":"qf-mod30-hilbert-distance/{\"left\":[-7,-2],\"matrix\":[[7,2],[2,5]],\"right\":[7,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0057","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-distance；資料={\"matrix\":[[8,0],[0,1]],\"left\":[-6,-1],\"right\":[6,3]}。請依 [left,right,difference,distance_square,distance] 欄序回答。","answer":{"kind":"expression","value":"([-6, -1], [6, 3], [-12, -4], 1168, sqrt(1168))","explanation_zh":"Wave4第四層核對=([-6, -1], [6, 3], [-12, -4], 1168, sqrt(1168))；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-distance","seed":202610302057,"parameters":{"matrix":[[8,0],[0,1]],"left":[-6,-1],"right":[6,3],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-distance","semantic_template":"qf-mod30-hilbert-distance/{\"left\":[-6,-1],\"matrix\":[[8,0],[0,1]],\"right\":[6,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0058","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-distance；資料={\"matrix\":[[9,1],[1,2]],\"left\":[-5,0],\"right\":[5,-3]}。請依 [left,right,difference,distance_square,distance] 欄序回答。","answer":{"kind":"expression","value":"([-5, 0], [5, -3], [-10, 3], 858, sqrt(858))","explanation_zh":"Wave4第四層核對=([-5, 0], [5, -3], [-10, 3], 858, sqrt(858))；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-distance","seed":202610302058,"parameters":{"matrix":[[9,1],[1,2]],"left":[-5,0],"right":[5,-3],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-distance","semantic_template":"qf-mod30-hilbert-distance/{\"left\":[-5,0],\"matrix\":[[9,1],[1,2]],\"right\":[5,-3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0059","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-distance；資料={\"matrix\":[[10,2],[2,5]],\"left\":[-4,1],\"right\":[4,-2]}。請依 [left,right,difference,distance_square,distance] 欄序回答。","answer":{"kind":"expression","value":"([-4, 1], [4, -2], [-8, 3], 589, sqrt(589))","explanation_zh":"Wave4第四層核對=([-4, 1], [4, -2], [-8, 3], 589, sqrt(589))；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-distance","seed":202610302059,"parameters":{"matrix":[[10,2],[2,5]],"left":[-4,1],"right":[4,-2],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-distance","semantic_template":"qf-mod30-hilbert-distance/{\"left\":[-4,1],\"matrix\":[[10,2],[2,5]],\"right\":[4,-2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0060","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-distance；資料={\"matrix\":[[11,0],[0,1]],\"left\":[-3,2],\"right\":[3,-1]}。請依 [left,right,difference,distance_square,distance] 欄序回答。","answer":{"kind":"expression","value":"([-3, 2], [3, -1], [-6, 3], 405, sqrt(405))","explanation_zh":"Wave4第四層核對=([-3, 2], [3, -1], [-6, 3], 405, sqrt(405))；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-distance","seed":202610302060,"parameters":{"matrix":[[11,0],[0,1]],"left":[-3,2],"right":[3,-1],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-distance","semantic_template":"qf-mod30-hilbert-distance/{\"left\":[-3,2],\"matrix\":[[11,0],[0,1]],\"right\":[3,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0061","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-distance；資料={\"matrix\":[[12,1],[1,2]],\"left\":[-2,-2],\"right\":[2,0]}。請依 [left,right,difference,distance_square,distance] 欄序回答。","answer":{"kind":"expression","value":"([-2, -2], [2, 0], [-4, -2], 216, sqrt(216))","explanation_zh":"Wave4第四層核對=([-2, -2], [2, 0], [-4, -2], 216, sqrt(216))；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-distance","seed":202610302061,"parameters":{"matrix":[[12,1],[1,2]],"left":[-2,-2],"right":[2,0],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-distance","semantic_template":"qf-mod30-hilbert-distance/{\"left\":[-2,-2],\"matrix\":[[12,1],[1,2]],\"right\":[2,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0062","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-distance；資料={\"matrix\":[[13,2],[2,5]],\"left\":[-1,-1],\"right\":[1,1]}。請依 [left,right,difference,distance_square,distance] 欄序回答。","answer":{"kind":"expression","value":"([-1, -1], [1, 1], [-2, -2], 88, sqrt(88))","explanation_zh":"Wave4第四層核對=([-1, -1], [1, 1], [-2, -2], 88, sqrt(88))；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-distance","seed":202610302062,"parameters":{"matrix":[[13,2],[2,5]],"left":[-1,-1],"right":[1,1],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-distance","semantic_template":"qf-mod30-hilbert-distance/{\"left\":[-1,-1],\"matrix\":[[13,2],[2,5]],\"right\":[1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0063","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-distance；資料={\"matrix\":[[14,0],[0,1]],\"left\":[0,0],\"right\":[0,2]}。請依 [left,right,difference,distance_square,distance] 欄序回答。","answer":{"kind":"expression","value":"([0, 0], [0, 2], [0, -2], 4, sqrt(4))","explanation_zh":"Wave4第四層核對=([0, 0], [0, 2], [0, -2], 4, sqrt(4))；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-distance","seed":202610302063,"parameters":{"matrix":[[14,0],[0,1]],"left":[0,0],"right":[0,2],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-distance","semantic_template":"qf-mod30-hilbert-distance/{\"left\":[0,0],\"matrix\":[[14,0],[0,1]],\"right\":[0,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0064","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-distance；資料={\"matrix\":[[15,1],[1,2]],\"left\":[1,1],\"right\":[-1,3]}。請依 [left,right,difference,distance_square,distance] 欄序回答。","answer":{"kind":"expression","value":"([1, 1], [-1, 3], [2, -2], 60, sqrt(60))","explanation_zh":"Wave4第四層核對=([1, 1], [-1, 3], [2, -2], 60, sqrt(60))；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-distance","seed":202610302064,"parameters":{"matrix":[[15,1],[1,2]],"left":[1,1],"right":[-1,3],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-distance","semantic_template":"qf-mod30-hilbert-distance/{\"left\":[1,1],\"matrix\":[[15,1],[1,2]],\"right\":[-1,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0065","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-distance；資料={\"matrix\":[[16,2],[2,5]],\"left\":[2,2],\"right\":[-2,-3]}。請依 [left,right,difference,distance_square,distance] 欄序回答。","answer":{"kind":"expression","value":"([2, 2], [-2, -3], [4, 5], 461, sqrt(461))","explanation_zh":"Wave4第四層核對=([2, 2], [-2, -3], [4, 5], 461, sqrt(461))；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-distance","seed":202610302065,"parameters":{"matrix":[[16,2],[2,5]],"left":[2,2],"right":[-2,-3],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-distance","semantic_template":"qf-mod30-hilbert-distance/{\"left\":[2,2],\"matrix\":[[16,2],[2,5]],\"right\":[-2,-3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0066","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-distance；資料={\"matrix\":[[17,0],[0,1]],\"left\":[3,-2],\"right\":[-3,-2]}。請依 [left,right,difference,distance_square,distance] 欄序回答。","answer":{"kind":"expression","value":"([3, -2], [-3, -2], [6, 0], 612, sqrt(612))","explanation_zh":"Wave4第四層核對=([3, -2], [-3, -2], [6, 0], 612, sqrt(612))；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-distance","seed":202610302066,"parameters":{"matrix":[[17,0],[0,1]],"left":[3,-2],"right":[-3,-2],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-distance","semantic_template":"qf-mod30-hilbert-distance/{\"left\":[3,-2],\"matrix\":[[17,0],[0,1]],\"right\":[-3,-2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0067","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-distance；資料={\"matrix\":[[18,1],[1,2]],\"left\":[4,-1],\"right\":[-4,-1]}。請依 [left,right,difference,distance_square,distance] 欄序回答。","answer":{"kind":"expression","value":"([4, -1], [-4, -1], [8, 0], 1152, sqrt(1152))","explanation_zh":"Wave4第四層核對=([4, -1], [-4, -1], [8, 0], 1152, sqrt(1152))；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-distance","seed":202610302067,"parameters":{"matrix":[[18,1],[1,2]],"left":[4,-1],"right":[-4,-1],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-distance","semantic_template":"qf-mod30-hilbert-distance/{\"left\":[4,-1],\"matrix\":[[18,1],[1,2]],\"right\":[-4,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0068","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-distance；資料={\"matrix\":[[19,2],[2,5]],\"left\":[5,0],\"right\":[-5,0]}。請依 [left,right,difference,distance_square,distance] 欄序回答。","answer":{"kind":"expression","value":"([5, 0], [-5, 0], [10, 0], 1900, sqrt(1900))","explanation_zh":"Wave4第四層核對=([5, 0], [-5, 0], [10, 0], 1900, sqrt(1900))；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-distance","seed":202610302068,"parameters":{"matrix":[[19,2],[2,5]],"left":[5,0],"right":[-5,0],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-distance","semantic_template":"qf-mod30-hilbert-distance/{\"left\":[5,0],\"matrix\":[[19,2],[2,5]],\"right\":[-5,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0069","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-distance；資料={\"matrix\":[[20,0],[0,1]],\"left\":[6,1],\"right\":[-6,1]}。請依 [left,right,difference,distance_square,distance] 欄序回答。","answer":{"kind":"expression","value":"([6, 1], [-6, 1], [12, 0], 2880, sqrt(2880))","explanation_zh":"Wave4第四層核對=([6, 1], [-6, 1], [12, 0], 2880, sqrt(2880))；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-distance","seed":202610302069,"parameters":{"matrix":[[20,0],[0,1]],"left":[6,1],"right":[-6,1],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-distance","semantic_template":"qf-mod30-hilbert-distance/{\"left\":[6,1],\"matrix\":[[20,0],[0,1]],\"right\":[-6,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0070","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-distance；資料={\"matrix\":[[21,1],[1,2]],\"left\":[7,2],\"right\":[-7,2]}。請依 [left,right,difference,distance_square,distance] 欄序回答。","answer":{"kind":"expression","value":"([7, 2], [-7, 2], [14, 0], 4116, sqrt(4116))","explanation_zh":"Wave4第四層核對=([7, 2], [-7, 2], [14, 0], 4116, sqrt(4116))；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-distance","seed":202610302070,"parameters":{"matrix":[[21,1],[1,2]],"left":[7,2],"right":[-7,2],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-distance","semantic_template":"qf-mod30-hilbert-distance/{\"left\":[7,2],\"matrix\":[[21,1],[1,2]],\"right\":[-7,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0071","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-distance；資料={\"matrix\":[[22,2],[2,5]],\"left\":[8,-2],\"right\":[-8,3]}。請依 [left,right,difference,distance_square,distance] 欄序回答。","answer":{"kind":"expression","value":"([8, -2], [-8, 3], [16, -5], 5437, sqrt(5437))","explanation_zh":"Wave4第四層核對=([8, -2], [-8, 3], [16, -5], 5437, sqrt(5437))；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-distance","seed":202610302071,"parameters":{"matrix":[[22,2],[2,5]],"left":[8,-2],"right":[-8,3],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-distance","semantic_template":"qf-mod30-hilbert-distance/{\"left\":[8,-2],\"matrix\":[[22,2],[2,5]],\"right\":[-8,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0072","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-distance；資料={\"matrix\":[[23,0],[0,1]],\"left\":[9,-1],\"right\":[-9,-3]}。請依 [left,right,difference,distance_square,distance] 欄序回答。","answer":{"kind":"expression","value":"([9, -1], [-9, -3], [18, 2], 7456, sqrt(7456))","explanation_zh":"Wave4第四層核對=([9, -1], [-9, -3], [18, 2], 7456, sqrt(7456))；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-distance","seed":202610302072,"parameters":{"matrix":[[23,0],[0,1]],"left":[9,-1],"right":[-9,-3],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-distance","semantic_template":"qf-mod30-hilbert-distance/{\"left\":[9,-1],\"matrix\":[[23,0],[0,1]],\"right\":[-9,-3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0073","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-distance；資料={\"matrix\":[[24,1],[1,2]],\"left\":[10,0],\"right\":[-10,-2]}。請依 [left,right,difference,distance_square,distance] 欄序回答。","answer":{"kind":"expression","value":"([10, 0], [-10, -2], [20, 2], 9688, sqrt(9688))","explanation_zh":"Wave4第四層核對=([10, 0], [-10, -2], [20, 2], 9688, sqrt(9688))；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-distance","seed":202610302073,"parameters":{"matrix":[[24,1],[1,2]],"left":[10,0],"right":[-10,-2],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-distance","semantic_template":"qf-mod30-hilbert-distance/{\"left\":[10,0],\"matrix\":[[24,1],[1,2]],\"right\":[-10,-2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0074","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-distance；資料={\"matrix\":[[25,2],[2,5]],\"left\":[11,1],\"right\":[-11,-1]}。請依 [left,right,difference,distance_square,distance] 欄序回答。","answer":{"kind":"expression","value":"([11, 1], [-11, -1], [22, 2], 12296, sqrt(12296))","explanation_zh":"Wave4第四層核對=([11, 1], [-11, -1], [22, 2], 12296, sqrt(12296))；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-distance","seed":202610302074,"parameters":{"matrix":[[25,2],[2,5]],"left":[11,1],"right":[-11,-1],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-distance","semantic_template":"qf-mod30-hilbert-distance/{\"left\":[11,1],\"matrix\":[[25,2],[2,5]],\"right\":[-11,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0075","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-distance；資料={\"matrix\":[[26,0],[0,1]],\"left\":[12,2],\"right\":[-12,0]}。請依 [left,right,difference,distance_square,distance] 欄序回答。","answer":{"kind":"expression","value":"([12, 2], [-12, 0], [24, 2], 14980, sqrt(14980))","explanation_zh":"Wave4第四層核對=([12, 2], [-12, 0], [24, 2], 14980, sqrt(14980))；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-distance","seed":202610302075,"parameters":{"matrix":[[26,0],[0,1]],"left":[12,2],"right":[-12,0],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-distance","semantic_template":"qf-mod30-hilbert-distance/{\"left\":[12,2],\"matrix\":[[26,0],[0,1]],\"right\":[-12,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0076","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-pythagorean；資料={\"matrix\":[[2,0],[0,1]],\"x\":[1,-2],\"y\":[0,-2]}。請依 [x,y,inner,lhs,rhs,result] 欄序回答。","answer":{"kind":"expression","value":"([1, -2], [0, -2], 4, 18, 10, orthogonality_premise_false)","explanation_zh":"Wave4第四層核對=([1, -2], [0, -2], 4, 18, 10, orthogonality_premise_false)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-pythagorean","seed":202610302076,"parameters":{"matrix":[[2,0],[0,1]],"x":[1,-2],"y":[0,-2],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-pythagorean","semantic_template":"qf-mod30-hilbert-pythagorean/{\"matrix\":[[2,0],[0,1]],\"x\":[1,-2],\"y\":[0,-2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0077","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-pythagorean；資料={\"matrix\":[[3,1],[1,2]],\"x\":[1,-1],\"y\":[1,-3]}。請依 [x,y,inner,lhs,rhs,result] 欄序回答。","answer":{"kind":"expression","value":"([1, -1], [1, -3], 5, 28, 18, orthogonality_premise_false)","explanation_zh":"Wave4第四層核對=([1, -1], [1, -3], 5, 28, 18, orthogonality_premise_false)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-pythagorean","seed":202610302077,"parameters":{"matrix":[[3,1],[1,2]],"x":[1,-1],"y":[1,-3],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-pythagorean","semantic_template":"qf-mod30-hilbert-pythagorean/{\"matrix\":[[3,1],[1,2]],\"x\":[1,-1],\"y\":[1,-3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0078","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-pythagorean；資料={\"matrix\":[[4,2],[2,5]],\"x\":[1,0],\"y\":[2,-4]}。請依 [x,y,inner,lhs,rhs,result] 欄序回答。","answer":{"kind":"expression","value":"([1, 0], [2, -4], 0, 68, 68, 是)","explanation_zh":"Wave4第四層核對=([1, 0], [2, -4], 0, 68, 68, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-pythagorean","seed":202610302078,"parameters":{"matrix":[[4,2],[2,5]],"x":[1,0],"y":[2,-4],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-pythagorean","semantic_template":"qf-mod30-hilbert-pythagorean/{\"matrix\":[[4,2],[2,5]],\"x\":[1,0],\"y\":[2,-4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0079","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-pythagorean；資料={\"matrix\":[[5,0],[0,1]],\"x\":[1,1],\"y\":[0,-5]}。請依 [x,y,inner,lhs,rhs,result] 欄序回答。","answer":{"kind":"expression","value":"([1, 1], [0, -5], -5, 21, 31, orthogonality_premise_false)","explanation_zh":"Wave4第四層核對=([1, 1], [0, -5], -5, 21, 31, orthogonality_premise_false)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-pythagorean","seed":202610302079,"parameters":{"matrix":[[5,0],[0,1]],"x":[1,1],"y":[0,-5],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-pythagorean","semantic_template":"qf-mod30-hilbert-pythagorean/{\"matrix\":[[5,0],[0,1]],\"x\":[1,1],\"y\":[0,-5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0080","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-pythagorean；資料={\"matrix\":[[6,1],[1,2]],\"x\":[1,2],\"y\":[1,-6]}。請依 [x,y,inner,lhs,rhs,result] 欄序回答。","answer":{"kind":"expression","value":"([1, 2], [1, -6], -22, 40, 84, orthogonality_premise_false)","explanation_zh":"Wave4第四層核對=([1, 2], [1, -6], -22, 40, 84, orthogonality_premise_false)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-pythagorean","seed":202610302080,"parameters":{"matrix":[[6,1],[1,2]],"x":[1,2],"y":[1,-6],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-pythagorean","semantic_template":"qf-mod30-hilbert-pythagorean/{\"matrix\":[[6,1],[1,2]],\"x\":[1,2],\"y\":[1,-6]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0081","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-pythagorean；資料={\"matrix\":[[7,2],[2,5]],\"x\":[1,-2],\"y\":[2,-7]}。請依 [x,y,inner,lhs,rhs,result] 欄序回答。","answer":{"kind":"expression","value":"([1, -2], [2, -7], 62, 360, 236, orthogonality_premise_false)","explanation_zh":"Wave4第四層核對=([1, -2], [2, -7], 62, 360, 236, orthogonality_premise_false)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-pythagorean","seed":202610302081,"parameters":{"matrix":[[7,2],[2,5]],"x":[1,-2],"y":[2,-7],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-pythagorean","semantic_template":"qf-mod30-hilbert-pythagorean/{\"matrix\":[[7,2],[2,5]],\"x\":[1,-2],\"y\":[2,-7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0082","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-pythagorean；資料={\"matrix\":[[8,0],[0,1]],\"x\":[1,-1],\"y\":[0,-8]}。請依 [x,y,inner,lhs,rhs,result] 欄序回答。","answer":{"kind":"expression","value":"([1, -1], [0, -8], 8, 89, 73, orthogonality_premise_false)","explanation_zh":"Wave4第四層核對=([1, -1], [0, -8], 8, 89, 73, orthogonality_premise_false)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-pythagorean","seed":202610302082,"parameters":{"matrix":[[8,0],[0,1]],"x":[1,-1],"y":[0,-8],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-pythagorean","semantic_template":"qf-mod30-hilbert-pythagorean/{\"matrix\":[[8,0],[0,1]],\"x\":[1,-1],\"y\":[0,-8]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0083","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-pythagorean；資料={\"matrix\":[[9,1],[1,2]],\"x\":[1,0],\"y\":[1,-9]}。請依 [x,y,inner,lhs,rhs,result] 欄序回答。","answer":{"kind":"expression","value":"([1, 0], [1, -9], 0, 162, 162, 是)","explanation_zh":"Wave4第四層核對=([1, 0], [1, -9], 0, 162, 162, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-pythagorean","seed":202610302083,"parameters":{"matrix":[[9,1],[1,2]],"x":[1,0],"y":[1,-9],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-pythagorean","semantic_template":"qf-mod30-hilbert-pythagorean/{\"matrix\":[[9,1],[1,2]],\"x\":[1,0],\"y\":[1,-9]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0084","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-pythagorean；資料={\"matrix\":[[10,2],[2,5]],\"x\":[1,1],\"y\":[2,-10]}。請依 [x,y,inner,lhs,rhs,result] 欄序回答。","answer":{"kind":"expression","value":"([1, 1], [2, -10], -46, 387, 479, orthogonality_premise_false)","explanation_zh":"Wave4第四層核對=([1, 1], [2, -10], -46, 387, 479, orthogonality_premise_false)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-pythagorean","seed":202610302084,"parameters":{"matrix":[[10,2],[2,5]],"x":[1,1],"y":[2,-10],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-pythagorean","semantic_template":"qf-mod30-hilbert-pythagorean/{\"matrix\":[[10,2],[2,5]],\"x\":[1,1],\"y\":[2,-10]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0085","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-pythagorean；資料={\"matrix\":[[11,0],[0,1]],\"x\":[1,2],\"y\":[0,-11]}。請依 [x,y,inner,lhs,rhs,result] 欄序回答。","answer":{"kind":"expression","value":"([1, 2], [0, -11], -22, 92, 136, orthogonality_premise_false)","explanation_zh":"Wave4第四層核對=([1, 2], [0, -11], -22, 92, 136, orthogonality_premise_false)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-pythagorean","seed":202610302085,"parameters":{"matrix":[[11,0],[0,1]],"x":[1,2],"y":[0,-11],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-pythagorean","semantic_template":"qf-mod30-hilbert-pythagorean/{\"matrix\":[[11,0],[0,1]],\"x\":[1,2],\"y\":[0,-11]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0086","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-pythagorean；資料={\"matrix\":[[12,1],[1,2]],\"x\":[1,-2],\"y\":[1,-12]}。請依 [x,y,inner,lhs,rhs,result] 欄序回答。","answer":{"kind":"expression","value":"([1, -2], [1, -12], 46, 384, 292, orthogonality_premise_false)","explanation_zh":"Wave4第四層核對=([1, -2], [1, -12], 46, 384, 292, orthogonality_premise_false)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-pythagorean","seed":202610302086,"parameters":{"matrix":[[12,1],[1,2]],"x":[1,-2],"y":[1,-12],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-pythagorean","semantic_template":"qf-mod30-hilbert-pythagorean/{\"matrix\":[[12,1],[1,2]],\"x\":[1,-2],\"y\":[1,-12]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0087","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-pythagorean；資料={\"matrix\":[[13,2],[2,5]],\"x\":[1,-1],\"y\":[2,-13]}。請依 [x,y,inner,lhs,rhs,result] 欄序回答。","answer":{"kind":"expression","value":"([1, -1], [2, -13], 61, 929, 807, orthogonality_premise_false)","explanation_zh":"Wave4第四層核對=([1, -1], [2, -13], 61, 929, 807, orthogonality_premise_false)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-pythagorean","seed":202610302087,"parameters":{"matrix":[[13,2],[2,5]],"x":[1,-1],"y":[2,-13],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-pythagorean","semantic_template":"qf-mod30-hilbert-pythagorean/{\"matrix\":[[13,2],[2,5]],\"x\":[1,-1],\"y\":[2,-13]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0088","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-pythagorean；資料={\"matrix\":[[14,0],[0,1]],\"x\":[1,0],\"y\":[0,-14]}。請依 [x,y,inner,lhs,rhs,result] 欄序回答。","answer":{"kind":"expression","value":"([1, 0], [0, -14], 0, 210, 210, 是)","explanation_zh":"Wave4第四層核對=([1, 0], [0, -14], 0, 210, 210, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-pythagorean","seed":202610302088,"parameters":{"matrix":[[14,0],[0,1]],"x":[1,0],"y":[0,-14],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-pythagorean","semantic_template":"qf-mod30-hilbert-pythagorean/{\"matrix\":[[14,0],[0,1]],\"x\":[1,0],\"y\":[0,-14]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0089","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-pythagorean；資料={\"matrix\":[[15,1],[1,2]],\"x\":[1,1],\"y\":[1,-15]}。請依 [x,y,inner,lhs,rhs,result] 欄序回答。","answer":{"kind":"expression","value":"([1, 1], [1, -15], -29, 396, 454, orthogonality_premise_false)","explanation_zh":"Wave4第四層核對=([1, 1], [1, -15], -29, 396, 454, orthogonality_premise_false)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-pythagorean","seed":202610302089,"parameters":{"matrix":[[15,1],[1,2]],"x":[1,1],"y":[1,-15],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-pythagorean","semantic_template":"qf-mod30-hilbert-pythagorean/{\"matrix\":[[15,1],[1,2]],\"x\":[1,1],\"y\":[1,-15]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0090","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-pythagorean；資料={\"matrix\":[[16,2],[2,5]],\"x\":[1,2],\"y\":[2,-16]}。請依 [x,y,inner,lhs,rhs,result] 欄序回答。","answer":{"kind":"expression","value":"([1, 2], [2, -16], -152, 956, 1260, orthogonality_premise_false)","explanation_zh":"Wave4第四層核對=([1, 2], [2, -16], -152, 956, 1260, orthogonality_premise_false)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-pythagorean","seed":202610302090,"parameters":{"matrix":[[16,2],[2,5]],"x":[1,2],"y":[2,-16],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-pythagorean","semantic_template":"qf-mod30-hilbert-pythagorean/{\"matrix\":[[16,2],[2,5]],\"x\":[1,2],\"y\":[2,-16]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0091","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-pythagorean；資料={\"matrix\":[[17,0],[0,1]],\"x\":[1,-2],\"y\":[0,-17]}。請依 [x,y,inner,lhs,rhs,result] 欄序回答。","answer":{"kind":"expression","value":"([1, -2], [0, -17], 34, 378, 310, orthogonality_premise_false)","explanation_zh":"Wave4第四層核對=([1, -2], [0, -17], 34, 378, 310, orthogonality_premise_false)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-pythagorean","seed":202610302091,"parameters":{"matrix":[[17,0],[0,1]],"x":[1,-2],"y":[0,-17],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-pythagorean","semantic_template":"qf-mod30-hilbert-pythagorean/{\"matrix\":[[17,0],[0,1]],\"x\":[1,-2],\"y\":[0,-17]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0092","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-pythagorean；資料={\"matrix\":[[18,1],[1,2]],\"x\":[1,-1],\"y\":[1,-18]}。請依 [x,y,inner,lhs,rhs,result] 欄序回答。","answer":{"kind":"expression","value":"([1, -1], [1, -18], 35, 718, 648, orthogonality_premise_false)","explanation_zh":"Wave4第四層核對=([1, -1], [1, -18], 35, 718, 648, orthogonality_premise_false)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-pythagorean","seed":202610302092,"parameters":{"matrix":[[18,1],[1,2]],"x":[1,-1],"y":[1,-18],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-pythagorean","semantic_template":"qf-mod30-hilbert-pythagorean/{\"matrix\":[[18,1],[1,2]],\"x\":[1,-1],\"y\":[1,-18]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0093","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-pythagorean；資料={\"matrix\":[[19,2],[2,5]],\"x\":[1,0],\"y\":[2,-19]}。請依 [x,y,inner,lhs,rhs,result] 欄序回答。","answer":{"kind":"expression","value":"([1, 0], [2, -19], 0, 1748, 1748, 是)","explanation_zh":"Wave4第四層核對=([1, 0], [2, -19], 0, 1748, 1748, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-pythagorean","seed":202610302093,"parameters":{"matrix":[[19,2],[2,5]],"x":[1,0],"y":[2,-19],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-pythagorean","semantic_template":"qf-mod30-hilbert-pythagorean/{\"matrix\":[[19,2],[2,5]],\"x\":[1,0],\"y\":[2,-19]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0094","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-pythagorean；資料={\"matrix\":[[20,0],[0,1]],\"x\":[1,1],\"y\":[0,-20]}。請依 [x,y,inner,lhs,rhs,result] 欄序回答。","answer":{"kind":"expression","value":"([1, 1], [0, -20], -20, 381, 421, orthogonality_premise_false)","explanation_zh":"Wave4第四層核對=([1, 1], [0, -20], -20, 381, 421, orthogonality_premise_false)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-pythagorean","seed":202610302094,"parameters":{"matrix":[[20,0],[0,1]],"x":[1,1],"y":[0,-20],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-pythagorean","semantic_template":"qf-mod30-hilbert-pythagorean/{\"matrix\":[[20,0],[0,1]],\"x\":[1,1],\"y\":[0,-20]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0095","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-pythagorean；資料={\"matrix\":[[21,1],[1,2]],\"x\":[1,2],\"y\":[1,-21]}。請依 [x,y,inner,lhs,rhs,result] 欄序回答。","answer":{"kind":"expression","value":"([1, 2], [1, -21], -82, 730, 894, orthogonality_premise_false)","explanation_zh":"Wave4第四層核對=([1, 2], [1, -21], -82, 730, 894, orthogonality_premise_false)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-pythagorean","seed":202610302095,"parameters":{"matrix":[[21,1],[1,2]],"x":[1,2],"y":[1,-21],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-pythagorean","semantic_template":"qf-mod30-hilbert-pythagorean/{\"matrix\":[[21,1],[1,2]],\"x\":[1,2],\"y\":[1,-21]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0096","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-pythagorean；資料={\"matrix\":[[22,2],[2,5]],\"x\":[1,-2],\"y\":[2,-22]}。請依 [x,y,inner,lhs,rhs,result] 欄序回答。","answer":{"kind":"expression","value":"([1, -2], [2, -22], 212, 2790, 2366, orthogonality_premise_false)","explanation_zh":"Wave4第四層核對=([1, -2], [2, -22], 212, 2790, 2366, orthogonality_premise_false)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-pythagorean","seed":202610302096,"parameters":{"matrix":[[22,2],[2,5]],"x":[1,-2],"y":[2,-22],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-pythagorean","semantic_template":"qf-mod30-hilbert-pythagorean/{\"matrix\":[[22,2],[2,5]],\"x\":[1,-2],\"y\":[2,-22]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0097","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-pythagorean；資料={\"matrix\":[[23,0],[0,1]],\"x\":[1,-1],\"y\":[0,-23]}。請依 [x,y,inner,lhs,rhs,result] 欄序回答。","answer":{"kind":"expression","value":"([1, -1], [0, -23], 23, 599, 553, orthogonality_premise_false)","explanation_zh":"Wave4第四層核對=([1, -1], [0, -23], 23, 599, 553, orthogonality_premise_false)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-pythagorean","seed":202610302097,"parameters":{"matrix":[[23,0],[0,1]],"x":[1,-1],"y":[0,-23],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-pythagorean","semantic_template":"qf-mod30-hilbert-pythagorean/{\"matrix\":[[23,0],[0,1]],\"x\":[1,-1],\"y\":[0,-23]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0098","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-pythagorean；資料={\"matrix\":[[24,1],[1,2]],\"x\":[1,0],\"y\":[1,-24]}。請依 [x,y,inner,lhs,rhs,result] 欄序回答。","answer":{"kind":"expression","value":"([1, 0], [1, -24], 0, 1152, 1152, 是)","explanation_zh":"Wave4第四層核對=([1, 0], [1, -24], 0, 1152, 1152, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-pythagorean","seed":202610302098,"parameters":{"matrix":[[24,1],[1,2]],"x":[1,0],"y":[1,-24],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-pythagorean","semantic_template":"qf-mod30-hilbert-pythagorean/{\"matrix\":[[24,1],[1,2]],\"x\":[1,0],\"y\":[1,-24]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0099","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-pythagorean；資料={\"matrix\":[[25,2],[2,5]],\"x\":[1,1],\"y\":[2,-25]}。請依 [x,y,inner,lhs,rhs,result] 欄序回答。","answer":{"kind":"expression","value":"([1, 1], [2, -25], -121, 2817, 3059, orthogonality_premise_false)","explanation_zh":"Wave4第四層核對=([1, 1], [2, -25], -121, 2817, 3059, orthogonality_premise_false)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-pythagorean","seed":202610302099,"parameters":{"matrix":[[25,2],[2,5]],"x":[1,1],"y":[2,-25],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-pythagorean","semantic_template":"qf-mod30-hilbert-pythagorean/{\"matrix\":[[25,2],[2,5]],\"x\":[1,1],\"y\":[2,-25]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261030-r02-real-hilbert-space-0100","batch_id":"batch-20261014-advance-20261030-r02-real-hilbert-space-foundations","language":"zh-Hant","source_mko_ids":["mko-real-hilbert-space"],"title_zh":"實Hilbert空間","stem_zh":"inner product norm metric carrier completeness綁同一空間。操作=hilbert-pythagorean；資料={\"matrix\":[[26,0],[0,1]],\"x\":[1,2],\"y\":[0,-26]}。請依 [x,y,inner,lhs,rhs,result] 欄序回答。","answer":{"kind":"expression","value":"([1, 2], [0, -26], -52, 602, 706, orthogonality_premise_false)","explanation_zh":"Wave4第四層核對=([1, 2], [0, -26], -52, 602, 706, orthogonality_premise_false)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實Hilbert空間"]},"variant":{"family_id":"qf-mod30-hilbert-pythagorean","seed":202610302100,"parameters":{"matrix":[[26,0],[0,1]],"x":[1,2],"y":[0,-26],"domain":"Named_finite_dimensional_real_Hilbert_space","operation":"hilbert-pythagorean","semantic_template":"qf-mod30-hilbert-pythagorean/{\"matrix\":[[26,0],[0,1]],\"x\":[1,2],\"y\":[0,-26]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}